Chapter 10, Problem 59E

(a)

To determine

Find the value of $V_o/V_s$ and show $V_o/V_s\to -j\omega C_1{R}_f$ as $A\to \infty$.

Answer to Problem 59E

The value of $V_o/V_s$ is $\underset{\_}{-\frac{-1}{\left(\frac{-j}{\omega C_1}\right)\left(\frac{1}{A{R}_f}+\frac{1}{R_f}+\frac{1}{A\left(\frac{-j}{\omega C_1}\right)}\right)}}$.

Explanation of Solution

Given data:

$A=-\frac{V_o}{V_i}$        (1)

$Z_f=R_f$

Calculation:

Rearrange equation (1) as follows.

$V_i=-\frac{V_o}{A}$

Refer to Figure in the respective question.

Apply KVL at node $V_i$ of given figure.

$\frac{V_i-V_o}{Z_f}=\frac{V_s-V_i}{\left(\frac{-j}{\omega C_1}\right)}$

Simplify the equation as follows.

$\begin{array}{l}\frac{V_i-V_o}{Z_f}=\frac{V_s-V_i}{\left(\frac{-j}{\omega C_1}\right)}\\ \frac{V_i}{Z_f}-\frac{V_o}{Z_f}=\frac{V_s}{\left(\frac{-j}{\omega C_1}\right)}-\frac{V_i}{\left(\frac{-j}{\omega C_1}\right)}\\ \frac{V_s}{\left(\frac{-j}{\omega C_1}\right)}=\frac{V_i}{Z_f}-\frac{V_o}{Z_f}+\frac{V_i}{\left(\frac{-j}{\omega C_1}\right)}\end{array}$

Substitute $-\frac{V_o}{A}$ for $V_i$ and $R_f$ for $Z_f$ as follows.

$\begin{array}{l}\frac{V_s}{\left(\frac{-j}{\omega C_1}\right)}=\frac{\left(-\frac{V_o}{A}\right)}{R_f}-\frac{V_o}{R_f}+\frac{\left(-\frac{V_o}{A}\right)}{\left(\frac{-j}{\omega C_1}\right)}\\ \frac{V_s}{\left(\frac{-j}{\omega C_1}\right)}=\frac{-V_o}{A{R}_f}-\frac{V_o}{R_f}+\frac{\left(-\frac{V_o}{A}\right)}{\left(-\frac{j}{\omega C_1}\right)}\\ \frac{V_s}{\left(\frac{-j}{\omega C_1}\right)}=-V_o\left(\frac{1}{A{R}_f}+\frac{1}{R_f}+\frac{1}{A\left(\frac{-j}{\omega C_1}\right)}\right)\\ \frac{V_s}{V_o}=-\left(\frac{-j}{\omega C_1}\right)\left(\frac{1}{A{R}_f}+\frac{1}{R_f}+\frac{1}{A\left(\frac{-j}{\omega C_1}\right)}\right)\end{array}$

Rearrange the equation as follows.

$\frac{V_o}{V_s}=\frac{-1}{\left(\frac{-j}{\omega C_1}\right)\left(\frac{1}{A{R}_f}+\frac{1}{R_f}+\frac{1}{A\left(\frac{-j}{\omega C_1}\right)}\right)}$        (2)

Thus, the expression $\frac{V_o}{V_s}$ is $\frac{V_o}{V_s}=\frac{-1}{\left(\frac{-j}{\omega C_1}\right)\left(\frac{1}{A{R}_f}+\frac{1}{R_f}+\frac{1}{A\left(\frac{-j}{\omega C_1}\right)}\right)}$.

Substitute $\infty$ for A in equation (2).

$\begin{array}{c}\frac{V_o}{V_s}=\frac{-1}{\left(\frac{-j}{\omega C_1}\right)\left(\frac{1}{\infty \left(R_f\right)}+\frac{1}{R_f}+\frac{1}{\infty \left(\frac{-j}{\omega C_1}\right)}\right)}\\ =\frac{-1}{\left(\frac{-j}{\omega C_1}\right)\left(\frac{1}{R_f}\right)}\\ =\frac{1}{\left(\frac{j}{\omega C_1{R}_f}\right)}\\ =-j\omega C_1{R}_f\end{array}$

Conclusion:

Thus, the value of $V_o/V_s$ is $\underset{\_}{-\frac{-1}{\left(\frac{-j}{\omega C_1}\right)\left(\frac{1}{A{R}_f}+\frac{1}{R_f}+\frac{1}{A\left(\frac{-j}{\omega C_1}\right)}\right)}}$ and $V_o/V_s$ for $A\to \infty$ is $-j\omega C_1{R}_f$ is obtained.

(b)

To determine

Find the value of $V_o/V_s$ and show $V_o/V_s\to -j\omega C_1{R}_f/\left(1+j\omega C_f{R}_f\right)$ as $A\to \infty$.

Answer to Problem 59E

The value of $V_o/V_s$ is $\underset{\_}{\frac{-1}{\left(\frac{-j}{\omega C_1}\right)\left(\frac{1}{A\left(\frac{R_f}{j\omega C_f{R}_f+1}\right)}+\frac{1}{\left(\frac{R_f}{j\omega C_f{R}_f+1}\right)}+\frac{1}{A\left(\frac{-j}{\omega C_1}\right)}\right)}}$.

Explanation of Solution

Calculation:

Since $Z_f$ represents $C_f$ parallel with $R_f$, therefore.

