Author Topic: Why little math required for electronics?  (Read 4275 times)

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Offline Kyl8145

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Why little math required for electronics?
« on: April 28, 2019, 02:18:04 am »
So one thing I have been curious about as a beginner, but I have read in The Art of Electronics and in Practical Electronics for Inventors that you don't need to know high-level math to design good circuits, just algebra basically. Now The Art of Electronics is considered a major reference in electronic design, but it doesn't seem to get into things like advanced calculus-based math. My question then, is why does acquiring an electrical engineering degree require so much math? Is it just to understand the underlying theory regarding WHY circuits work the way that they do, but otherwise isn't actually needed to create the circuits, or is it really needed for high-level circuit design? Or is it that the kind of circuit design performed by electrical engineers is different somehow...? :-//
 
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Offline jeremy

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Re: Why little math required for electronics?
« Reply #1 on: April 28, 2019, 02:52:16 am »
Math is a tool in your toolbox, just like a multimeter. You need different tools for different things (in fact, sometimes you don’t even need a multimeter). You can do a lot of circuit design simply without using maths because there is a lot of empirical data used (data sheet specs, look up tables, rules of thumb) which exists in part to avoid this complicated maths. Simulators like LTSpice also hide the complex maths from you, so things can look simpler than they really are.

Complicated maths is taught (I assume) because university is supposed to give you a broad base of knowledge, not job specific knowledge. Someone who designs digital circuits their whole life might think the maths is pointless, but someone who designs laser diodes would use it every day (or a simulator that uses it)

 

Online hamster_nz

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Re: Why little math required for electronics?
« Reply #2 on: April 28, 2019, 02:58:09 am »
In my shallow experience, simple math is used to design circuits, but hard math is used to work out why they don't work.

Things like stability analysis and filters design use transforms that I am only just starting to be able to grock.

Then there is also the issues that real world components don't behave like their ideal models, so theory and practice never match unless you design with lots of margin for real world tollorances, making near enough good enough in a lot of cases.

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Online NiHaoMike

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Re: Why little math required for electronics?
« Reply #3 on: April 28, 2019, 03:20:28 am »
I think they should cover more about the intricacies of digital math, e.g. fixed point arithmetic. It's really useful to know when dealing with microcontrollers.
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Offline schmitt trigger

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Re: Why little math required for electronics?
« Reply #4 on: April 28, 2019, 03:51:30 am »
Math is strength-training your brain.

Similar to a football player will do laps and pushups even before touching a ball, math will strengthen your mental agility to solve engineering problems.
« Last Edit: April 28, 2019, 03:53:29 am by schmitt trigger »
 

Offline basinstreetdesign

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Re: Why little math required for electronics?
« Reply #5 on: April 28, 2019, 04:00:16 am »
why does acquiring an electrical engineering degree require so much math?
All of the math taught at university is to demonstrate the validity of the principles they want you to learn.  You need to know how to manipulate Maxwell's equations to understand why clamp-on ammeters work, why transformers, motors and generators work like they do and why antennas work at all.  You need to do the calculus involved in the physics of semiconductor materials before you see the validity of the simpler algebra involved in biasing a transistor.  You have to do the calculus involved in Fourier transforms to know what different effects come from signal modulation and  Laplace transforms to know that you can avoid all that messy calculus in circuit analysis and replace it with much easier algebra.
The main effect of all this is to reduce the level of abstraction involved in all of this so that intelligent approximations can be used with resorting to any of that godawfull stuff.

I did all that in school but in my career I never used anything more sophisticated than a 4 banger calculator.  I got through half of my undergrad years with only a slide rule.
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Offline coppercone2

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Re: Why little math required for electronics?
« Reply #6 on: April 28, 2019, 04:05:53 am »
the main reason is because the real world does not really work well with theory, so you end up significantly adding to your already rough calculations to make something reliable.

Even if you do advanced mathematical analysis on electrical circuits, once you include mechanical and thermal effects on a circuit, and the effects they have on the electronics, and the effects those effects have on the thermal and electrical effects and so on (to get something exactly) it becomes extremely unweidly to build something more then a single resistor standard (and thats just a resistor). Then factor in a HUMAN and you find out the biggest problem is the buttons are too weak because someone is slamming on it like balboa

So 1) you simply the math by getting rid of all the 'non-affects (they really effect things but its just sub PPM and there is so many variables it essentially looks like its random, another factor is dominating, etc.. its basically hunting a fart in a tornado. and its all time consuming and expensive to measure. oh yea and simulators go out the window, you will need to make your own simulator algorithm that might link like 3 different simulation programs together to form a experimental guideline, that you need to test, that there wont be cheap equipment or trustworthy equipment or equipment with bandwidth required or stability required etc)
2) you make the circuit tougher then it needs to be to circumvent a potential defect

3) you know 1) and 2) so you save yourself the time and just aim high with a guess, if it fails you use some math to aim a bit higher.

4) worry if pursuing this relentlessly will make you feel accomplished in life, if it will help anyone, etc. is it just a psychotic obsession at this point? is this what you want? did you play any videogames, smell some flowers, watch some TV, bake a cake recently?

If you do high-order analysis of circuits and parts and stuff you will get insight but for many things that are being sold, its not difficult to figure out how to make it better (but it wont fit a PRICE or development time model that's considered realistic). perhaps you over heard the board room talk about baseball going on for 4 hours and you realized... i should take it easy rather then crush my brain for these guys... lets take it easy on youtube for a while.. is my hair turning grey??

Even with the art of electronics, if you studied something like Sedra Smith in school, then you read AOE after (since its not part of corriculums) you will likely want to go back to the 'classic hard' literature to follow up on something, then that might make you go back to the AOE and back and forth and back and forth... you kind of just get interesting ideas you want to follow up on
« Last Edit: April 28, 2019, 04:18:58 am by coppercone2 »
 

Online CatalinaWOW

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Re: Why little math required for electronics?
« Reply #7 on: April 28, 2019, 04:15:48 am »
All of the above answers are correct.  Another analogy may help. 

Basketball is now a worldwide sport and many, many people play.  You can learn and play basketball at a hoop in your backyard and may well develop the ability to shoot well and dribble well.  And those skills may serve you well in the local pickup game.  At that pickup game you may learn of the need to play defense and learn to keep your center of gravity low watch your opponents center of gravity.  Later you may get into organized basketball and learn elements of team play such as the pick and roll, zone defense and others.

At each level of play there is a higher and higher level of sophistication, requiring additional skills to succeed.  If you are playing in the local city league you may do quite well without employing any of the more sophisticated tools.  If you are playing in the Olympics in the medal rounds you will have to be versed in all that is known about basketball (and physically gifted as well).

Same general idea applies in engineering.  On the biggest stages you need all the tools there are to succeed.  Not just math.  Not just the basics taught in the Art of Electronics.  But both and many more.

 

Offline golden_labels

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Re: Why little math required for electronics?
« Reply #8 on: April 28, 2019, 04:26:34 am »
You need maths to understand details of how things work. But not to use those things in a useful manner.

If you would try to precisely model any real-world circuits of complexity equivalent to more than a few elements, you would soon face the need to symbolically solve arbitrary integrals. Good luck with that — there is no known general way to deal with integrals. Most probably you would also encounter problems equivalent to SAT for hundreds of parameters. Solve that efficiently for just 60 inputs and you will become the legend of comp-sci :D. Oh, did I mentioned 100-dimensional shapes representing probabilistic distributions of the solutions? ;)

It is much more efficient to divide the problem to easily digestible bits, which are approximated with simple (but good enough) models. And then adjust them, if something is not exactly as it should be. And that requires very little maths beyond arithmetic and a few other tools.

If you wish to see the problem yourself, try modeling the most basic thing in electronics: a voltage divider consisting of two resistors. But for real-world elements, not theoretical, simplified ones. Compare that to doing the same for the usual way and consider, if the effort was worth it for a typical scenario.
« Last Edit: April 28, 2019, 04:30:26 am by golden_labels »
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Offline Kyl8145

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Re: Why little math required for electronics?
« Reply #9 on: April 28, 2019, 04:28:29 am »
What I was finding confusing was that in some forums when asked, a lot of engineers were saying that they haven't used calculus since college. And it was just making me wonder if, other than in certain applications, you technically could have a person who didn't know any calculus or physics work in the same position as the engineer who had to learn all that stuff and yet design circuits just as well.

BTW, this question is out of curiosity, I have no issue with math or physics or any of that, I actually love calculus and calculus-based math in fact.
 

Offline Kyl8145

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Re: Why little math required for electronics?
« Reply #10 on: April 28, 2019, 04:31:40 am »
You need maths to understand details of how things work. But not to use those things in a useful manner.

If you would try to precisely model any real-world circuits of complexity equivalent to more than a few elements, you would soon face the need to symbolically solve arbitrary integrals. Good luck with that — there is no known general way to deal with integrals. Most probably you would also encounter problems equivalent to SAT for hundreds of parameters. Solve that efficiently for just 60 inputs and you will become the legend of comp-sci :D. Oh, did I mentioned 100-dimensional shapes representing probabilistic distributions of the solutions? ;)

It is much more efficient to divide the problem to easily digestible bits, which are approximated with simple (but good enough) models. And then adjust them, if something is not exactly as it should be. And that requires very little maths beyond arithmetic and a few other tools.

I see; well all of that is way over my head right now, but I'll get there :D But I see your point, that is what I figured, that they want you to know the underlying theory and also for applications where such knowledge is needed. However, what happens if there is a situation where the knowledge is needed, but the engineer has forgotten it because it's been awhile?
 

