Thevenin intuitive approach

[nForce]

Hi,

this is the method: https://en.wikipedia.org/wiki/Th%C3%A9venin%27s_theorem

if I understand correctly the method: We calculate the voltage on AB terminals, because this is the voltage we get when rearranging the circuit with just only one voltage source and in series connected resistor? The same is for the resistor.

How do we calculate if we have inductors and capacitors and not just only resistors? Do we consider impedance, let's say for the inductor = jwL? The problem is that we now have complex numbers. How do we solve it then?

Thanks.

[T3sl4co1l]

Yes, the same method (and formulas) works for AC steady state, in the complex domain.

Tim

[rstofer]

The only difference is the math gets ugly for AC circuits:

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/acthev.html

It sure wasn't fun with a slide rule back in the day!  Today I would use wxMaxima, Octave or Matlab.  Even with a calculator, I would try to avoid doing it by hand.

I would like to increase my use of Matlab so:
https://www.amazon.com/exec/obidos/ASIN/0982497008/themathworks

[MrAl]

Hi,

this is the method: https://en.wikipedia.org/wiki/Th%C3%A9venin%27s_theorem

if I understand correctly the method: We calculate the voltage on AB terminals, because this is the voltage we get when rearranging the circuit with just only one voltage source and in series connected resistor? The same is for the resistor.

How do we calculate if we have inductors and capacitors and not just only resistors? Do we consider impedance, let's say for the inductor = jwL? The problem is that we now have complex numbers. How do we solve it then?

Thanks.

Hi there,

A little prerequisite for AC circuits is the theory of complex numbers.  If you have dealt with them before then you will find AC circuits to be exceptionally EASY.  Much, much, much easier than more general transient circuits where you can have signals that are other than sinusoidal.  You will be very happy to learn about complex numbers.  AC circuit theory will look easy after that.

The main difference is not about the individual numbers involved, but how we handle them and how to do the main operations like multiply, divide, add, and subtract.  I'll give a quick intro here...

Addition of two complex numbers a+b*j and c+d*j:
(a+b*j)+(c+d*j)=a+c+(b+d)*j

So you see all we do is keep the real and imaginary parts separated.
Subtraction is the same, just subtract, so we end up with:
a-c+(b-d)*j

Multiplication get a little more tricky, but just follows the rules of algebra:
(a+b*j)*(c+d*j)=a*c+c*b*j+a*d*j+b*d*j^2

and then we can note that j^2=-1 because sqrt(-1) is j.

Division is a little harder, but not too bad.

The main idea in any of these is that we have to do more than one operation for a pair of complex numbers rather than just one operation like addition.  In the end we get another complex number as the result.

Knowing just how to add, subtract, multiply, and divide complex numbers, can get you pretty far into AC circuit   theory.  Just add a small table of powers of j and you will find it is not that hard.

Also, there is automatic math software that can get you going faster, and advanced calculators that you can use to make this short work.

Questions welcome, i'll try to get back soon.

For the addition above a quick example is to add 1+2*j plus 3+4*j:
1+3+(2+4)*j=4+6*j

See how easy that was?

[T3sl4co1l]

The wider-picture view is that, complex numbers are the completion of everything you've learned about arithmetic up to this point -- there is no operation which produces a result outside the domain of complex numbers, that also operates from within them.  In the reals, you can only take sqrt(positive values), but there is no complex number you cannot take the sqrt of, or any other root, or any exponent or trig function (indeed, on further study, you learn why exp and trig functions are actually the same thing in the end -- isn't that just a tease?).

And if you've taken linear algebra, it's just that, all over again.  You have a row or block of numbers, and you operate on that block as if it's a single number, subject to basic arithmetic operations.  Complex numbers can be defined as a 2x2 matrix with redundant values (namely, a + j*b ~= [a, b; -b, a], I think it is?), or a 1x2 vector with slightly different definitions for multiplication, and admitting a division operation (technically, linear algebra does not have a division operator, but one can be defined for specific types of blocks; this happens to be one such case!).

The seamy side of it is, you've been lied to all your life, about not being able to take sqrt(-1), or certain division operations, solutions of equations and so on.  It's rather unfortunate, and I suspect the labeling of "imaginary" is almost as much a mental block to students as "real" numbers (which are anything but real, but that's a separate and wonderful exploration which I will not go into here! ).  Please proceed bravely and without regard for such blocks, if possible!

