Engineering RLC circuit -- Phasor Transfer Function calculation

AI Thread Summary
The discussion focuses on calculating the transfer function for an RLC circuit using phasors, specifically seeking to derive an expression for the voltage ratio V_R/V_in and identify the resonant frequency. Participants emphasize the importance of expressing the transfer function in terms of amplitude and phase difference, noting that the circuit behaves like a bandpass filter. The resonant frequency condition is confirmed as X_L = X_C, although the impact of resistance on the calculations is acknowledged as minimal for high Q systems. The conversation highlights the need for algebraic analysis to simplify the transfer function expression effectively. Overall, the thread provides insights into frequency response analysis and the derivation of transfer functions in RLC circuits.
Franklie001
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Homework Statement
For the RLC circuit shown above and sinusoidal input voltage, derive the theoretical expression for Vout/Vin (frequency response) using phasors. Also, give the expression for the resonant frequency of the circuit.
Calculate the resonant frequency for L=10mH, C=10nF, R=50 ohm
Relevant Equations
Vout/Vin
Good morning,

I need some help solving those two question. I've attached my attempted solution below. Could i solve the transfer function any further?

Thank you for your help
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Hi,

From the "using phasors" in the problem statement, I suspect you are asked to derive an expression that clearly shows amplitude (i.e. ## \left | {V_{R}\over V_{in}} \right | ## ) and phase difference.

##\ ##
 
Frequency response analysis in Finite Element Analysis (FEA) is used to calculate the steady-state response due to a sinusoidal load applied to a structure at a single frequency. It is a specialized type of transient response analysis that is extremely efficient to solve a very specific type of model.
I think in RLC circuit the equation has to be:
VL+VR+VC=f(t) if no load is out. f(t)=F1sin(ωt)+F2cos (ωt)={F}e^iωt
Then the derivate will be Ld2i/dt2+Rdi/dt+i/C=-ω^2{F}e^iωt
 
thanks for your response. Could i ask why you derivate the expression? I'm sure that question asks to solve for the frequency response Vout/Vin of the circuit and its resonant frequency.
I had an attempt at it but unsure how to derive the expression for the resonant frequency?
Also, in the second question, I've been given a value of the resistor to calculate the resonant frequency. Could i ask why? Isn't the resonant frequency condition is XL = XC ?

Thanks again
 
The circuit will have a maximum ## \left | {V_{R}\over V_{in}} \right | ## at resonance.
Work out that expression and ##\omega_{res}## can be found. Rather easily.

Franklie001 said:
Could i ask why?
Perhaps to confuse you ?

##\ ##
 
BvU said:
The circuit will have a maximum ## \left | {V_{R}\over V_{in}} \right | ## at resonance.
Work out that expression and ##\omega_{res}## can be found. Rather easily.Perhaps to confuse you ?

##\ ##
Hi, thank again for your help. I really appreciate it
Just to clarify ..isn't Vr/Vin what i was trying to find?
Also, It looks to me to be a bandpass filter from the expression found previously.
Also, how would you simply the expression for Vout/Vin for transfer function?
 
Franklie001 said:
Isn't the resonant frequency condition is XL = XC ?
Yes, this is the common answer and it works well for highly resonant systems (i.e. high Q, low damping). The real answer does depend a bit on the losses (resistance), but I'm not sure they really intend for you to account for that. For the values given the difference is quite insignificant.

https://en.wikipedia.org/wiki/RLC_circuit#Overdamped_response
 
Franklie001 said:
Just to clarify ..isn't Vr/Vin what i was trying to find?
Yes. From the "using phasors" in the problem statement I conclude your textbook (or notes) should have something along those lines. Also, ##V_{in}## is given as a phasor: ##V_{in}\ \angle \ 0^\circ##.
So an answer along these lines (or any suitable other) seems indicated.
Franklie001 said:
Also, It looks to me to be a bandpass filter from the expression found previously.
Yes. The frequency characteristic ##\left | {V_{R}\over V_{in}} \right |## as a function of ##\omega## shows a peak at the resonance frequency.

Franklie001 said:
Also, how would you simply the expression for Vout/Vin for transfer function?
That, for me, is the essence of the job (your job) at hand ! :smile:

And your link in #8 is not very useful in this context.

Your work in post #1 is more or less the start of an algebraic analysis (an approach that I strongly support; much better in an introductory stage (correct me if I am wrong, but I estimate you are still in that stage).

So you have $$ {V_{R}\over V_{in}} = {R\over R + j\left ( \omega L - \displaystyle {1\over \omega C} \right ) }
$$
How can you write this in the form the exercise seems to prefer: ##A\ \angle\ \phi## ? ( Note that I tried to help a little bit by writing the numerator in this way :wink: )

##\ ##
 
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DaveE said:
Yes, this is the common answer and it works well for highly resonant systems (i.e. high Q, low damping). The real answer does depend a bit on the losses (resistance), but I'm not sure they really intend for you to account for that. For the values given the difference is quite insignificant.

https://en.wikipedia.org/wiki/RLC_circuit#Overdamped_response
Thank you for the help I ll read this as soon as I can
 

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