Offline coppercone2

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Re: Why little math required for electronics?
« Reply #11 on: April 28, 2019, 04:34:25 am »
consultant, team member, complain to boss, post on forum, send letter to lead company/designer, publish research in a journal and hope someone else decides to research it (playing the 20 year game), etc

you don't need to be a one man army

if something is such a complete dog then why are you building it? do you even know what solution you want?

oh yea, but if you want to misuse something, to save money, thats always what you get.  :-DD
« Last Edit: April 28, 2019, 04:36:45 am by coppercone2 »
 

Offline jeremy

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Re: Why little math required for electronics?
« Reply #12 on: April 28, 2019, 06:21:27 am »
And it was just making me wonder if, other than in certain applications, you technically could have a person who didn't know any calculus or physics work in the same position as the engineer who had to learn all that stuff and yet design circuits just as well.

Yes, you can. I think the point everyone is trying to make is that “electrical engineering” or “circuit design” do not refer to niche areas of knowledge; they are enormous, and many people who have jobs in these area will be doing completely different things (and possibly not even realise that others exist). For example: an “LED lighting engineer” may be someone who installs lots of LED lights with off the shelf drivers (minimal maths), designs LED drivers (moderate maths) or designs the LED die itself (knee deep in it).
 

Offline soldar

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Re: Why little math required for electronics?
« Reply #13 on: April 28, 2019, 07:38:22 am »
you don't need to know high-level math to design good circuits, just algebra basically.


Depending on what you are designing you might need Fourier analysis, Nyquist polar plot analysis, ...

Heck, how about the classic "how fast does a capacitor charge through a resistor". You can't get much simpler than that and you are already looking at an exponential. 

Designing transformers isn't even "electronics" and you will be dealing with harmonics.

Calculus is a useful tool which a good engineer has in his toolbox. He might use it more or less but it is a tool he should have.

Like any other engineer or scientist some tools get used more than others depending on what you are doing. 

Calculus, transforms and other tools help analyze mathematical models of physical things.

Higher math is everywhere. Just to give an example, MP3, MPG, NTSC, are all developed with much math analysis... not to mention all modern forms of cell phone, tv etc modulations and transmissions.

It gets to be annoying and frustrating when you hear someone with no preparation analyze or describe a circuit in terms which are not really engineering but just woo.

Why is a linear power supply oscillating? Why is an audio amp oscillating?  Why is my oscillator NOT oscillating?  You can do a good model analysis or you can fumble in the dark all day and get nowhere.
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Offline Benta

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Re: Why little math required for electronics?
« Reply #14 on: April 28, 2019, 11:02:23 am »
Another place where you'll really need higher maths and physics is at the edge of the electronic realm:
Real-life interfacing, more specifically sensors and transducers.

 

Online dmills

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Re: Why little math required for electronics?
« Reply #15 on: April 28, 2019, 01:22:14 pm »
It is somewhat rare to have to really get your math on, but knowing the mathematics helps to warn you when you are approaching one of those areas.

You have to know enough to know when the simplified models have reached their limits, and to know how to handle it when that is the case.

It is possible to go quite a long way just gluing datasheet and app note circuits together, but that hardly counts as engineering, sensor interfacing, design of oscillators and PLLs, Filters, Stability, the whole world of control theory really, electromagnetism, all of the fun stuff needs you to get your maths on.

Sure, you can usually buy a chip that does something complex for you that you could have done more cheaply if you fully grokked the theory, and sometimes that is a valid choice, but uni should teach the theory (Otherwise, where will the designers of chips that make the hard stuff go away come from?). 

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Offline GopherT

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Re: Why little math required for electronics?
« Reply #16 on: April 28, 2019, 01:24:23 pm »
There is a growing push to teach statistics or more linear algebra instead of calculus to engineers.



« Last Edit: April 28, 2019, 01:25:58 pm by GopherT »
 

Offline hans

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Re: Why little math required for electronics?
« Reply #17 on: April 28, 2019, 02:04:49 pm »
Stochastic research is getting more important in modern technologies. Think machine learning, but also high-level dependable networks such as car-to-car communication or other opportunistic networks.

Nevertheless, I think the main calculus in bachelor/master is still essential to be able to dive deep if you need to. Maths are useful to prove a problem. If you are trained well in modeling your problem and let loose some calculus on it, it's potentially far easier to demonstrate an issue.

For example: consider the problem of L+C inrush (in particular for low ESR situations, e.g. ceramic caps), where hotplugging may create an upsweep of the input voltage and potentially exceed absolute max specs of some components.

A measurement could show the problem, but since this may be a destructive test could be expensive and time consuming.
A numerical analysis (simulation) of this circuit can also showcase the problem, and is not destructive.
Finally, an analytical model (circuit theory) could show the real upper bound of the voltage upswing, i.e. Vpeak = 2 * Vdc, and use that result to dimension the input protection, component specification or realistically constrain your product specifications.

In the other cases, you would either perform a manual search of the problem, learn to avoid problems from experience or try to be on the too cautious side (overkill engineering).
 

Offline tggzzz

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Re: Why little math required for electronics?
« Reply #18 on: April 28, 2019, 02:28:41 pm »
My question then, is why does acquiring an electrical engineering degree require so much math? Is it just to understand the underlying theory regarding WHY circuits work the way that they do, but otherwise isn't actually needed to create the circuits, or is it really needed for high-level circuit design?

It is needed to create circuits that work predictably and reliably.

It is needed to understand where circuits won't work, could fail.

If you don't care about reliabilty or predictability, or you don't care about whether failure might occur and what the consequences might be, then you can just throw things together, cross your fingers, and hope. Hillbilly "engineering" at its "finest".

Far too much software falls into the latter camp.

The other point is that "the best result of maths is that you don't have to use it". That can happen because understanding the maths allows you to create simple models that are sufficient in some circumstances. But you need to know the preconditions necessary for validity, so that you can understand when you are breaking them.

Simple example: please go and buy me a resistor that can be used at 10GHz (or even 1GHz). You can't, because any real resistor also has associated inductance (unless it is zero length) and capacitance (unless the terminals are infinitely far apart). Those "added components" are critical at microwave frequencies.

Simple example: define and understand what you see on a medium speed scope (i.e. >100MHz) if you probe a digital signal with a standard *10 "high" impedance scope probe with a 6" ground lead.

Simple example: you have built a circuit and are looking at its output. It isn't behaving as you expect. Is that because of a faulty design, faulty components, or faulty measurement technique? Maths helps you eliminate possibilities.
There are lies, damned lies, statistics - and ADC/DAC specs.
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Offline free_electron

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Re: Why little math required for electronics?
« Reply #19 on: April 28, 2019, 02:53:15 pm »
why does acquiring an electrical engineering degree require so much math?
All of the math taught at university is to demonstrate the validity of the principles they want you to learn.  You need to know how to manipulate Maxwell's equations to understand why clamp-on ammeters work, why transformers, motors and generators work like they do and why antennas work at all.  You need to do the calculus involved in the physics of semiconductor materials before you see the validity of the simpler algebra involved in biasing a transistor.  You have to do the calculus involved in Fourier transforms to know what different effects come from signal modulation and  Laplace transforms to know that you can avoid all that messy calculus in circuit analysis and replace it with much easier algebra.
The main effect of all this is to reduce the level of abstraction involved in all of this so that intelligent approximations can be used with resorting to any of that godawfull stuff.

I did all that in school but in my career I never used anything more sophisticated than a 4 banger calculator.  I got through half of my undergrad years with only a slide rule.

you can also simply accept the stuff that has already been proven over and over and move on ...

Every year millions of students get questions on their finals to prove this or that theorem. Why ? Do math teachers really have no imagination ? Those theorems are sometimes thousands of years old and have been proven by billions of students through the ages. Do they really think there is need to prove it once again ? are they still unsure about them ?
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Offline apis

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Re: Why little math required for electronics?
« Reply #20 on: April 28, 2019, 03:36:09 pm »
You can build a bridge or a house without knowing any maths, but if you want to be sure it will be safe and long lasting while at the same time not using more materials than you need to meet the specs, then you need the math and theory to calculate strength and resonance modes, etc.

Still, you can get pretty far without any maths, consider medieval cathedrals and castles for example (or even older structures like the Pantheon). But many cathedrals collapsed shortly after completion, those that still stand today were the ones that got it right through trial and error, and a bit of luck.
« Last Edit: April 28, 2019, 03:53:41 pm by apis »
 

Offline GopherT

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Re: Why little math required for electronics?
« Reply #21 on: April 28, 2019, 05:53:16 pm »

Still, you can get pretty far without any maths, consider medieval cathedrals and castles for example (or even older structures like the Pantheon). But many cathedrals collapsed shortly after completion, those that still stand today were the ones that got it right through trial and error, and a bit of luck.

Or, like many of the cathedrals that are still standing, they are significantly over engineered and, therefore, so expensive that it took a century or more to fund to completion.
 

Offline Dave

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Re: Why little math required for electronics?
« Reply #22 on: April 28, 2019, 06:37:47 pm »
Being familiar with somewhat advanced maths topics (fourier/laplace transforms, differential equations, etc.) can only be beneficial. It's another tool in your arsenal that you take out when you need to tackle slightly more complex problems.

From personal experience, the people that speak loudest against "wasting time" to learn them either:
a) never learned them themselves, or
b) took a class and memorized procedures without ever fully grasping them, therefore never applying the knowledge on real life problems.