Tim

[rstofer]

From time to time you need to be able to convert from rectangular (a+jb) to polar (magnitude,angle) coordinates and back again.  It's pretty easy to deal with complex numbers in rectangular form but when it comes time to add/subtract polar coordinates it is often easier to first convert to rectangular, perform the operations and then convert back to polar form.

https://www.allaboutcircuits.com/textbook/alternating-current/chpt-2/polar-rectangular-notation/

https://www.allaboutcircuits.com/textbook/alternating-current/chpt-2/complex-vector-addition/

[MrAl]

Hi,

Yes good point where we often have to find the phase angle.

In this case i recommend starting with using the function atan2():
Angle=atan2(i,r)

This computes the phase angle from the complex number imaginary part 'i' and real part 'r'.
That works in all cases except those where we have to know the multi cycle phase shift, but that is much more rare.

A simple example is if real part is 1 and imaginary part is also 1 we have:
Angle=atan2(1,1)=pi/4 rads
and convert to degrees:
Angle=(pi/4)*180/pi=45 degrees

You can look up the function atan2() on the web.

[rstofer]

When you use atan2, be certain to check the documentation.  Some variants define it as atan2(x,y) {like every other function under the sun} and others define it as atan2(y,x).  Note that the arguments are swapped!  Additionally, check to see what it does as atan2(0,0)  Some systems return NaN (not a number) and others return 0.

It is always better to plot the vector before using atan2().  Then you know what order of magnitude the result should be and all you're getting from a calculator is more precision.

arctan() and atan2() are complicated because tan(+ or - 90 degrees) is undefined.  This is awkward because a pure inductor will have a phase angle of +90 degrees and a pure capacitor will have a phase angle of -90 degrees.  If necessary, add a little resistance in series with the reactive component to get the phase angle away from +- 90 degrees.  It doesn't need to be much, just keep the real component (resistance) away from zero.

Just when you think you have complex numbers worked out, try Kirchhoff's Voltage Law on 4 meshes.  A 4x4 matrix is bad enough when it is just real numbers, complex numbers gets completely out of hand.

Use a tool!

We didn't have anything but slide rules when I took the course.  What a PITA!

[MrAl]

When you use atan2, be certain to check the documentation.  Some variants define it as atan2(x,y) {like every other function under the sun} and others define it as atan2(y,x).  Note that the arguments are swapped!  Additionally, check to see what it does as atan2(0,0)  Some systems return NaN (not a number) and others return 0.

It is always better to plot the vector before using atan2().  Then you know what order of magnitude the result should be and all you're getting from a calculator is more precision.

arctan() and atan2() are complicated because tan(+ or - 90 degrees) is undefined.  This is awkward because a pure inductor will have a phase angle of +90 degrees and a pure capacitor will have a phase angle of -90 degrees.  If necessary, add a little resistance in series with the reactive component to get the phase angle away from +- 90 degrees.  It doesn't need to be much, just keep the real component (resistance) away from zero.

Just when you think you have complex numbers worked out, try Kirchhoff's Voltage Law on 4 meshes.  A 4x4 matrix is bad enough when it is just real numbers, complex numbers gets completely out of hand.

Use a tool!

We didn't have anything but slide rules when I took the course.  What a PITA!

Hi,

Yes there are some nuances to using atan2() and it is good to know when you have to think about it vs when you can just do the nard and fast calculation.

For the most part though it works for plotting especially when the plot function knows how to reject infinite values.
I use it a lot to plot phase responses.

The other problem which i may not have made clear is that an angle plus 360 degrees is taken to be that same angle:
ph=ph+360

and this is really:
ph=ph+360*n

where n is a positive integer or zero.  For many applications like power line stuff and filters this does not matter, but for something like a phase shift oscillator it may start to show up during the startup calculation for example.  The delay may not really be 1/2 cycle or whatever, it may be 1/2 cycle plus another full cycle or any number of full cycles.  As i said though, this rarely becomes an issue in beginning studies or even with many applications in real life.

[nForce]

What about this part of the question:

if I understand correctly the method: We calculate the voltage on AB terminals, because this is the voltage we get when rearranging the circuit with just only one voltage source and in series connected resistor?

Do I understand correctly?

[rstofer]

Yes, you calculate the open circuit voltage and that is the voltage on the AB terminals.  You mess around with all the internal impedances and current sources to come up with an equivalent internal impedance.  In the case of resistors only, this will turn out to be a simple resistor.

