# Homework Help: Transfer Function and State Space Analysis of Op-Amp circuit

1. Jan 27, 2014

### GreenPrint

1. The problem statement, all variables and given/known data

[PLAIN]http://imagizer.imageshack.us/v2/800x600q90/690/0tab.png [Broken] [Broken]
I'm asked to find the transfer function and then find the state space representation and seem to be stuck.

2. Relevant equations

3. The attempt at a solution

My textbook states that for a noninverting operational amplifier shown below

http://imagizer.imageshack.us/v2/800x600q90/10/zmir.png [Broken]

$\frac{V_{o}(s)}{V_{i}(s)} = \frac{A}{1 + frac{AZ_{1}(s)}{Z_{1}(s) + Z_{2}(s)}}$

and for large A

$\frac{V_{o}(s)}{V_{i}(s)} = \frac{Z_{1}(s) + Z_{2}(s)}{Z_{1}(s)}$

For my circuit

[PLAIN]http://imagizer.imageshack.us/v2/800x600q90/690/0tab.png [Broken] [Broken]

$R_{4} = 110 KΩ$
$C_{2} = 4 μF$
$R_{3} = 600 KΩ$
$R_{2} = 400 KΩ$
$C_{1} = 4 μF$
$R_{1} = 600 KΩ$

Now then

$Z_{1}(s) = R_{2} + (C_{1}s + \frac{1}{R_{1}})^{-1}$
$Z_{2}(s) = R_{3} + (C_{2}s + \frac{1}{R_{4}})^{-1}$

Hence for my circuit the transfer function is

$\frac{V_{o}(s)}{V_{i}(s)} = \frac{Z_{1}(s) + Z_{2}(s)}{Z_{1}(s)} = \frac{R_{2} + (C_{1}s + \frac{1}{R_{1}})^{-1} + R_{3} + (C_{2}s + \frac{1}{R_{4}})^{-1}}{R_{2} + (C_{1}s + \frac{1}{R_{1}})^{-1}}$

I can rearrange and get

$V_{o}(s)(R_{2} + (C_{1}s + \frac{1}{R_{1}})^{-1}) = V_{i}(s)(R_{2} + (C_{1}s + \frac{1}{R_{1}})^{-1} + R_{3} + (C_{2}s + \frac{1}{R_{4}})^{-1})$
$R_{2}V_{o}(s) + \frac{R_{1}}{C_{1}R_{1}s + 1}V_{o}(s) = R_{2}V_{i}(s) + \frac{R_{1}}{C_{1}R_{1}s + 1}V_{i}(s) + R_{3}V_{i}(s) + \frac{R_{4}}{C_{2}R_{1}s + 1}V_{i}(s)$
$L^{-1}(R_{2}V_{o}(s)) + \frac{\frac{1}{C_{1}}}{s + \frac{1}{C_{1}R_{1}}}V_{o}(s) = L^{-1}(R_{2}V_{i}(s) + R_{3}V_{i}(s)) + \frac{\frac{1}{C_{1}}}{s + \frac{1}{C_{1}R_{1}}}V_{i}(s) + \frac{\frac{1}{C_{2}}}{s + \frac{1}{C_{2}R_{4}}}V_{i}(s)$
$R_{2}V_{o}(t) + \frac{1}{C_{1}}L^{-1}(\frac{V_{o}(s)}{s + \frac{1}{C_{1}R_{1}}}) = (R_{2} + R_{3})V_{i}(t) + \frac{1}{C_{1}}L^{-1}(\frac{V_{i}(s)}{s + \frac{1}{C_{1}R_{1}}}) + \frac{1}{C_{2}}L^{-1}(\frac{V_{i}(s)}{s + \frac{1}{C_{2}R_{4}}})$

I'm not sure how to evaluate this further

Last edited by a moderator: May 6, 2017
2. Jan 27, 2014

### donpacino

starting here

R2Vo(s)+$\frac{R1}{C1R1s+1}$Vo(s)=R2Vi(s)+$\frac{R1}{C1R1s+1}$Vi(s)+R3Vi(s)+$\frac{R4}{C2R1s+1}$Vi(s)

I would get everything on one side of the equation under a common denominator. I'll do the left half of an equation.

$\frac{R2(C1R1s+1)}{C1R1s+1}$Vo(s)+$\frac{R1}{C1R1s+1}$Vo(s)=....

$\frac{R2(C1R1s+1)+R2}{C1R1s+1}$Vo(s)=....

You do the right half. Then you can get it in the standard form for a transfer function.

We'll address the state space work when you get there