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Offline 2N3055

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Re: Why little math required for electronics?
« Reply #23 on: April 28, 2019, 07:48:56 pm »
At the risk of starting the riot, I will postulate that many here are confusing mathematics with physics.
You absolutely need to thoroughly understand both physics behind your circuit and that of outside world your circuit is connected to (robot, vehicle, printer, sensor).
To do so you will need to calculate things according to models, and you do that using math. But nowadays, there are many computer tools that can do arduous math for you. But you still need to understand enough to be able to set up calculations correctly.
Today, you probably don't need to be really proficient in calculating something by hand over 20 pages.. But you need to understand concepts and logic behind it.
 

Offline coppercone2

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Re: Why little math required for electronics?
« Reply #24 on: April 28, 2019, 08:04:49 pm »
I actually think calculus education is useful. It does not even get hard until you get to multivariable equations.

If you want to make engineering boring then sure, skip the DE and MV and just teach linear algebra. :barf: I hated it.
 

Offline coppercone2

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Re: Why little math required for electronics?
« Reply #25 on: April 28, 2019, 08:06:53 pm »
There is a growing push to teach statistics or more linear algebra instead of calculus to engineers.



You take random signals class.. it teaches you all the stuff with PDF etc.

My suggestion: do the 4 year degree in 5 years and spread your classes out. That will reduce the math induced rage most students get because they do something crazy like take two far-out math classes in 1 semester (say multivariable calculus, programming C++ and RF theory in the same semester).

Planning actually makes a good engineer. That means going to the school office and spending some time interviewing the secretaries about when classes will be available and stuff (its a good primer for the real engineering world where you have to interrogate people constantly about status of things). And developing contingencies too (i.e. you fail a class, does that mean your gonna do some crazy scheduling next semester? or did you plan ahead to see where you can re-take hard classes so you don't get a double wammy (like do research and tell yourself realistically you have a 70% of passing that thing without destroying your life).
« Last Edit: April 28, 2019, 08:11:53 pm by coppercone2 »
 

Offline NorthGuy

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Re: Why little math required for electronics?
« Reply #26 on: April 28, 2019, 08:30:17 pm »
My question then, is why does acquiring an electrical engineering degree require so much math?

Education is for the benefit of the student, not necessarily to provide a set of skills which are immediately usable. Most people who go to work after a college need to do quite a bit of specific learning directly at their place of work. If they have a broad educational base, they can learn new things faster and more effectively.
 

Offline tggzzz

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Re: Why little math required for electronics?
« Reply #27 on: April 28, 2019, 08:52:21 pm »
Being familiar with somewhat advanced maths topics (fourier/laplace transforms, differential equations, etc.) can only be beneficial. It's another tool in your arsenal that you take out when you need to tackle slightly more complex problems.

From personal experience, the people that speak loudest against "wasting time" to learn them either:
a) never learned them themselves, or
b) took a class and memorized procedures without ever fully grasping them, therefore never applying the knowledge on real life problems.

You have no shortage of mediocre engineers in the world. Strive to become better.

Agreed, but I'd add:

c) took a class, failed it, managed to get a job where other people did the necessary theory, therefore think that theory isn't necessary and that practical skills are all that counts
There are lies, damned lies, statistics - and ADC/DAC specs.
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Offline tggzzz

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Re: Why little math required for electronics?
« Reply #28 on: April 28, 2019, 08:56:28 pm »
I actually think calculus education is useful. It does not even get hard until you get to multivariable equations.

Calculus is fun and has direct relevance to everyday life - and is easy.

I was taught differentiation from first principles in 2 1.5 hour lessons, and integration in another pair of 1.5 hour lessons. We were 14 at the time.

OK, it was only polynomials except 1/x. The transcendental functions had to wait until we were 15 :)
There are lies, damned lies, statistics - and ADC/DAC specs.
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Offline blueskull

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Re: Why little math required for electronics?
« Reply #29 on: April 28, 2019, 08:59:39 pm »
TAoE is a good practical handbook, not a formal textbook. From an academic perspective, it is at very low level, far from the state of the art.

It's designed to be a quick reference book, but it's not designed to give you the required insight to be a good researcher in electronics.
 

Offline tggzzz

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Re: Why little math required for electronics?
« Reply #30 on: April 28, 2019, 09:05:32 pm »
My question then, is why does acquiring an electrical engineering degree require so much math?

Education is for the benefit of the student, not necessarily to provide a set of skills which are immediately usable. Most people who go to work after a college need to do quite a bit of specific learning directly at their place of work. If they have a broad educational base, they can learn new things faster and more effectively.

Exactly.

If university spits out people that have all the knowledge for a job, then it is a trade school producing technicians not a university producing engineers. (ObCaveat: nothing wrong with that).

For a neat example of that distinction by one of the masters of science fiction, Isaac Asimov, spead read "Profession" at http://www.abelard.org/asimov.php
There are lies, damned lies, statistics - and ADC/DAC specs.
Glider pilot's aphorism: "there is no substitute for span". Retort: "There is a substitute: skill+imagination. But you can buy span".
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Offline tggzzz

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Re: Why little math required for electronics?
« Reply #31 on: April 28, 2019, 09:12:57 pm »
TAoE is a good practical handbook, not a formal textbook. From an academic perspective, it is at very low level, far from the state of the art.

It's designed to be a quick reference book, but it's not designed to give you the required insight to be a good researcher in electronics.

That's a good point.

TAoE is aimed at people that have been taught the fundamental theory, and are having to use electronics to get their job done. Hence it is a documentation of the practical art of how to do things in 2015. Compare that with theory, which lasts more than an entire career.

Of course it goes into considerable depth, so it is also useful for practicing electronic engineers as well as other people using electronics.
There are lies, damned lies, statistics - and ADC/DAC specs.
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Offline chickenHeadKnob

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Re: Why little math required for electronics?
« Reply #32 on: April 28, 2019, 09:36:39 pm »
Exactly.

If university spits out people that have all the knowledge for a job, then it is a trade school producing technicians not a university producing engineers. (ObCaveat: nothing wrong with that).

For a neat example of that distinction by one of the masters of science fiction, Isaac Asimov, spead read "Profession" at http://www.abelard.org/asimov.php

Yesss, tggzzz (and Blueskull) homes in on the real reason. A proper university is not a trade school. Their function is not to create job ready tradesmen or even engineers, but rather the next generation of professors/researchers.
 
This type of thread occurs with some regularity on this forum. Almost invariably what is behind it is some combination of math angst, or a need to feel superior to academics, or a need to overcome a deep-seated inferiority complex which then leads to the preceding. "My professor is so useless, couldn't design a working circuit if his life depended on it", ect.

Math is tough, lets go shopping, says Barbie girl.

I am not pointing fingers at the original poster. I am one of the few who revelled in the Math and other arcane subjects during my Uni days even though I knew I wouldn’t stay in the academic track. Enjoy the academic life while you can. before the real world sucks the life out of you.
 

Offline Kyl8145

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Re: Why little math required for electronics?
« Reply #33 on: April 28, 2019, 10:30:18 pm »

you can also simply accept the stuff that has already been proven over and over and move on ...

Every year millions of students get questions on their finals to prove this or that theorem. Why ? Do math teachers really have no imagination ? Those theorems are sometimes thousands of years old and have been proven by billions of students through the ages. Do they really think there is need to prove it once again ? are they still unsure about them ?

Well there are multiple reasons I'd say:

1) Having students prove things is so that the students gain a true understanding of the WHY behind the mathematics. Just as the high-level math and physics is needed to understand the theory behind engineering, within the math itself there is the surface level stuff and then the theory behind it. For example, I am working my way through a calculus textbook right now. The book explains the concept but then shows the proof. Now to understand how to apply the concept is one thing, and that can often be done easily, but to understand the proof behind it can be a lot harder and take more thinking. More applied calculus classes will not focus much on proofs, but rather on using the calculus for various real-world problems, whereas the more pure math calculus classes, often graduate-level, will work through calculus a second time, but this time, teaching all the proofs, so that students gain a really deep, true understanding of it.

It is debated from what I understand in the teaching of calculus regarding proofs, because some are of the opinion that proofs should more be left for after teaching calculus and how to use it, letting the knowledge sink in, and then re-teaching it, this time going hardcore into proofs (graduate-level calc).

2) A reason to teach proofs though I think also depends on the particular calculus class. If the university uses the same calculus classes for teaching the initial mathematics to physics, engineering, chemistry, biology, statistics, computer science, etc...and of course mathematics majors, then they often want proofs to help students more deeply understand the math itself and also, if a math major, to prepare the student for more pure math, as knowing how to work out proofs is essential to advance the field of mathematics. 
 

Offline Kyl8145

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Re: Why little math required for electronics?
« Reply #34 on: April 28, 2019, 10:30:58 pm »
TAoE is a good practical handbook, not a formal textbook. From an academic perspective, it is at very low level, far from the state of the art.

It's designed to be a quick reference book, but it's not designed to give you the required insight to be a good researcher in electronics.

Well it looks very advanced to my ultra beginner self, so have I got a lot to learn :)
 

Offline Kyl8145

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Re: Why little math required for electronics?
« Reply #35 on: April 28, 2019, 10:38:27 pm »

Yesss, tggzzz (and Blueskull) homes in on the real reason. A proper university is not a trade school. Their function is not to create job ready tradesmen or even engineers, but rather the next generation of professors/researchers.
 
This type of thread occurs with some regularity on this forum. Almost invariably what is behind it is some combination of math angst, or a need to feel superior to academics, or a need to overcome a deep-seated inferiority complex which then leads to the preceding. "My professor is so useless, couldn't design a working circuit if his life depended on it", ect.