Finally, you have the equivalent of a battery with a series resistor hidden inside a black box.

You can measure (calculate) the value of the equivalent resistor by loading the output such that the voltage drops in half (exactly).  In this situation, the voltage dropped by the internal resistor is identical to the voltage dropped by the external resistor and since they see the same current, they must have the same value.

[nForce]

Ok, let's say for the inductor. How do we get the impedance jwl?

We take L and calculate Fourier transform? And at the end we get jwL?

[rstofer]

Ok, let's say for the inductor. How do we get the impedance jwl?

We take L and calculate Fourier transform? And at the end we get jwL?

It's nowhere near that hard!  Try this page:
http://www.electronics-tutorials.ws/inductor/ac-inductors.html

X_L = 2*pi*f*L or  wL

Note that a perfect inductor won't have resistance so it's reactance vector is at 90 degrees (up).  A pure capacitance has a reactance vector at -90 degrees.

This thread will be over the top for your question but it will give you a chance to see 'school' problems using capacitors and inductors.  Pay careful attention to any replies by Orolo - he is very good at the math.

https://www.eevblog.com/forum/beginners/mesh-analysis/

You can see where Simon's schematic deals in terms of + or - j values for the reactive components.

Somewhere in the several pages, I posted at least one computer solution to the math.

[nForce]

Ok, thank you. But I would like to see the hard way. I know what Fourier transform is, but I don't know what to insert instead of f(t).

[rstofer]

I guess I don't see what Fourier has to do with Thevenin Equivalent Circuits for DC or a simple sine wave.  All we want to find is an equivalent voltage and impedance and, in the case of AC, at some frequency.

Using a non-sinusoidal generator might require a Fourier Transform but that's way beyond the math skills assumed for a Beginner's forum.  Nevertheless, Google has references:

http://www.springer.com/cda/content/document/cda_downloaddocument/9780387490618-c1.pdf?SGWID=0-0-45-379507-p173700539

In terms of a simple sinusoidal generator, the frequencies to be plugged in are a) well below the corner frequency, b) at the corner frequency and c) at several frequencies beyond the corner frequency (several samples over a couple of decades for example).  The roll off of a single pole low pass filter is 20 dB per decade.

The problem with a very low corner frequency (like 1.59 Hz) is that a frequency one decade lower is 0.159 Hz and that is just too slow to watch.  It's better to work in the kHz region when 'well below' might be 100 Hz or so.

[IanB]

Ok, thank you. But I would like to see the hard way. I know what Fourier transform is, but I don't know what to insert instead of f(t).

There isn't a "hard way". Complex impedances are not a physical thing, and they are not derived from frequency analysis.

The use of complex numbers in AC circuit analysis is nothing more than a mathematical trick. You convert the problem to complex numbers, do the complex number arithmetic, and then extract the real answers (magnitude and phase angle) from the complex number result.

Many people who use this trick have not necessarily seen or have derived the proof of why it works, they just take it as a given and use the results.

[T3sl4co1l]

There isn't a "hard way". Complex impedances are not a physical thing, and they are not derived from frequency analysis.

The use of complex numbers in AC circuit analysis is nothing more than a mathematical trick. You convert the problem to complex numbers, do the complex number arithmetic, and then extract the real answers (magnitude and phase angle) from the complex number result.

Many people who use this trick have not necessarily seen or have derived the proof of why it works, they just take it as a given and use the results.

Well.  I would dare say complex numbers are real, to the extent that their algebra is equivalent to the real phenomena we describe with them.

I always hated the term "imaginary".  Regular numbers are bad enough -- abstract -- what are we supposed to do with them when they are naught but ghosts in our head!?  ("Real" numbers are my second most hated term, but that's a subject for another time.)

There are geometric applications of complex numbers as well.  Suppose you had a right triangle with side lengths z, 5, and hypotenuse of 4?  No, not a 3-4-5 triangle, a 3-5-4.  Well, that's patently silly, right?  But what if we plug it in anyway?  The Pythagorean theorem is perfectly valid on the complex numbers, giving z = +/- sqrt(4^2 - 5^2) = +/- 3*i.  Clearly it's not a right triangle anymore, but if you draw it out anyway, what does it tell you about the "angle" of the i factor?

Tim

[MrAl]

What about this part of the question:

if I understand correctly the method: We calculate the voltage on AB terminals, because this is the voltage we get when rearranging the circuit with just only one voltage source and in series connected resistor?

