Question

1. Plot or sketch the magnitude vs. frequency and the phase vs. frequency curves in a linear or log scale.
2. Indicate if the circuit is a low pass, high pass, band pass or band reject filter.

1 C1 R2 C2 Vout in

Answer 1

The given circuit can be represented in the Laplace domain as follows.

2 (の 2 2,0の 乙.

Let $Z_2(s)$ be the impedance representing the parallel combination of the resistor $R_2$ and the capacitor $C_2$ .

Then,

$Z_2(s)=\frac{R_2\cdot\frac{1}{sC_2}}{R_2+\frac{1}{sC_2}}=\frac{R_2}{1+sC_2R_2}$

Now according the Voltage division principle, the output voltage is

$V_{out}(s)=\frac{Z_2(s)}{R_1+\frac{1}{sC_1}+Z_2(s)}V_{in}(s)$

$\Rightarrow V_{out}(s)=\frac{\frac{R_2}{1+sC_2R_2}}{R_1+\frac{1}{sC_1}+\frac{R_2}{1+sC_2R_2}}V_{in}(s)$

$\Rightarrow V_{out}(s)=\frac{sC_1R_2}{R_1(sC_1)(1+sC_2R_2)+(1+sC_2R_2)+(sC_1)R_2}V_{in}(s)$

$\Rightarrow V_{out}(s)=\frac{sC_1R_2}{s^2R_1R_2C_2+s(R_1C_1+R_2C_2+R_2C_1)+1}V_{in}(s)$

So, the transfer function is given by

$\frac{V_{out}(s)}{V_{in}(s)} =\frac{sC_1R_2}{s^2R_1R_2C_2+s(R_1C_1+R_2C_2+R_2C_1)+1}$

Given the circuit elements are

$R_1=1K\Omega;\ R_2=10K\Omega;\ C_1=0.01\mu F\ \&\ C_2=0.0056\mu F$

The transfer function is written as

$\frac{V_{out}(s)}{V_{in}(s)} =\frac{10^{-4}s}{0.056s^2+0.000166s+1}$

$\frac{V_{out}(s)}{V_{in}(s)} =\frac{s}{560s^2+1.66s+10^4}$

The following MATLAB code is used to generate the frequency response plots.

%values of circuit elements
R1=1*10^3;
R2=10*10^3;
C1=0.01*10^(-6);
C2=0.0056*10^(-6);
%coefficients of numerator
num=[C1*R2 0];
%coefficients of denominator
den=[R1*R2*C2 R1*C1+R2*C2+R2*C1 1];
%plotting the frequency response using the command 'freqs'
freqs(num,den)

The frequency response plot obtained is

100 10 CD -4 E 10 .6 10 10-2 10-1 100 Frequency (rad/s) 10 100 50 E -50 -100 10-2 10-1 100 Frequency (rad/s) 10

The frequency response plot represents the response of the band-pass filter.