Math is tough, lets go shopping, says Barbie girl.

I am not pointing fingers at the original poster. I am one of the few who revelled in the Math and other arcane subjects during my Uni days even though I knew I wouldn’t stay in the academic track. Enjoy the academic life while you can. before the real world sucks the life out of you.

I want to be clear again that I have no math angst nor am trying to find an excuse to not do it or anything like that. I actually love mathematics and am very much enjoying working through calculus right now. I then plan to work through linear algebra, differential equations, and beyond. My question was purely curiosity in terms of how much math is actually needed in the real world of engineering is all, because I had encountered some threads on Reddit with engineers saying they haven't used it since college and because some electronics books had said it isn't much needed to design good circuits.

IN FACT, I was also wondering because I found it a bit depressing that, if you are a person who loves math and science and the arcane engineering stuff (of which electrical engineering and electronics has tons of), you wouldn't actually have an excuse to use it in your real-world job ;)
 

Offline rstofer

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Re: Why little math required for electronics?
« Reply #36 on: April 28, 2019, 10:41:59 pm »
What I was finding confusing was that in some forums when asked, a lot of engineers were saying that they haven't used calculus since college.
There is a difference between not using math very often and not knowing it at all.  Furthermore, I would bet those engineers were using it intuitively without even thinking about it.  For a fact, I used Fourier Analysis just one time in my career.  It was fun to play with and the project was important but it just turned out that I never needed it before or after.  But I did know how it worked!
Quote
And it was just making me wonder if, other than in certain applications, you technically could have a person who didn't know any calculus or physics work in the same position as the engineer who had to learn all that stuff and yet design circuits just as well.
Highly unlikely!  Furthermore, it is unlikely the lesser educated technician would ever be given an opportunity to find out.  I worked for a very large company and it always seemed to me that those with MS degrees were higher ranked and higher paid than those with BS level degrees.  All of the big kids had MS degrees or higher.
Quote
BTW, this question is out of curiosity, I have no issue with math or physics or any of that, I actually love calculus and calculus-based math in fact.
Math is a lot easier in the age of the computer.  I graduated at a time where the HP35 had JUST been released and the slide rule was king.  Poles and Zeros?  Use a spirule!

Today we have some very nice tools like MATLAB and wxMaxima.  MATLAB also has MANY add-on packages for specialized applications.  Simulink for MATLAB!  This is a great tool for simulating control systems - the kind of thing that simply didn't exist back in the early '70s.  Great stuff!
 

Offline rstofer

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Re: Why little math required for electronics?
« Reply #37 on: April 28, 2019, 10:56:59 pm »
IN FACT, I was also wondering because I found it a bit depressing that, if you are a person who loves math and science and the arcane engineering stuff (of which electrical engineering and electronics has tons of), you wouldn't actually have an excuse to use it in your real-world job ;)

I take a different point of view about engineering.  I don't want to DO engineering, I want to get paid to BUY engineering.  There's more money in managing engineering than there is in doing it.

 

Offline tggzzz

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Re: Why little math required for electronics?
« Reply #38 on: April 28, 2019, 11:28:43 pm »
TAoE is a good practical handbook, not a formal textbook. From an academic perspective, it is at very low level, far from the state of the art.

It's designed to be a quick reference book, but it's not designed to give you the required insight to be a good researcher in electronics.

Well it looks very advanced to my ultra beginner self, so have I got a lot to learn :)

In any interesting career, that will continue to be true throughout the career.

The alternative is stagnation, where nothing has changed since graduation. Some prefer that. I don't. You might care to decide what is right for you. Your choices can legitimately vary over time.

As Frank Herbert put it in the Dune saga, "the easy path that leads ever downwardstowards stagnation".
There are lies, damned lies, statistics - and ADC/DAC specs.
Glider pilot's aphorism: "there is no substitute for span". Retort: "There is a substitute: skill+imagination. But you can buy span".
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Offline tggzzz

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Re: Why little math required for electronics?
« Reply #39 on: April 28, 2019, 11:35:02 pm »
IN FACT, I was also wondering because I found it a bit depressing that, if you are a person who loves math and science and the arcane engineering stuff (of which electrical engineering and electronics has tons of), you wouldn't actually have an excuse to use it in your real-world job ;)

I take a different point of view about engineering.  I don't want to DO engineering, I want to get paid to BUY engineering.  There's more money in managing engineering than there is in doing it.

I explicitly considered such a career path, and explicitly decided not to.

There is, of course, nothing reprehensible about selling your soul in that way.
There are lies, damned lies, statistics - and ADC/DAC specs.
Glider pilot's aphorism: "there is no substitute for span". Retort: "There is a substitute: skill+imagination. But you can buy span".
Having fun doing more, with less
 

Offline Kyl8145

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Re: Why little math required for electronics?
« Reply #40 on: April 29, 2019, 02:39:26 am »
IN FACT, I was also wondering because I found it a bit depressing that, if you are a person who loves math and science and the arcane engineering stuff (of which electrical engineering and electronics has tons of), you wouldn't actually have an excuse to use it in your real-world job ;)

I take a different point of view about engineering.  I don't want to DO engineering, I want to get paid to BUY engineering.  There's more money in managing engineering than there is in doing it.

If you have no passion for engineering, then sure, go to whichever makes the most money. But me personally, I could never go into management.
 

Online 0culus

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Re: Why little math required for electronics?
« Reply #41 on: April 29, 2019, 03:12:40 am »
TAoE is a good practical handbook, not a formal textbook. From an academic perspective, it is at very low level, far from the state of the art.

It's designed to be a quick reference book, but it's not designed to give you the required insight to be a good researcher in electronics.

Well it looks very advanced to my ultra beginner self, so have I got a lot to learn :)

The best way I've seen it explained is that AoE is a great handbook (as others have mentioned) for doing practical things with just enough theory to get by (I think the only really deep mathematics in AoE show up when they discuss reactance, complex impedance, etc.) If you need more, you'll be better served by finding a specific textbook on the subject you need greater depth in.
 

Online 0culus

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Re: Why little math required for electronics?
« Reply #42 on: April 29, 2019, 03:16:27 am »

Yesss, tggzzz (and Blueskull) homes in on the real reason. A proper university is not a trade school. Their function is not to create job ready tradesmen or even engineers, but rather the next generation of professors/researchers.
 
This type of thread occurs with some regularity on this forum. Almost invariably what is behind it is some combination of math angst, or a need to feel superior to academics, or a need to overcome a deep-seated inferiority complex which then leads to the preceding. "My professor is so useless, couldn't design a working circuit if his life depended on it", ect.

Math is tough, lets go shopping, says Barbie girl.

I am not pointing fingers at the original poster. I am one of the few who revelled in the Math and other arcane subjects during my Uni days even though I knew I wouldn’t stay in the academic track. Enjoy the academic life while you can. before the real world sucks the life out of you.

I want to be clear again that I have no math angst nor am trying to find an excuse to not do it or anything like that. I actually love mathematics and am very much enjoying working through calculus right now. I then plan to work through linear algebra, differential equations, and beyond. My question was purely curiosity in terms of how much math is actually needed in the real world of engineering is all, because I had encountered some threads on Reddit with engineers saying they haven't used it since college and because some electronics books had said it isn't much needed to design good circuits.

IN FACT, I was also wondering because I found it a bit depressing that, if you are a person who loves math and science and the arcane engineering stuff (of which electrical engineering and electronics has tons of), you wouldn't actually have an excuse to use it in your real-world job ;)

The point of studying the mathematics isn't that you'll use it again (whether you do or not depends on where you go with your career), but the way it teaches you to think. I have a math degree and a computer science degree (adding an MS in CS very soon). The heavy math that shows up in the upper divisions of both are not what I want to do with my life. But, learning that way of thinking is invaluable IMO. I like to think of it as a license to learn. I now have the toolbox to crack open a technical textbook of my choice and learn it on my own.
 

Online hamster_nz

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Re: Why little math required for electronics?
« Reply #43 on: April 29, 2019, 03:23:58 am »
IN FACT, I was also wondering because I found it a bit depressing that, if you are a person who loves math and science and the arcane engineering stuff (of which electrical engineering and electronics has tons of), you wouldn't actually have an excuse to use it in your real-world job ;)

I take a different point of view about engineering.  I don't want to DO engineering, I want to get paid to BUY engineering.  There's more money in managing engineering than there is in doing it.

If you have no passion for engineering, then sure, go to whichever makes the most money. But me personally, I could never go into management.

I am old(ish) but by no means rich... but old enough that I can offer a tidbit of advice that might help somebody, or then again may be completely useless.

Even if you have a passion for engineering you will never make a fortune only off of the results of your own effort, To make your way up the  ladder (if that is your aim) then you have to do it by taking a small cut of the effort of many others. That might mean team leader, or manager, or CEO, otherwise you will never get to be an richer than your hourly charge out rate (less everybody else's cut of your efforts) dictates.

That is why the Uber founders will be rich, and their drivers, opps sorry, independent contractors will be on or below  minimum wage and working 60+ hour weeks.
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Offline Kyl8145

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Re: Why little math required for electronics?
« Reply #44 on: April 29, 2019, 04:11:31 am »

I am old(ish) but by no means rich... but old enough that I can offer a tidbit of advice that might help somebody, or then again may be completely useless.

Even if you have a passion for engineering you will never make a fortune only off of the results of your own effort, To make your way up the  ladder (if that is your aim) then you have to do it by taking a small cut of the effort of many others. That might mean team leader, or manager, or CEO, otherwise you will never get to be an richer than your hourly charge out rate (less everybody else's cut of your efforts) dictates.