Do I understand correctly?

Hi,

The thing about linear circuits is that they respond the same with any amplitude excitation except for the amplitude itself, but that amplitude is proportional to the excitation.  So you can often find the open circuit voltage and short circuit current and then the equivalent impedance is:
Z=Voc/Ioc

To do this with inductors and capacitors, you just have to be able to calculate the impedances of each element and combine them.

[MrAl]

Ok, thank you. But I would like to see the hard way. I know what Fourier transform is, but I don't know what to insert instead of f(t).

There isn't a "hard way". Complex impedances are not a physical thing, and they are not derived from frequency analysis.

The use of complex numbers in AC circuit analysis is nothing more than a mathematical trick. You convert the problem to complex numbers, do the complex number arithmetic, and then extract the real answers (magnitude and phase angle) from the complex number result.

Many people who use this trick have not necessarily seen or have derived the proof of why it works, they just take it as a given and use the results.

Hi,

Not sure what you are trying to say here.  Two numbers that form a vector have amplitude and angle.  If we draw a triangle we can show the relationship.  We can find triangular structures in real life.

[IanB]

Not sure what you are trying to say here.  Two numbers that form a vector have amplitude and angle.  If we draw a triangle we can show the relationship.  We can find triangular structures in real life.

Complex numbers arise from trying to find the roots of polynomials when some or all of the roots are not real numbers. The result is the invention of a new kind of number that can be expressed as an ordered pair of reals (x, y) and some definitions for how to do arithmetic with these numbers. An ordered pair of reals can also be used to represent points on a 2D plane and this presents a way of visualizing complex numbers.

It is not obvious that solving linear problems in AC circuit theory has anything to do with finding the roots of polynomials, however it was discovered that if such problems are expressed as complex numbers then the rules of complex arithmetic are the right way to solve these AC circuit problems. Again, it is not obvious why the rules of complex arithmetic are the right way to go here rather than other rules (e.g. dot products, cross products, or combinations thereof), but serendipitously or otherwise, they are.

[Vtile]

Not sure what you are trying to say here.  Two numbers that form a vector have amplitude and angle.  If we draw a triangle we can show the relationship.  We can find triangular structures in real life.

Complex numbers arise from trying to find the roots of polynomials when some or all of the roots are not real numbers. The result is the invention of a new kind of number that can be expressed as an ordered pair of reals (x, y) and some definitions for how to do arithmetic with these numbers.

Actually the invention were multi-staged as is most our (the human kind) inventions. Euler made the last adjustments to them to be useful outside pure scientific object. The phasor notation used today (|Z|L(angle) instead of r*e^angle*i) is from somewhere WW1 to WW2 era **. 

The note is that complex / phasor calculus is for steady state AC as widely used.

** My assumption, based on book of Finnish professor from 1950 or close by. Where he notes this new notation which he says: "have gained popularity in recent years", this notation were angle aka phasor notation. The claim is for notation, not for the use of phasors for AC steadystate calculus as it were the norm.

Edit. Grammar and clarification of the last evenings write-up.

[Syntax_Error]

Circuits, Devices, and Systems attributes the phasor diagram representation to Charles Steinmetz. Here's a page instead of me quoting it all.

p.s. Love the caption for the graphic and the cigar and expression. Baller status.

[Vtile]

Circuits, Devices, and Systems attributes the phasor diagram representation to Charles Steinmetz. Here's a page instead of me quoting it all.

p.s. Love the caption for the graphic and the cigar and expression. Baller status.

thx a lot! That doesn't reveal the notation he used, I pressume the more mathematical r×e^j....

[IanB]

The angle notation makes a lot of sense for engineering use. Surely better to write \$5\angle 30^\circ\$ than to write \$5e^{i\pi/6}\$ ?

[Vtile]

The angle notation makes a lot of sense for engineering use. Surely better to write \$5\angle 30^\circ\$ than to write \$5e^{i\pi/6}\$ ?

Indeed, much more expressive for significant information, but could be hard at start to follow if not properly opened for who ever it is to observe it for a first time.

For non-steady-state analysis. I did personally find after side note of the before mentioned professor ("they are actually spirals" or something along those lines) and my long continued intuitive irritation (and faint wondering if it could work with non-steady-state) that something is wrong with phasors logic. I found the work of Sakae Yamamura and specifically Spiral Vector Theory, but I still haven't studied that tool through. It is surprising it is not heard more often.