Chosing component values for LC tuned circuits

[urlkrueger]

I'm trying to get a better understanding of LC tuned circuit construction as used in perhaps an oscillator or radio IF circuit.

From the basic formula for resonance



if I set the frequency to be constant and rearrange I can get



where
 

This clearly points out that as C gets smaller L must increase to maintain the constant frequency.  But in the real world at some point C would approach the stray capacitance of the circuit and things would get a little unpredictable or L would become a medium size boat anchor, neither of which are highly desirable.   Or if you go the other way C becomes a space hog and L becomes a short length of PCB trace. 

Since within these limits there are still an infinite number of values of C (or L) that could be used how does one know where to start?

I've read quite a few construction articles where they may say use toroid core xxx with 15 turns of number 30 wire or such, but they never explain why they chose those particular values.

Do you have any insight on this to share?  Or is it one of those things that if you do it enough you just develop an intuition of where to start?

Thanks...
earl...

[w2aew]

Usually, it comes down to the parasitics that determine where the bounds are. Self-resonant frequency of the cap and inductor play a big role in selecting good starting points. The required Q of the tuned circuit also plays a role. Often, once these boundaries are applied, there may still be an order of magnitude range that you can "pick" from. Many times, it's the availability of particular values that helps settle the final values.

[salbayeng]

Decades ago we had the ARRL handbook which walked you through all this.

----- basically its all due to the physics of the universe ----
One needs to consider the two flavors of energy : Dissipative and Stored
Dissipative is the one that gets hot.
You can store energy as potential ( a brick on top of the cupboard)
or as a Kinetic , (a brick swung horizontally on a piece of string)
you can swap back and forth ( a brick swung in a vertical loop)
In all cases when the brick hits you , energy is dissipated according to its velocity, and the amount of reaction force.

Lets drop a rubber ball from 5ft ,  it bounces back to 4ft,
Lets drop a gob of putty from 5ft , it doesnt bounce.
Lets drop a massive steel ball 5ft ,  it bounces back to 4ft,
Ok whats the same, and whats different?
The rubber ball with initial energy 5units, split this off into one unit of dissipation, and 4of stored energy.
The ball is obviously a higher quality bouncer than putty, lets define the quality factor of the ball
as the ratio of stored energy to dissipated energy, i.e 4ft/1ft = 4 so Q = 4
What about the Steel ball ? obviously Q = 4  for it also,
The putty, Q = 1 
But the ball is "harder" right,  so lets define an impedance Z as the ratio of force/velocity
Both balls bounce at the same frequency so this frequency is independant of Q or Z
All of the above is true throughout the physical universe, be it planets, or balls, or compressed air, or plastic rulers, springs, tyres,  magnets, motors, electrical circuits,
I may have skipped 2pi here and there , refer here http://en.wikipedia.org/wiki/Q_factor

---- Reactance-Frequency Chart -----
Get one of these, it's is the only piece of paper (except for birth certificate) that I've kept for the last 50yrs.
Here is a typical one http://www.rfcafe.com/references/electrical/frequency-reactance-nomograph.htm
 
------ R L C -----
There are two very important concepts Q=quality factor , and Z = impedance.

As Q is the ratio of reactive/dissipative it is also the ratio of the resistive impedance to the reactive impedance.
Consider a 1000uF capacitor in series with a 1ohm resistor , being used to filter a power supply that has ripple at 160Hz
So at 160Hz, the impedance of the resistor is 1 ohm , the reactance of the capacitor is -j1ohm , the Q = 1/1 = 1  at 160Hz
Lets make it more interesting and throw a 1mH filter inductor in there , surprise it has an impedance of j1ohm at 160Hz,
so there we have a resonant circuit with Q=1 , at 160Hz, the total series resistance is 1 +J1 -J1  = 1ohm .
Oddly enough it doesn't matter how you wire up these three components, you always get the same Q.
Now this Q is inversely proportional to your k , e.g. Q = 1/(2k)     
Q is also the ratio of bandwidth to resonant frequency.
Q is approximately the ratio of the energy lost from successive cycles.

So its a two step process figure out the impedance of the parts at the frequency of interest
Then work out the Q from the ratio. Once you have these, everything else fits together.

Lets build your radio transmitter now, let's make it operate at 1.6MHz , so take your chart, look horizontally along at 1.6MHz , there are hundreds of possible choices of L and C combinations (the 45deg lines) , each corresponds to a certain impedance (the X axis).  Lets assume we have a certain output transistor which has an effective resistance of 10ohm, That kind of suggests 10nf and 1uH as the resonant parts , but we really want the transmitter tank to be swinging on its own with just a little push from the transistor , typical transmitter tanks  run with a Q of 12-20 , lets make it Q=10 ,   So to make a Q of 10 with a 10ohm transistor we need 100ohms of reactive components, which is 1000pF and 10uH.
So in keeping with Q=10 , we might input 10W from the transistor, and have 100W bouncing around inside the tank, i.e. there will be 1amp of collector current and 10A circulating.  The other reason to have some Q at the collector is to suck up the second harmonic , the second harmonic is at twice the frequency, so L and C will be 20ohm and 50hm , so the "gain" at 3.2MHz is only 0.5 , while the "gain" at 1.6MHz is 10,