That is why the Uber founders will be rich, and their drivers, opps sorry, independent contractors will be on or below  minimum wage and working 60+ hour weeks.

Certainly; my thinking was more, if I could either work and make say $60K/yr as an engineer doing a job I love or make $300K/yr doing a job I don't much care about or even hate, I'd prefer the $60K engineering job. But yes, to make really serious money, you need to either go into management or be a successful entrepreneur.
 

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Re: Why little math required for electronics?
« Reply #45 on: April 29, 2019, 04:54:23 am »
My question then, is why does acquiring an electrical engineering degree require so much math?

Education is for the benefit of the student, not necessarily to provide a set of skills which are immediately usable. Most people who go to work after a college need to do quite a bit of specific learning directly at their place of work. If they have a broad educational base, they can learn new things faster and more effectively.

Exactly.


If university spits out people that have all the knowledge for a job, then it is a trade school producing technicians not a university producing engineers. (ObCaveat: nothing wrong with that).

For a neat example of that distinction by one of the masters of science fiction, Isaac Asimov, spead read "Profession" at http://www.abelard.org/asimov.php

Nope! That is what the modern fake technical education does---"monkey see, monkey do".
The old Technician training system was wide ranging, & albeit at a lower level, taught a wide gamut of theory.

Re:- calculus.
It is easy to learn, but many students, (like me) were lost in the breakneck speed at which it was taught in the past at some institutions.

I started the Class, (night school after work, alongside several electronics subjects), had the flu, & missed a week.
We had just been introduced to  simple "Functions of x", or whatever, but when I came back it was "functions of a function".

I bobbed along in the wake for a few weeks,then gave up!

The best thing I got out of that, was the text book, which I kept for years, dipping into it from time to time, but getting scared off.

Many years later, I had occasion to learn calculus, as part of an "in house" bridging training scheme.
It was pretty easy, for the most part, but when I couldn't understand the supplied text, the old book was gold!

Since that time, I haven't had any occasion to use calculus, so it has slowly drifted away.
As I am now retired, I have to learn how to sleep in front of the TV, & drool down my cardigan, so I doubt I will ever have the spare time to revise any branch of mathematics.

PS:I think my iPad is after me!
It substituted "die" for "sleep" in the above paragraph!
 

Offline tggzzz

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Re: Why little math required for electronics?
« Reply #46 on: April 29, 2019, 08:23:04 am »
My question then, is why does acquiring an electrical engineering degree require so much math?

Education is for the benefit of the student, not necessarily to provide a set of skills which are immediately usable. Most people who go to work after a college need to do quite a bit of specific learning directly at their place of work. If they have a broad educational base, they can learn new things faster and more effectively.

Exactly.


If university spits out people that have all the knowledge for a job, then it is a trade school producing technicians not a university producing engineers. (ObCaveat: nothing wrong with that).

For a neat example of that distinction by one of the masters of science fiction, Isaac Asimov, spead read "Profession" at http://www.abelard.org/asimov.php

Nope! That is what the modern fake technical education does---"monkey see, monkey do".
The old Technician training system was wide ranging, & albeit at a lower level, taught a wide gamut of theory.

There is and always has been a wide range of education. Asimov's story was written in 1957 and is as valid then as now.

Quote
Since that time, I haven't had any occasion to use calculus, so it has slowly drifted away.
As I am now retired, I have to learn how to sleep in front of the TV, & drool down my cardigan, so I doubt I will ever have the spare time to revise any branch of mathematics.

PS:I think my iPad is after me!
It substituted "die" for "sleep" in the above paragraph!

I too really love auto-corrupt entry text systems.
There are lies, damned lies, statistics - and ADC/DAC specs.
Glider pilot's aphorism: "there is no substitute for span". Retort: "There is a substitute: skill+imagination. But you can buy span".
Having fun doing more, with less
 

Offline johnwa

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Re: Why little math required for electronics?
« Reply #47 on: April 29, 2019, 09:11:07 am »


Since that time, I haven't had any occasion to use calculus, so it has slowly drifted away.
As I am now retired, I have to learn how to sleep in front of the TV, & drool down my cardigan, so I doubt I will ever have the spare time to revise any branch of mathematics.


It seems that differential and integral calculus in particular is one of the things that is frequently labelled as 'useless'. Now, if you are only talking about finding derivatives and integrals of symbolic expressions, this is something that a lot of people will only need to do relatively rarely - I needed to use the quotient rule the other day for the first time in years, had to look it up. But there is so much more to calculus than this. Understanding the ideas behind finite difference approximations and numerical integration, related rates, graph sketching - I use this all the time. Especially in signal processing, it is a key skill to be able to think "the frequency looks like this as a function of time, what does the phase look like? What will it look like after we differentiate it/integrate it/multiply it with this other signal?"
 

Offline Kyl8145

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Re: Why little math required for electronics?
« Reply #48 on: April 29, 2019, 10:00:36 am »


Since that time, I haven't had any occasion to use calculus, so it has slowly drifted away.
As I am now retired, I have to learn how to sleep in front of the TV, & drool down my cardigan, so I doubt I will ever have the spare time to revise any branch of mathematics.


It seems that differential and integral calculus in particular is one of the things that is frequently labelled as 'useless'. Now, if you are only talking about finding derivatives and integrals of symbolic expressions, this is something that a lot of people will only need to do relatively rarely - I needed to use the quotient rule the other day for the first time in years, had to look it up. But there is so much more to calculus than this. Understanding the ideas behind finite difference approximations and numerical integration, related rates, graph sketching - I use this all the time. Especially in signal processing, it is a key skill to be able to think "the frequency looks like this as a function of time, what does the phase look like? What will it look like after we differentiate it/integrate it/multiply it with this other signal?"

I've read signal processing is basically a branch of applied mathematics in its own right.
 

Offline emece67

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Re: Why little math required for electronics?
« Reply #49 on: April 29, 2019, 12:03:38 pm »
I actually think calculus education is useful. It does not even get hard until you get to multivariable equations.

If you want to make engineering boring then sure, skip the DE and MV and just teach linear algebra. :barf: I hated it.

Precisely it was linear algebra what I found most rewarding when coping with hard problems. But, as a student, I found it, as you did, really boring.
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Online vk6zgo

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Re: Why little math required for electronics?
« Reply #50 on: April 29, 2019, 03:02:21 pm »
My question then, is why does acquiring an electrical engineering degree require so much math?

Education is for the benefit of the student, not necessarily to provide a set of skills which are immediately usable. Most people who go to work after a college need to do quite a bit of specific learning directly at their place of work. If they have a broad educational base, they can learn new things faster and more effectively.

Exactly.


If university spits out people that have all the knowledge for a job, then it is a trade school producing technicians not a university producing engineers. (ObCaveat: nothing wrong with that).

For a neat example of that distinction by one of the masters of science fiction, Isaac Asimov, spead read "Profession" at http://www.abelard.org/asimov.php

Nope! That is what the modern fake technical education does---"monkey see, monkey do".
The old Technician training system was wide ranging, & albeit at a lower level, taught a wide gamut of theory.

There is and always has been a wide range of education. Asimov's story was written in 1957 and is as valid then as now.

In Australia, there has been a definite move away from teaching Electronics as a complete subject in the "Technical & Further Education"(TAFE) system, moving to a system of "Outcomes education" where the student does a course to perform a single function, or number of functions, & if they can do them, all is supposedly well.
The problem with this approach is that, if some new development arrives, the "Tech" does not have the technical knowledge to adapt to it, & needs to do another "short course".

Even supposedly "more in depth" courses are shallow.
I have run across people holding "Advanced Diplomas in Electronics" to whom Ohm's Law is a new & startling concept.
Many have admitted they got through on their IT knowledge---it seems this is given a great loading, as "We all know, everything is done with computers".  |O
It is as if the decisions about the content of these courses are made by people who themselves have no Electronics knowledge.

Back in 1959, when I started at Tech School, there were no "Outcomes" based courses.
You learnt Electronics to an appropriate level, then you specialised in whatever area you were working.

About Asimov--- I've always enjoyed his writings, both fiction & non-fiction.

I have half--read "The Caves of Steel" twice, once when I was 16, & again a few years back.
I don't why i didn't finish the first time, but this latest time, I found the paperback at a secondhand bookshop in Carnarvon WA, read it on the plane back to Perth, & left it on the taxi home.

The local library didn't have it on their database last time I looked, so it looks like I will have to prowl around the secondhand bookshops again.
Quote

Quote
Since that time, I haven't had any occasion to use calculus, so it has slowly drifted away.
As I am now retired, I have to learn how to sleep in front of the TV, & drool down my cardigan, so I doubt I will ever have the spare time to revise any branch of mathematics.

PS:I think my iPad is after me!
It substituted "die" for "sleep" in the above paragraph!

I too really love auto-corrupt entry text systems.
« Last Edit: April 29, 2019, 03:04:55 pm by vk6zgo »
 

Offline NorthGuy

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Re: Why little math required for electronics?
« Reply #51 on: April 29, 2019, 05:00:36 pm »
Asimov's story was written in 1957 and is as valid then as now.

It's a trend. It started long time ago and it is moving along slowly. I'm sure many thing which would look absurd to Azimov, look quite normal now. The trend will continue until it comes to total absurd, at which point you probably should expect some sort of shake up.
 