$\begin{array}{c}{Z}_f=\frac{R_f\left(\frac{1}{j\omega C_f}\right)}{R_f+\frac{1}{j\omega C_f}}\\ =\frac{\left(\frac{R_f}{j\omega C_f}\right)}{R_f+\frac{1}{j\omega C_f}}\\ =\frac{R_f}{j\omega C_f{R}_f+1}\end{array}$

Refer to Figure in the respective question.

Apply KVL at node $V_i$ of given figure.

$\frac{V_i-V_o}{Z_f}=\frac{V_s-V_i}{\left(\frac{-j}{\omega C_1}\right)}$

Simplify the equation as follows.

$\begin{array}{l}\frac{V_i-V_o}{Z_f}=\frac{V_s-V_i}{\left(\frac{-j}{\omega C_1}\right)}\\ \frac{V_i}{Z_f}-\frac{V_o}{Z_f}=\frac{V_s}{\left(\frac{-j}{\omega C_1}\right)}-\frac{V_i}{\left(\frac{-j}{\omega C_1}\right)}\\ \frac{V_s}{\left(\frac{-j}{\omega C_1}\right)}=\frac{V_i}{Z_f}-\frac{V_o}{Z_f}+\frac{V_i}{\left(\frac{-j}{\omega C_1}\right)}\end{array}$

Substitute $-\frac{V_o}{A}$ for $V_i$ and $\frac{R_f}{j\omega C_f{R}_f+1}$ for $Z_f$ as follows.

$\begin{array}{l}\frac{V_s}{\left(\frac{-j}{\omega C_1}\right)}=\frac{\left(-\frac{V_o}{A}\right)}{\left(\frac{R_f}{j\omega C_f{R}_f+1}\right)}-\frac{V_o}{\left(\frac{R_f}{j\omega C_f{R}_f+1}\right)}+\frac{\left(-\frac{V_o}{A}\right)}{\left(\frac{-j}{\omega C_1}\right)}\\ \frac{V_s}{\left(\frac{-j}{\omega C_1}\right)}=\frac{-V_o}{A\left(\frac{R_f}{j\omega C_f{R}_f+1}\right)}-\frac{V_o}{\left(\frac{R_f}{j\omega C_f{R}_f+1}\right)}+\frac{\left(-\frac{V_o}{A}\right)}{\left(-\frac{j}{\omega C_1}\right)}\\ \frac{V_s}{\left(\frac{-j}{\omega C_1}\right)}=-V_o\left(\frac{1}{A\left(\frac{R_f}{j\omega C_f{R}_f+1}\right)}+\frac{1}{\left(\frac{R_f}{j\omega C_f{R}_f+1}\right)}+\frac{1}{A\left(\frac{-j}{\omega C_1}\right)}\right)\\ \frac{V_s}{V_o}=-\left(\frac{-j}{\omega C_1}\right)\left(\frac{1}{A\left(\frac{R_f}{j\omega C_f{R}_f+1}\right)}+\frac{1}{\left(\frac{R_f}{j\omega C_f{R}_f+1}\right)}+\frac{1}{A\left(\frac{-j}{\omega C_1}\right)}\right)\end{array}$

Rearrange the equation as follows.

$\frac{V_o}{V_s}=\frac{-1}{\left(\frac{-j}{\omega C_1}\right)\left(\frac{1}{A\left(\frac{R_f}{j\omega C_f{R}_f+1}\right)}+\frac{1}{\left(\frac{R_f}{j\omega C_f{R}_f+1}\right)}+\frac{1}{A\left(\frac{-j}{\omega C_1}\right)}\right)}$        (3)

Thus, the expression $\frac{V_o}{V_s}$ is $\frac{-1}{\left(\frac{-j}{\omega C_1}\right)\left(\frac{1}{A\left(\frac{R_f}{j\omega C_f{R}_f+1}\right)}+\frac{1}{\left(\frac{R_f}{j\omega C_f{R}_f+1}\right)}+\frac{1}{A\left(\frac{-j}{\omega C_1}\right)}\right)}$.

Substitute $\infty$ for A in equation (2).

$\begin{array}{c}\frac{V_o}{V_s}=\frac{-1}{\left(\frac{-j}{\omega C_1}\right)\left(\frac{1}{\infty \left(\frac{R_f}{j\omega C_f{R}_f+1}\right)}+\frac{1}{\left(\frac{R_f}{j\omega C_f{R}_f+1}\right)}+\frac{1}{\infty \left(\frac{-j}{\omega C_1}\right)}\right)}\\ =\frac{-1}{\left(\frac{-j}{\omega C_1}\right)\left(\frac{j\omega C_f{R}_f+1}{R_f}\right)}\\ =\frac{-1}{\left(\frac{-j}{\omega C_1}\right)\left(\frac{j\omega C_f{R}_f+1}{R_f}\right)}\\ =\frac{-1}{\left(\frac{j\omega C_f{R}_f+1}{j\omega C_1{R}_f}\right)}\end{array}$

$\frac{V_o}{V_s}=\frac{-j\omega C_1{R}_f}{1+j\omega C_f{R}_f}$

Conclusion:

Thus, the value of $V_o/V_s$ is $\underset{\_}{\frac{-1}{\left(\frac{-j}{\omega C_1}\right)\left(\frac{1}{A\left(\frac{R_f}{j\omega C_f{R}_f+1}\right)}+\frac{1}{\left(\frac{R_f}{j\omega C_f{R}_f+1}\right)}+\frac{1}{A\left(\frac{-j}{\omega C_1}\right)}\right)}}$ and $V_o/V_s$ for $A\to \infty$ is $\frac{-j\omega C_1{R}_f}{1+j\omega C_f{R}_f}$ is obtained.