Finding the Transfer Function of an RC Circuit in the Laplace domain

In summary: Remember to combine like terms and isolate I1. Once you have I1, you can solve for Vo/Vi by using the relation I1 = Vo/R2.In summary, the conversation discusses a circuit with an input Vi(s) across a capacitor C and resistor R1 in parallel, and another resistor R2 in series. The question asks to show the transfer function of the circuit, where the laplace voltage across each component is given. The conversation also discusses using mesh analysis to find the equivalent equation for Vi and Vo, and using the voltage divider concept to simplify the transfer function. The speaker also suggests using conductance and susceptance instead of resistance and impedance for easier calculation. Finally, the conversation concludes with advice on solving for I
  • #1
zaka
3
0
Hi,

Basically I have circuit with an input Vi (s) across a capacitor C which is in parallel with a resistor R1. And these 2 components are in series with another resistor R2 (please see attached drawing).

The question states:

Show that the transfer function of the circuit is:

Vo/Vi = (s + (1/(R1.C))) / (s + (R1+R2)/(R1.R2.C)) (Hope that's clear)

Where the laplace voltage (drop) across each individual component is as follows:

V(s) = I(s) / sC [laplace of capacitor]

V(s) = I(s) . R [laplace of a resistor]



In a previous worked example, the Transfer function was found by using mesh analysis and Kirchhoff's second law to find an equivalent equation for Vi and Vo. Therefore I thought I could use that here.

In the attachment I've redrawn the circuit given in the question (which is on the left labeled (1)), so that there are 2 clear loops to use mesh analysis on (the right labeled (2)). Is this correct?

(1) is the diagram given in the question
and (2) is what I've redrawn (1) as.
(This is the first time I've come across a resistor in parallel with a capacitor and I'm guessing they both have the same voltage drop across them).



Anyway so what I did was as follows (using my attached equivalent drawing (2):

loop 1) Vi (s) = [I1(s) . R1] + [I1(s) . R2] - [I2(s) . R1]

loop 2) 0 = [I2(s) . R1] + [I2(s) / sC] - [I1(s) . R1]

I rearranged loop 2 equation to get I2(s) in terms of I1(s) and substituted that into loop 1 equation. I think this would be Vi. As for Vo I assumed that would just be the voltage drop across R2 which is I1(s) . R2.

However, I could not simplify my answer (whatsoever) to what the question gave, making me think that I've done something wrong.

Any help would be much appriciated. I hope everything above makes sense.
 

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  • #2
Why write these loop equations?

If you have a voltage divider with R1 and R2, R1 input to output & R2 output to ground, the transfer function is R2/(R1 + R2), right? So use the same idea except you now have Z's instead of R's.

What is Z1?
What is Z2?
Then Vo/Vi = Z2/(Z1 + Z2).

It's much easier in this case to use conductance and susceptance instead of resistance and impedance but I'm guessing that would not help you at this stage.
 
  • #3
zaka said:
Anyway so what I did was as follows (using my attached equivalent drawing (2):

loop 1) Vi (s) = [I1(s) . R1] + [I1(s) . R2] - [I2(s) . R1]

loop 2) 0 = [I2(s) . R1] + [I2(s) / sC] - [I1(s) . R1]

I rearranged loop 2 equation to get I2(s) in terms of I1(s) and substituted that into loop 1 equation. I think this would be Vi. As for Vo I assumed that would just be the voltage drop across R2 which is I1(s) . R2.

However, I could not simplify my answer (whatsoever) to what the question gave, making me think that I've done something wrong.
Your work looks fine so far. You're probably just making an algebra mistake when solving for I1.
 

1. What is a transfer function?

A transfer function is a mathematical representation of the relationship between the input and output of a system. In the context of an RC circuit in the Laplace domain, it describes how the input voltage affects the output voltage.

2. How do you find the transfer function of an RC circuit in the Laplace domain?

To find the transfer function of an RC circuit in the Laplace domain, you can use the formula Vout(s)/Vin(s) = 1/(1 + RCs), where Vout(s) is the output voltage and Vin(s) is the input voltage. This formula is derived from the Laplace transform of the differential equation that describes the RC circuit.

3. What is the significance of finding the transfer function in the Laplace domain?

The Laplace domain is a mathematical representation that allows for easier analysis and manipulation of complex systems, such as an RC circuit. By finding the transfer function in the Laplace domain, we can use algebraic methods to analyze and design the circuit, rather than relying on differential equations.

4. How does the transfer function of an RC circuit change with different values of resistance and capacitance?

The transfer function of an RC circuit is affected by the values of resistance and capacitance in the circuit. A higher resistance value will result in a slower response to changes in the input voltage, while a higher capacitance value will result in a faster response. The transfer function can also be adjusted by changing the values of resistance and capacitance to achieve desired circuit behavior.

5. Can the transfer function of an RC circuit in the Laplace domain be used to predict the behavior of the circuit in the time domain?

Yes, the transfer function of an RC circuit in the Laplace domain can be converted back to the time domain using the inverse Laplace transform. This allows us to predict the output voltage of the circuit over time based on the input voltage and the transfer function.

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