Offline EEVblog

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Re: Why little math required for electronics?
« Reply #52 on: May 01, 2019, 01:47:33 pm »
So one thing I have been curious about as a beginner, but I have read in The Art of Electronics and in Practical Electronics for Inventors that you don't need to know high-level math to design good circuits, just algebra basically. Now The Art of Electronics is considered a major reference in electronic design, but it doesn't seem to get into things like advanced calculus-based math. My question then, is why does acquiring an electrical engineering degree require so much math? Is it just to understand the underlying theory regarding WHY circuits work the way that they do, but otherwise isn't actually needed to create the circuits, or is it really needed for high-level circuit design? Or is it that the kind of circuit design performed by electrical engineers is different somehow...? :-//

The majority of electronics design work would be applied engineering (I know the term applied engineering can have a specific meaning, but it's just a point), or practical engineering if you will.
This means that most solutions are solved by practical means without much recourse to anything beyond fairly basic applied theory and math. e.g. ohms law, the concepts of integration etc, rather than actually having to derive ohms law or solve an integral.
Solving problems and troubleshooting is also usually approached (at least initially) from an applied practical aspect as well, again without much recourse to anything beyond fairly basic applied theory and math. Usually the first pass practical approach will solve your issue and you move on, rarely if ever having to resort back to the heavy math to solve something.

e.g. there are hundreds of thousands of chips out there, all with highly details practical datasheets on how to implement that part and complete systems. And if something goes wrong you don't immediately resort back to Maxwell's equations to try and figure out what went wrong, you try more basic stuff first.

This is why it's so common to hear career engineers say they have never needed to practically apply the advanced math they were taught.

I highlighted the word "majority" above because there are obviously some (many) fields and jobs where you'd use the heavy math all the time, but it's certainly the exception rather than the rule.

Of course that's not any sort of argument for whether or not the advanced math should be taught, it's just pointing out that yes, as per the OP "Why little math required for electronics?" can be true.

As for the argument should it be taught or not, of course it should. There are basically three levels of engineering education:
1) Trade level - Very little to no math.
2) Diploma level - Usually some advanced math, but mostly more practical stuff.
3) Degree level - You go the Full Maxwell

Choose your own adventure, all of them (or even none) can lead to a successful career in electronics design.

And within those there are some degrees that go much more advanced than others, and likewise for diploma and trades.
« Last Edit: May 01, 2019, 02:08:23 pm by EEVblog »
 

Offline EEVblog

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Re: Why little math required for electronics?
« Reply #53 on: May 01, 2019, 02:17:43 pm »
You can build a bridge or a house without knowing any maths, but if you want to be sure it will be safe and long lasting while at the same time not using more materials than you need to meet the specs, then you need the math and theory to calculate strength and resonance modes, etc.

I get the purpose of the analogy here, and don't want to take away form that message, but want to add that it's not always applicable to electronics design.
In many cases you don't ensure a product is safe and long lasting (or whatever) by going into the mathematical theory and modelling of the components and the construction. Instead you usually follow rigorous practical testing methodology, probably some statistics, and looking at and analysing existing data.
One could even argue that statistics is a more valuable skill in practical electronics design that any of the physics theory.
 

Offline tggzzz

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Re: Why little math required for electronics?
« Reply #54 on: May 01, 2019, 07:21:41 pm »
One could even argue that statistics is a more valuable skill in practical electronics design that any of the physics theory.

That way lies the "because the computer says so" justification for a decision. Even now that can be a real problem because sometimes nobody knows the reason by design. Think "machine learning", which is currently fashionable because it can be cheaper than people.

There are such systems in the USA that cause people to be locked up in prison, and nobody can specify why it makes such recommendations. Yes, it is probably racist.

From a engineering standpoint, the problem is that nobody knows how close a system is operating to the edge of the envelope. Inevitably such faults will be passed off as operator error.

Summary: beware of using stats as a drunkard uses a lamp post - for support rather than illumination.
There are lies, damned lies, statistics - and ADC/DAC specs.
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Offline rstofer

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Re: Why little math required for electronics?
« Reply #55 on: May 01, 2019, 07:22:57 pm »
Certainly; my thinking was more, if I could either work and make say $60K/yr as an engineer doing a job I love or make $300K/yr doing a job I don't much care about or even hate, I'd prefer the $60K engineering job. But yes, to make really serious money, you need to either go into management or be a successful entrepreneur.

I would ALWAYS go for the money!  Working for somebody else is just another form of prostitution, I might as well get paid.  I never wanted to work hard enough to be an entrepreneur.

The even better news about going for the money is the fact that you can put a bunch away for retirement.  That's why, 15 years into retirement, my life style hasn't diminished at all.  Arguably, I am better off after retirement.

Even though my education is primarily in electronics, I never worked a day in that particular part of the electrical engineering sandbox.  Instead, I worked on power systems in industrial applications and managed some high tech facilities.  A good career in a related field while still allowing me to enjoy the hobby of electronics without confusing it with making a living, working for somebody else.

Everybody has their own approach to what they want to do for a living and how far up the chain they want to go.  It's just a choice to make...
 

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Re: Why little math required for electronics?
« Reply #56 on: May 02, 2019, 03:34:24 am »
One could even argue that statistics is a more valuable skill in practical electronics design that any of the physics theory.

That way lies the "because the computer says so" justification for a decision. Even now that can be a real problem because sometimes nobody knows the reason by design.

Sure but in the majority of the time in practical electronics design you are simply not using advanced math and physics level stuff to analyse if your circuit/project is going to work or not, you just likely aren't for practical reasons, design are so complex combining so many complex chips and other components that you can't possible do that except on niche stuff. YMMV of course, we are talking majority applications here.
Hence why I said that statistics is arguably a more valuable skill in practical electronics design that any of the physics theory, because it potentially gets used in more situations in practical product design.
 

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Re: Why little math required for electronics?
« Reply #57 on: May 02, 2019, 03:36:30 am »
Everybody has their own approach to what they want to do for a living and how far up the chain they want to go.  It's just a choice to make...

Some people just aren't going to have a personality that is compatible with climbing that ladder  ;D
 

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Re: Why little math required for electronics?
« Reply #58 on: May 02, 2019, 04:21:50 am »
One could even argue that statistics is a more valuable skill in practical electronics design that any of the physics theory.

That way lies the "because the computer says so" justification for a decision. Even now that can be a real problem because sometimes nobody knows the reason by design.

Sure but in the majority of the time in practical electronics design you are simply not using advanced math and physics level stuff to analyse if your circuit/project is going to work or not, you just likely aren't for practical reasons, design are so complex combining so many complex chips and other components that you can't possible do that except on niche stuff. YMMV of course, we are talking majority applications here.
Hence why I said that statistics is arguably a more valuable skill in practical electronics design that any of the physics theory, because it potentially gets used in more situations in practical product design.

I agree that statistics are more frequently applied than vector calculus - in circuit design.  But serious statistics and noise requires advanced math also to be sure that it is applied correctly.  Not to do, that is more often just algebra, but knowing when the algebra is appropriate requires more background. 

Math requirements vary with area of focus, and if you are like most your area of focus will change over time, either due to your interests or when jobs evaporate in one area and you move on to another.  Antennas and waveguides - again the day to day stuff is just algebra, but understanding why that algebra is applicable or not is vector calculus.  Doing thermal design can be done with computer programs, or in simple cases with algebra, but most programs will give stupid answers if you don't understand boundary conditions, the impact of time sampling and a few other things which can come with either a lot of experience, or for most people a substantially shorter time learning the math.  This pattern repeats over a great many subsets of the engineering problem set.

You don't have to have advanced math to do a great many engineering jobs, but it has been my experience that those tools enhance understanding and help avoid errors.  Even if you are going to have a computer do all the grunt work you need to know what to tell it to do.  And when the computer answer doesn't converge the tools for understanding why not, and what to do about it are maybe not necessary, but they are useful.  Cut and try can be slow and frustrating.
 

Offline GregDunn

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Re: Why little math required for electronics?
« Reply #59 on: May 02, 2019, 05:01:49 am »

you can also simply accept the stuff that has already been proven over and over and move on ...

Every year millions of students get questions on their finals to prove this or that theorem. Why ? Do math teachers really have no imagination ? Those theorems are sometimes thousands of years old and have been proven by billions of students through the ages. Do they really think there is need to prove it once again ? are they still unsure about them ?

If you are an athlete (say triathlon or body builder) - why lift those weights?  Surely there's a forklift or device to do that already.  Why run training laps on a track?  You're going nowhere and just burning energy.  Is there really a need to prove that you can do something millions of other people have already done before?

No, the whole point is to improve yourself by using techniques which have worked for hundreds of years.  Endurance and strength will give you an edge against the competition - math and familiarity with the processes of solving problems will enable you to do designs and handle complex issues without struggling to find a specific answer which already has been worked out.
 

Offline tggzzz

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Re: Why little math required for electronics?
« Reply #60 on: May 02, 2019, 08:25:43 am »

you can also simply accept the stuff that has already been proven over and over and move on ...

Every year millions of students get questions on their finals to prove this or that theorem. Why ? Do math teachers really have no imagination ? Those theorems are sometimes thousands of years old and have been proven by billions of students through the ages. Do they really think there is need to prove it once again ? are they still unsure about them ?

If you are an athlete (say triathlon or body builder) - why lift those weights?  Surely there's a forklift or device to do that already.  Why run training laps on a track?  You're going nowhere and just burning energy.  Is there really a need to prove that you can do something millions of other people have already done before?

No, the whole point is to improve yourself by using techniques which have worked for hundreds of years.  Endurance and strength will give you an edge against the competition - math and familiarity with the processes of solving problems will enable you to do designs and handle complex issues without struggling to find a specific answer which already has been worked out.