So now we have a resonant circuit with Z=100ohm  but we have a 50ohm antenna to attach to it ,  we also need an output filter as our second harmonic is still too high. Lets make another  tank with 200pF // 50uH = 1.6Mhz = 500ohm, Q=10    , Where do we put the antenna wire? simple , make the 200pf as a series combination of 220pF + 2200pF , the bigger capacitor connects to ground, so what are the impedances?  Approximately -j50ohm and -J450ohm , so just attach the antenna across the 2200pF.  The antenna now also has 10W of real power, and 100W of reactive power (but the volts and amps are 5x and 1/5 that of the Transistor tank)

Now we need to connect the two tanks , obviously a straight piece of wire will detune both tanks,  there are numerous possibilities, here I will top couple the two coils, so a small capacitor between the tops of each coil, to keep our Q of 10 on the antenna tank I'll set Z capacitor = 10 x Z tank = 20pF , this is a pretty sloppy way to do it, but once you tune up each coil, all the power going in at 1.6MHz at the transistor will appear at the antenna, whereas only about 1% of the 3.2MHz will appear there.
So you can pretty much design the whole thing using just the chart and a pencil, Just like Marconi and Tesla.

[salbayeng]

OOps , forgot the original question.
The rule of thumb about inductor Q , this pretty much says that the best Q occurs with a solenoidal coil twice as long as its diameter.
Why is this so?
Consider a much longer solenoid, maybe 10 times its diameter of 10mm, arbitrarily has 100turns at 1mm pitch, so 100mm long,
now the magnetic field at the centre is generated by all 100 turns,
    the magnetic field at +/-10mm from the centre is generated by 80 turns,
    the  magnetic field at +/-20mm from the centre is generated by 60 turns,
    the magnetic field at +/-30mm from the centre is generated by 40 turns,
    and there is stuff-all field at the ends
The problem is that the resistance  = losses are spread uniformly along the wire ,  while the magnetic energy is all piled up in the middle. So if you unwind the last 10mm of each end and rewind it in the middle , the resistance is unchanged but the inductance goes up, so Q improves.
So you could do this again, and put more wire in the middle.  Some one dis this ~ 100years ago and determined that 2:1 gave the most inductance for a given length of wire.
So why 2:1 and not 1:1 or a pancake? This is related to the shape of the magnetic field of a dipole , a formula somewhere has a 2r sin(theta) + theta cos(theta)  in it , so there is the 2.
(or it's an inverse square law, and the calculus gives you the 2)

[K1JOS]

What a refreshingly clear and practical "refresher" learning.  You should send this in to QST magazine or Nuts and Volts, etc as a one-pager.  Also should be made a permanent sticky here.

Many thanks

Jerry K1JOS

[T3sl4co1l]

In short, filters are all about impedance and frequency, not L and C.  And, in turn, about power flow, and power transfer.

When working with the equations (should you be so adventurous as to evaluate the AC steady state yourself), one of the first things you want to do is rearrange the formula to use dimensionless ratios (of impedance and frequency), so you aren't shuffling around all those units (and inviting more mistakes!).

For example, the Q (selectivity) of an LC resonator is related to the losses in the circuit.  So, cap ESR, inductor DCR and parallel loss, etc.  But, that's all well and good for a standalone system, but you can't very well get anything into or out of it!  As soon as you add a source and a load, you have the equivalent resistance of them -- and they now determine Q along with losses.

It also doesn't matter much where the loss goes: whether in series or parallel, on one side of an LC ladder network or the other (or in the middle), the energy will find its way into that.  Because filters are not absolute things, energy is sloshing around back and forth all the time, eventually it'll find its way out (transmitted to the load, reflected back to the source, or absorbed as loss).

The main difference between filter characteristics (Butterworth, Chebyshev, etc.) is how the impedances and frequencies of each section are arranged.  By twisting the impedances and phase shifts at key frequencies, more or less energy is reflected (a lossless filter always either reflects power back to the source, or transmits it to the load), and by arranging the right superposition of those reflections, the profile can be shaped pretty much however you want.

Which has a direct analog in those pretty GHz-range traces (like in Dave's teardown of that portable Agilent spectrum analyzer -- I don't remember the vid #), where the impedance is the width of the trace (or how it spreads out, if not a simple straight shape), and the frequency or phase shift or delay is the length.  In fact, there's a theorem that allows you to compute one from the other -- transmission line versus RLC (lumped constants) forms.

Tim

[urlkrueger]

Looks like some great info there guys but it's going to take me a little while to digest it all. 

In trying to keep things simple I was ignoring Q and R but it looks like I can't do that anymore.


thanks again...
earl...