What Is Resonance in a Parallel-Series LCR Network?

[nathangrand]

I'm having some problems understanding resonance in an LCR network where the capaitor is in parallel to the inductor and resistor, which are in series. How would you go about deriving the equation for the transient response (to a square wave) and the steady state response to a sinusoidal wave?

For the transient response for a circuit where all the components are all in series, an energy method is used in my notes to find the differential equation:

E(total)= E(inductor) +E(capacitor) = Li²/2 +q²/2C

and then using the fact that the rate of energy 'loss' is -i²R

The differential equation that results is:
d²q/dt² +(R/L)dq/dt +q/LC = 0

Can the same analysis be used for the network I described to give the same differential equation?

If it is then I can solve the differential equation equation to find the freqeuncy of oscillation of the decay for the transient response.

What is the resonant frequency under steady state response? Is it w0={[tex]\sqrt{}(1/LC)-(R/L)²} as given on wikipedia?
http://en.wikipedia.org/wiki/RLC_circuit
Or is it the same as for the transient response? I get the impression there are different ways of defining the resonant frequency -what one should generally be used?

Any help massively appreciated!

 

[vela]

I'm having some problems understanding resonance in an LCR network where the capaitor is in parallel to the inductor and resistor, which are in series. How would you go about deriving the equation for the transient response (to a square wave) and the steady state response to a sinusoidal wave?

For the transient response for a circuit where all the components are all in series, an energy method is used in my notes to find the differential equation:

E(total)= E(inductor) +E(capacitor) = Li²/2 +q²/2C

and then using the fact that the rate of energy 'loss' is -i²R

The differential equation that results is:
d²q/dt² +(R/L)dq/dt +q/LC = 0

You didn't say if there a source driving the series circuit. I'm guessing there wasn't otherwise it would be supplying energy to the circuit and you'd have another term in the equation.

Can the same analysis be used for the network I described to give the same differential equation?

If it is then I can solve the differential equation equation to find the freqeuncy of oscillation of the decay for the transient response.

You'll get a different equation for the new circuit. How exactly is the source connected to the circuit? Is it a voltage source or a current source?

Did you really mean a square wave input (periodic) or did you mean a step input (goes from 0 to 1 at t=0 and stays at 1)?

What is the resonant frequency under steady state response? Is it w0={[tex]\sqrt{}(1/LC)-(R/L)²} as given on wikipedia?
http://en.wikipedia.org/wiki/RLC_circuit
Or is it the same as for the transient response? I get the impression there are different ways of defining the resonant frequency -what one should generally be used?

Any help massively appreciated!

The resonant frequency is the driving frequency that elicits the biggest response from the circuit.

 

[nathangrand]

Thanks for the reply,

No source driving the series circuit so energy is just dissipated by the resistor.

In the set up a square voltage wave was used to study the transient response and a diode in series with the network meant that for half the period of the wave the circuit was essentially isolated, allowing the transient response to be observed. Does this make sense?

 

[vela]

I assume the voltage source was connected in parallel to the capacitor and the inductor-resistor combo. What quantity was considered the output of the circuit?

Are you familiar with the s-domain?

 

[nathangrand]

Transient output was the voltage across the LCR network.

Not familiar with S domain

 

[ehild]

Was the set-up like in the attachment? And the voltage between C and O was measured by an oscilloscope?

If so, the capacitor is charging up when the signal voltage is positive and therefore the diode is open. But you measured the transient response in the other half-period when the signal voltage was zero. In that case the diode was closed, and the capacitor discharged through the inductor (and its resistance). It is as if you had removed the generator and diode so you can consider the circuit as a simple series resonant one in that half-period.

ehild

 

[nathangrand]

Yes! Thank you!

That's the set up - so the equation I put in my original post is ok for the transient response in the 'other half period' as you say?

Any insight as to what the resonant frequency for the steady state response is?

 

[vela]

Yes! Thank you!

That's the set up - so the equation I put in my original post is ok for the transient response in the 'other half period' as you say?

Yup.

Any insight as to what the resonant frequency for the steady state response is?

I take it when you used the sine wave input, you removed the diode from the circuit. You'd want to apply Kirchoff's laws to the circuit to derive the differential equation that describes the circuit, and you'd find the resonant frequency would be given by the formula you found on Wikipedia.

 

[nathangrand]

Brilliant, I'd hoped that would be the case. Yes the diode was removed from the circuit. I have one final question - for the steady state response I found resonance curves by plotting the magnitude of the impedance/DC resistance as a function of frequency. As the resistance was increased the resonant curve got wider, occurred at a lower frequency and the value of magnitude of the impedance/DC resistance decreased. The first two are easy to explain with the the equation given on wikipedia and using the quality factor. However, is there a quick way of showing the final thing - I started to write out an equation for the impedance of the circuit and it got really messy very quickly!

 

[ehild]

Write the admittance instead (reciprocal impedance), find the magnitude and take the reciprocal. And show your work.

ehild

 

[nathangrand]

Ok...still not sure exactly how you'd go about it with admittances -- ie how to combine them in series/parallel but here's a start with impedancesL

Resistor: R
Inductor: iwl
Capacitor1/(iwc)

inductor and resisitor in series => iwl + R

This combination in parallel with capacitor => 1/(iwl + R) +(iwc) = 1/Ztot

so [1+(iwl + R)(iwc)]/(iwl+R) =1/Ztot

Thought I might rationalise it next by multiplying top and bottom by R - iwl

...

 

[ehild]

If the resonant frequency is defined with the frequency where the impedance/admittance is real you can find it by rationalizing the admittance you got. Do what you planned, multiply both the numerator and denominator by R-iwL, collect the imaginary terms and make the imaginary part zero.

See:
http://en.wikipedia.org/wiki/RLC_circuit#Other_configurations

ehild

 

[nathangrand]

Sorry for slow reply have been away!
That's great I got the equation found on wikipedia. To explain why the impedance at resonance decreases with increasing resistance I did the following:

At resonance frequency, where imaginary part of impedance = 0,
1/Ztot = R/[R²+w²L²] and then taking the reciprocal to find an expression for Ztot

substituting the resonant frequency from wikipedia into this equation for Ztot and cancelling eventually left me with Ztot(resonant)=L/CR which would explain why the magnitude of the impedance decreases with increasing resistance.

Can someone just confirm I've done this right??

Thanks again for all the help

 

[ehild]

Yes, you did it well, the impedance is simply L/(CR) at the resonant frequency.

You did not really need the formula from Wikipedia for the resonant frequency as you could have derived it yourself.

Eliminate the imaginary part from the denominator of the admittance and collect the imaginary terms in the numerator and make it equal to zero. It is better to find out something by yourself than picking up a formula from somewhere. Are you sure that Wikipedia was right?
I am sure you can do it. Try.

ehild

 

[nathangrand]

Sorry I didn't make myself clear - - I did derive the formula on wikipedia but just found it easier to write ''the formula from wikipedia'' rather than write out the equation! Not a fan of the Latex Reference! Thanks for everyone's help :)