I didn't make the point about stats  clearly enough.

Apart from that, I still disagree; a few examples of practical engineering....

Filter design needs and uses maths all the time, especially DSP filters.

Microwave engineering is all about visualising and then solving EM felds, with a computer doing the heavy lifting and the tuning.

EMI ditto, without the computer doing  the heavy lifting, because the problem's geometry is too poorly characterised.

Control theory in systems containing feedback is all about maths,  unless you don't care about stability or have.slugged the system so much that it is operating far more slowly than possible. That's true for purely digital or software systems as well, of course.

Whenever an analogue circuit misbehaves, having the maths guides the intuition towards finding the cause. (And almost all circuits are analogue, but that's a different discussion!)

And that's just a few examples from the top of my head.

Of course often it is possible to avoid maths in simple cases that are repititions of standard practice., But to think that maths isn't routinely used in practical engieering is simply false.
There are lies, damned lies, statistics - and ADC/DAC specs.
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Re: Why little math required for electronics?
« Reply #61 on: May 02, 2019, 08:29:21 am »
Computers are fundamentally "garbage in, garbage out" machines. Without the human who knows how the math works for debugging problems, they are pretty worthless.
 

Offline tggzzz

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Re: Why little math required for electronics?
« Reply #62 on: May 02, 2019, 08:33:38 am »
One could even argue that statistics is a more valuable skill in practical electronics design that any of the physics theory.

That way lies the "because the computer says so" justification for a decision. Even now that can be a real problem because sometimes nobody knows the reason by design.

Sure but in the majority of the time in practical electronics design you are simply not using advanced math and physics level stuff to analyse if your circuit/project is going to work or not, you just likely aren't for practical reasons, design are so complex combining so many complex chips and other components that you can't possible do that except on niche stuff. YMMV of course, we are talking majority applications here.
Hence why I said that statistics is arguably a more valuable skill in practical electronics design that any of the physics theory, because it potentially gets used in more situations in practical product design.

I agree that statistics are more frequently applied than vector calculus - in circuit design.  But serious statistics and noise requires advanced math also to be sure that it is applied correctly.  Not to do, that is more often just algebra, but knowing when the algebra is appropriate requires more background. 

Math requirements vary with area of focus, and if you are like most your area of focus will change over time, either due to your interests or when jobs evaporate in one area and you move on to another.  Antennas and waveguides - again the day to day stuff is just algebra, but understanding why that algebra is applicable or not is vector calculus.  Doing thermal design can be done with computer programs, or in simple cases with algebra, but most programs will give stupid answers if you don't understand boundary conditions, the impact of time sampling and a few other things which can come with either a lot of experience, or for most people a substantially shorter time learning the math.  This pattern repeats over a great many subsets of the engineering problem set.

You don't have to have advanced math to do a great many engineering jobs, but it has been my experience that those tools enhance understanding and help avoid errors.  Even if you are going to have a computer do all the grunt work you need to know what to tell it to do.  And when the computer answer doesn't converge the tools for understanding why not, and what to do about it are maybe not necessary, but they are useful.  Cut and try can be slow and frustrating.

Yes.

I'd summarise those points as "GIGO" and "blind fumbling'. Most people operate like that in most walks of life :(  I aspire to higher :)
There are lies, damned lies, statistics - and ADC/DAC specs.
Glider pilot's aphorism: "there is no substitute for span". Retort: "There is a substitute: skill+imagination. But you can buy span".
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Offline rstofer

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Re: Why little math required for electronics?
« Reply #63 on: May 02, 2019, 03:09:59 pm »
One could even argue that statistics is a more valuable skill in practical electronics design that any of the physics theory.

That way lies the "because the computer says so" justification for a decision. Even now that can be a real problem because sometimes nobody knows the reason by design.

Sure but in the majority of the time in practical electronics design you are simply not using advanced math and physics level stuff to analyse if your circuit/project is going to work or not, you just likely aren't for practical reasons, design are so complex combining so many complex chips and other components that you can't possible do that except on niche stuff. YMMV of course, we are talking majority applications here.
Hence why I said that statistics is arguably a more valuable skill in practical electronics design that any of the physics theory, because it potentially gets used in more situations in practical product design.

I don't know if it's still a thing but just before I retired 15 years ago Six Sigma was sweeping through industry.  They were dragging highly qualified engineers through the quagmire of statistics, kicking and screaming.  I'm pretty sure I can prove that pigs can fly using statistics.  Of course, there may be a vertical bias...

The math tools are so much better today!  I really had to struggle to get through Differential Equations back in the early '70s.  Slide rules just didn't help and the arithmetic soon got ugly.  Today you can solve a wide array of these problems with a computer using something like MATLAB or Octave (GNU opensource).  Now the student has more time to study the application and wastes less time on arithmetic.  When my grandson starts his DE course in the Fall, it will all be based around MATLAB.  And the University has a required MATLAB course, probably as a prerequisite.

These 3 lines of code produce a vector of 'y' values from dy/dt.  It doesn't get much easier than this
Code: [Select]
dydt = @(t,y) (t^3-2*y)/t;  % anonymous function y'=(t^3-2y)/t,y=4.2 at t=1
t = linspace(1,3,40);       % create a vector t values
[t,y] = ode45(dydt,t,4.2);  % ode45 returns a vector of y values from dydt
[/font]
« Last Edit: May 02, 2019, 03:11:50 pm by rstofer »
 

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Re: Why little math required for electronics?
« Reply #64 on: May 02, 2019, 03:23:34 pm »
I'm pretty sure I can prove that pigs can fly using statistics.  Of course, there may be a vertical bias...
It's statistically possible for an ice cube to spontaneously form in a pot of boiling water. The probability is extremely small and very close to zero, but still greater than zero.
Cryptocurrency has taught me to love math and at the same time be baffled by it.

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Re: Why little math required for electronics?
« Reply #65 on: May 02, 2019, 05:32:15 pm »
Many of those who where championing and teaching 6 sigma were examples of those who needed more math background.  There is much meat behind 6 sigma, but like any fad it got diluted, misapplied and otherwise degraded.
 

Offline NorthGuy

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Re: Why little math required for electronics?
« Reply #66 on: May 03, 2019, 03:16:18 pm »
It's statistically possible for an ice cube to spontaneously form in a pot of boiling water. The probability is extremely small and very close to zero, but still greater than zero.

Not really. You need external energy for that, and, as we all know, energy cannot come from nowhere.

And this is an example of how applying simple mathematical and physical principles can help you distinguish a hoax from a plausible scenario.
 

Offline tggzzz

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Re: Why little math required for electronics?
« Reply #67 on: May 03, 2019, 04:45:26 pm »
It's statistically possible for an ice cube to spontaneously form in a pot of boiling water. The probability is extremely small and very close to zero, but still greater than zero.

Not really. You need external energy for that, and, as we all know, energy cannot come from nowhere.

And this is an example of how applying simple mathematical and physical principles can help you distinguish a hoax from a plausible scenario.

Actually you don't need external energy, provided you are prepared to wait for many times the lifetime of the universe.
There are lies, damned lies, statistics - and ADC/DAC specs.
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Re: Why little math required for electronics?
« Reply #68 on: May 03, 2019, 04:59:33 pm »
why does acquiring an electrical engineering degree require so much math?
All of the math taught at university is to demonstrate the validity of the principles they want you to learn.  You need to know how to manipulate Maxwell's equations to understand why clamp-on ammeters work, why transformers, motors and generators work like they do and why antennas work at all.  You need to do the calculus involved in the physics of semiconductor materials before you see the validity of the simpler algebra involved in biasing a transistor.  You have to do the calculus involved in Fourier transforms to know what different effects come from signal modulation and  Laplace transforms to know that you can avoid all that messy calculus in circuit analysis and replace it with much easier algebra.
The main effect of all this is to reduce the level of abstraction involved in all of this so that intelligent approximations can be used with resorting to any of that godawfull stuff.

I did all that in school but in my career I never used anything more sophisticated than a 4 banger calculator.  I got through half of my undergrad years with only a slide rule.

you can also simply accept the stuff that has already been proven over and over and move on ...

Every year millions of students get questions on their finals to prove this or that theorem. Why ? Do math teachers really have no imagination ? Those theorems are sometimes thousands of years old and have been proven by billions of students through the ages. Do they really think there is need to prove it once again ? are they still unsure about them ?
Doesn't that logic mean that all engineering exams are pointless, as they only ask questions which have already been answered?
 

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Re: Why little math required for electronics?
« Reply #69 on: May 03, 2019, 06:13:55 pm »
So one thing I have been curious about as a beginner, but I have read in The Art of Electronics and in Practical Electronics for Inventors that you don't need to know high-level math to design good circuits, just algebra basically. Now The Art of Electronics is considered a major reference in electronic design, but it doesn't seem to get into things like advanced calculus-based math. My question then, is why does acquiring an electrical engineering degree require so much math? Is it just to understand the underlying theory regarding WHY circuits work the way that they do, but otherwise isn't actually needed to create the circuits, or is it really needed for high-level circuit design? Or is it that the kind of circuit design performed by electrical engineers is different somehow...? :-//
I always say that the 'AoE' is a book which shouldn't be on the desk of a good electronics engineer. It is good for managers of an electronics R&D department to get a small clue on what is going on  >:D

The level of math needed depends entirely on what kind of circuits you work on. Over the years I have used quite a lot of the math I learned. Usually solving equations but integrals and differentiation as well. Some Fourier and Z-transform where handy to know too.

