Some theory questions

[JimRayden]

Long time, no visit.

I have an exam coming up and would like a bit help with AC.

Firstly, could anyone give me a quick recap of an LC-circuit wired like a voltage divider (or a current divider with the components in parallel), I'm not entirely sure in the way the phase shifts communicate with one another. I do know that for RL and RC the total Z is calculated by using Pythagoros theorem but am confused by 180 degrees. Do they cancel each other out?

Thanks, I'll write more question as soon as I can think of any.

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Jimbo

[Thomeeque]

http://en.wikipedia.org/wiki/LC_circuit
http://en.wikipedia.org/wiki/RC_circuit
http://en.wikipedia.org/wiki/RL_circuit

[JimRayden]

I've been living in wikipedia for the past few weeks, have been helpful at some things but not so on others.

More particularly, most of the stuff i've found only include cutoff frequency calculations, as far as formulas go. My particular case calls for a transfer rate at a certain frequency.

So what I'm actually looking for are the transfer functions for an LC circuit in series and in parallel for both C and L. None of that in wiki somehow.


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Jimbo

[valdiorn]

The impedance of a capacitor is 1/(jwC)
The impedance of an inductor is jwL
Where w (omega ) is the radian frequency of the input signal (or part of the input signal)

With this kind of circuit you have a natural frequency that it will oscillate on.
Now, if you know your math this should be pretty straight forward to calculate, if not, read this (explains without using complex numbers)
http://www.allaboutcircuits.com/vol_2/index.html

[Ripthorn]

For phase shifts, I might just represent the system with an s-domain transfer function, which helps when you have a combination of resistors and capacitors/inductors.    For purely imaginary impedances, valdiorn makes it quite simple.

[JimRayden]

The impedance of a capacitor is 1/(jwC)
The impedance of an inductor is jwL
Where w (omega ) is the radian frequency of the input signal (or part of the input signal)

With this kind of circuit you have a natural frequency that it will oscillate on.
Now, if you know your math this should be pretty straight forward to calculate, if not, read this (explains without using complex numbers)
http://www.allaboutcircuits.com/vol_2/index.html

Have been using those formulas for a long while, so i can calculate their impedances, but they have opposing vectors so the phase thing confused me, still does. Is the resonant frequency relevant here? What's the method for calculating the attentuation it offers for a certain frequency?

Also, that page actually is the official studying material for this course and I have read it through many many times.


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Jimbo

[alanlan]

Well, the potential divider equation is:

(1/jwC) / ( (1/jwC) + jwL )

If you multiply throughout by (1/jwC), you get

1 / ( 1 + j^2.w^2.CL )

and since j^2 = -1

the transfer function is:

1 / ( 1 - w^2.CL )

So there is no phase shift at any frequency because there is no j term in the resulting equation and no imaginary component.

And the roll off is 2nd order i.e. 40dB/decade or 12dB/octave

turn over frequency will be given by 1 / (2.PI.sqrt(LC))

Hope this helps.

Similar results are had with a potential divider made of 2 caps or 2 inductors.

[Transmogrifox]

I'll take a more "what's going on" approach here:
Phase refers to the relationship between voltage and current.

Inductors -- Voltage leads the current.  For example, you apply a large voltage to an inductor, it takes a while before the current increases significantly.  For a simple RL circuit, you will observe that the waveform of the current lags behind the driving voltage source by 45 degrees at the 3dB frequency.  How do you see the waveform of the current?  Measure the voltage across the resistor and divide by the constant "R".

Capacitors --  Current leads the voltage.  You apply a current, and the voltage rises afterward.

Inductor in series with the capacitor:
Apply a voltage to the end of the inductor, and there is a delay before much current flows.  When current does flow, it begins to charge up the capacitor.  Now the voltage waveform goes negative, and the capacitor is now wanting to discharge through the inductor.  Depending on the forcing frequency, the sine wave at the capacitor may be left at its most positive peak while the forcing signal at the other end of the inductor has already cycled back to its negative peak.  Thats why you see to phase approaches 180 degrees as you go well above the resonant frequency.  Remember at the same time the amplitude response is getting ever so small as well.

At resonance, the capacitor and inductor exhibit complementary behavior with respect to time.  It takes the same amount of time to charge the capacitor with voltage as it does to charge the inductor with current.  As a result they dump power back and forth into eachother until the real-world parasitic resistances dissipate the power.  As a result, the input voltage source is getting current fed back out of the system when it's trying to pull it, and the system is sucking current into it when the voltage source is trying to push it.

This is why you see zero impedance on a series resonant LC system at resonance.  (of course all the while this system is storing all of that energy you're pumping into it and driving it to extremely high voltages.

[alanlan]

I didn't really interpret the equation very well:

At the resonant frequency i.e. when f = 1 / (2.PI.sqrt(LC)), the response is infinite i.e. you get the huge resonant peak.  In practice this is always dampened by some series resistance.

Also at resonance as Transmogrifox mentioned you have zero impedance across the 2 components anyway, so (at resonance) you would need a voltage source with a zero source impedance and infinite voltage to be able to drive the potential divider!

i.e. Z = sum of both components = (1/jwC) + jwL

Z = -j/wC + jwL, so at resonance, they cancel leaving zero total reactance.

[Transmogrifox]

Just a short addition to above from alanlan:

Z = 1/jwC + jwL = -j/wC + jwL  (per above)

Note this is a nice mnemonic for deriving the equation for the resonant frequency.  It clearly happens where 1/wC=wL
It's just algebra from there.
I'm a big fan of remembering a few key properties and learning how to derive the rest instead of trying to pack my head with a bunch of equations.

A classic example is power equations:
All you need to know:
1) V=IR
2)P =VI
You can get all of your I^2*R and V^2/R relationships from that, as well as determining I, R, or V given power and one of the other.  It also helps you derive formulas for the case when there are reactive elements in the circuit....and so on.

I was pretty excited (I know I'm a geek) when my college professor derived Ohm's law from Maxwell's equations.