If you are interested in good books on engineering maths then I can recommend the books 'Modern engineering mathematics' and 'Advanced modern engineering mathematics'.
« Last Edit: May 03, 2019, 08:23:15 pm by nctnico »
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Re: Why little math required for electronics?
« Reply #70 on: May 03, 2019, 08:30:37 pm »
I can tell you with absolute certainty that you will need all your math and even more when your grandson enrolls in an ME program.  You will want to add context to what is otherwise dry subject matter and provide real world examples of the applications for this material.  And if you don't remember the quotient rule, well, you better hit the books!

MATLAB is equally useful as a programming language and all of the ideas related to CS are helpful when writing the code.  Better get up to speed on wxMaxima as well.  And C, C++, FORTRAN, Java (?), Pascal, definitely Python and probably some others.

If you're going to coach your grandkids through STEM programs, you better bring your 'A' game.  Old age is not an acceptable excuse.
 

Offline Yansi

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Re: Why little math required for electronics?
« Reply #71 on: May 03, 2019, 08:51:12 pm »
Just my little grumpy bit to the plenum: Math is just a tool (one of many) universities use to fuck students with and kick them out of the school (...as soon as possible, because they get payed by the number of students applied to the school - many apply multiple times).

Does it sound downright stupid? Yes, of course. But if they would indeed want to teach the subject in the first place, they would not cut corners on merging a full 4 semester course into a single semester one light-speed course of all but nothing.

And if they mean I should've learn and practice myself, I shouldn't have attended the school in the first place, uh?

Please note that this grumpy opinion on (our local) school system does not by any means state my position against math as a subject. In fact I am quite pissed at the school, because they did not teach us math properly. They did not teach us what in fact may become useful for us to know in the future. Because of this, I still am wrestling badly with laplace transform, know sh!t about z-transform, we had not even single mention of discrete math and our course of control theory & stability analysis is barely usable in practice.

Instead we got many useless but mandatory courses of history, theology, ecology, economy, while the technical subjects were dumbed down to become boring wasted time for many of us.
 

Offline tggzzz

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Re: Why little math required for electronics?
« Reply #72 on: May 03, 2019, 09:32:03 pm »
They did not teach us what in fact may become useful for us to know in the future. Because of this, I still am wrestling badly with laplace transform, know sh!t about z-transform, we had not even single mention of discrete math and our course of control theory & stability analysis is barely usable in practice.

No course will ever teach you everything you need to know in your career - unless your career is so boring that everything is static and preordained.

A course should teach you how to continue to educate yourself throughout your career about the specific things that are relevant to your career.
There are lies, damned lies, statistics - and ADC/DAC specs.
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Offline rstofer

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Re: Why little math required for electronics?
« Reply #73 on: May 03, 2019, 10:46:41 pm »
The Internet is full of tutorials.  Every subject you could possibly name is covered somewhere.  If you don't get the material in the lecture, get it from the Internet.

For those who are serious about math up through Linear Algebra and Differential Equations beginning with Algebra, consider subscribing to CalcWorkshop.com.

Khan Academy covers a LOT of material and there is an EE track.  There are multiple videos on Laplace Transforms with complete explanations.

Things are so much better than back in the '70s.  If you're willing to work, there are plenty of sources for help.  Chegg comes to mind as a 'for pay' source of homework answers.  I have never subscribed but they cover the popular textbooks like Stewart for Calculus.

Try 3blue1brown.com or https://www.mathtutordvd.com/

Calculus is easy, it's the PreCalc material that will eat you alive!
 

Offline rstofer

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Re: Why little math required for electronics?
« Reply #74 on: May 03, 2019, 10:52:04 pm »
If you are interested in good books on engineering maths then I can recommend the books 'Modern engineering mathematics' and 'Advanced modern engineering mathematics'.
I went looking at Alibris.com for both books and it seem like Glyn James should be the author except that I see another author for an identically named book and the cost is way up there!

https://www.alibris.com/booksearch?keyword=modern+engineering+mathematics

I thought I would add them to the library but I want to get the right book(s).
 

Online nctnico

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Re: Why little math required for electronics?
« Reply #75 on: May 03, 2019, 11:19:42 pm »
If you are interested in good books on engineering maths then I can recommend the books 'Modern engineering mathematics' and 'Advanced modern engineering mathematics'.
I went looking at Alibris.com for both books and it seem like Glyn James should be the author except that I see another author for an identically named book and the cost is way up there!

https://www.alibris.com/booksearch?keyword=modern+engineering+mathematics

I thought I would add them to the library but I want to get the right book(s).
I have the books written by Glyn James. I didn't know there where other books with the same title.
There are small lies, big lies and then there is what is on the screen of your oscilloscope.
 

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Re: Why little math required for electronics?
« Reply #76 on: May 04, 2019, 01:54:03 am »
Once you sign on to the math train you will find there are innumerable nooks and crannies to dive into.  There are purely theoretical approaches (theory of functions and others) and more directly practical approaches (perturbation methods and others).  All find their occasional application in solving what would be an otherwise tough problem.

A book I have found useful when reminding myself of terrain that I haven't been over for a while is:

Handbook of Applied Mathematics, Edited by Carl E Pearson.

There are always two problems in using mathematics in engineering.  The first is formidable - knowing and remembering that there is a tool available which has bearing on the problem at hand.  This book is helps with this, and also aids in the second problem - actually using the tool which is usually a more straightforward problem of looking up and absorbing or reabsorbing the information on the method.

There is a third problem - recognizing that a given tool can apply to a given problem.  That is usually beyond the province of us mere mortals.  The guys who solve these third problems have names like Maxwell, Heaviside and Dirac.
 
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Offline rstofer

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Re: Why little math required for electronics?
« Reply #77 on: May 04, 2019, 03:09:38 pm »
If you are interested in good books on engineering maths then I can recommend the books 'Modern engineering mathematics' and 'Advanced modern engineering mathematics'.
I went looking at Alibris.com for both books and it seem like Glyn James should be the author except that I see another author for an identically named book and the cost is way up there!

https://www.alibris.com/booksearch?keyword=modern+engineering+mathematics

I thought I would add them to the library but I want to get the right book(s).
I have the books written by Glyn James. I didn't know there where other books with the same title.

Thanks!  I have ordered both.
 

Offline mathsquid

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Re: Why little math required for electronics?
« Reply #78 on: May 05, 2019, 04:36:12 pm »
Just my little grumpy bit to the plenum: Math is just a tool (one of many) universities use to fuck students with and kick them out of the school (...as soon as possible, because they get payed by the number of students applied to the school - many apply multiple times).

I'm not sure which universities you are talking about (your country isn't showing), but that's not how it works for public universities in the USA. The vast majority of the budget for teaching comes from state funding and tuition.

Offline rstofer

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Re: Why little math required for electronics?
« Reply #79 on: May 16, 2019, 04:00:11 pm »
If you are interested in good books on engineering maths then I can recommend the books 'Modern engineering mathematics' and 'Advanced modern engineering mathematics'.
I went looking at Alibris.com for both books and it seem like Glyn James should be the author except that I see another author for an identically named book and the cost is way up there!

https://www.alibris.com/booksearch?keyword=modern+engineering+mathematics

I thought I would add them to the library but I want to get the right book(s).
I have the books written by Glyn James. I didn't know there where other books with the same title.

Thanks!  I have ordered both.


OK, I now have both volumes and I have spent some time going over the first book.  It is EXCELLENT!  The best book on math EVER!  In my unqualified opinion...  Great discussion, good examples and relevant problems. 

The authors recommend MATLAB or Maple for certain problems and provide code for both tools.  Both of these tools cost money, even for the student editions.  Perhaps GNU Octave is a workable substitute for MATLAB and there are others.

The 4th Edition books, in paperback, are pretty cheap at Alibris

Even though I have multiple books on the subjects, these are going to become my 'go-to' books.
 

Online coppice

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Re: Why little math required for electronics?
« Reply #80 on: May 16, 2019, 07:03:03 pm »
The authors recommend MATLAB or Maple for certain problems and provide code for both tools.  Both of these tools cost money, even for the student editions.  Perhaps GNU Octave is a workable substitute for MATLAB and there are others.
GNU Octave is an excellent tool, but its not 100% compatible with MATLAB. If you are writing your own Octave scripts you'll be pretty happy with the results. If you try to run other people's MATLAB scripts using Octave. you'll often get different results from using MATLAB.
 

Online 0culus

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Re: Why little math required for electronics?
« Reply #81 on: May 17, 2019, 03:31:54 am »
One of my current favorite books for getting a "one stop shopping" review of mathematics relevant to EE topics is Kreyszig's Advanced Engineering Mathematics. This is from the perspective of having a degree in math and reviewing to recover knowledge, however. It might be a little too terse for someone without some background already.
 

Online coppice

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Re: Why little math required for electronics?
« Reply #82 on: May 17, 2019, 05:24:30 am »
One of my current favorite books for getting a "one stop shopping" review of mathematics relevant to EE topics is Kreyszig's Advanced Engineering Mathematics. This is from the perspective of having a degree in math and reviewing to recover knowledge, however. It might be a little too terse for someone without some background already.
Kreyszig's Advanced Engineering Mathematics has been the standard undergraduate text in many engineering departments for decades. Its good if you like pure maths. Its not so good if you like your maths linked to the real world, so you can see some relevance to the maths. In our course we were taught maths very much in a pure maths way. The relevance was given later in the engineering courses which used that maths.
 

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Re: Why little math required for electronics?
« Reply #83 on: May 18, 2019, 02:09:50 pm »
That's what I said. I have a BS in math, so I enjoy the format of the book for reviewing key concepts.
 


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