1. Problem (a): Find the transfer function $\frac{e_o}{e_i}$ for the op amp circuit with resistors $R_1$, $R_2$, $R_3$ and capacitors $C_1$, $C_2$. Step 1: Identify the configuration and write the impedance expressions. - Let $Z_{C1} = \frac{1}{j\omega C_1}$ and $Z_{C2} = \frac{1}{j\omega C_2}$. Step 2: Write the node equations or use the voltage divider and feedback relations. Step 3: The transfer function is given by $$\frac{e_o}{e_i} = -\frac{Z_{f}}{Z_{in}}$$ where $Z_{in} = R_1$ and $Z_f$ is the parallel combination of $R_2$, $R_3$, and $C_2$ in series with $C_1$. Step 4: Calculate $Z_f$: $$Z_f = \left(\frac{1}{R_2} + \frac{1}{R_3} + j\omega C_2\right)^{-1} + \frac{1}{j\omega C_1}$$ Step 5: Substitute and simplify to get the transfer function. 2. Problem (b): Find the transfer function $\frac{v_o(t)}{v_i(t)}$ for the op amp circuit with resistors $R$, capacitor $C$, and feedback capacitor $C_i$. Step 1: Define impedances: $$Z_R = R, \quad Z_C = \frac{1}{j\omega C}, \quad Z_{C_i} = \frac{1}{j\omega C_i}$$ Step 2: Input impedance is $Z_{in} = R + R + Z_C = 2R + \frac{1}{j\omega C}$. Step 3: Feedback impedance is $Z_f = Z_{C_i} = \frac{1}{j\omega C_i}$. Step 4: Transfer function for inverting amplifier: $$\frac{v_o}{v_i} = -\frac{Z_f}{Z_{in}} = -\frac{\frac{1}{j\omega C_i}}{2R + \frac{1}{j\omega C}}$$ Step 5: Simplify denominator: $$2R + \frac{1}{j\omega C} = \frac{2R j\omega C + 1}{j\omega C}$$ Step 6: Thus, $$\frac{v_o}{v_i} = -\frac{\frac{1}{j\omega C_i}}{\frac{2R j\omega C + 1}{j\omega C}} = -\frac{1}{j\omega C_i} \cdot \frac{j\omega C}{2R j\omega C + 1} = -\frac{C}{C_i (1 + 2R j\omega C)}$$ 3. Problem (c): Find the transfer function $\frac{v_o(t)}{v_i(t)}$ for the op amp circuit with input resistors $2R$, capacitors $2C$, feedback resistor $R$, and capacitors $C$. Step 1: Input impedance: $$Z_{in} = 2R + \frac{1}{j\omega 2C}$$ Step 2: Feedback impedance: $$Z_f = R \parallel \frac{1}{j\omega C} \parallel \frac{1}{j\omega C}$$ Step 3: Calculate parallel of two capacitors $C$: $$Z_{C_{parallel}} = \frac{1}{j\omega C/2} = \frac{1}{j\omega (2C)}$$ Step 4: Then feedback impedance is: $$Z_f = \left( \frac{1}{R} + j\omega 2C \right)^{-1}$$ Step 5: Transfer function: $$\frac{v_o}{v_i} = -\frac{Z_f}{Z_{in}} = -\frac{\left( \frac{1}{R} + j\omega 2C \right)^{-1}}{2R + \frac{1}{j\omega 2C}}$$ 4. Problem (d): Electromechanical system with resistor $R_a$, inductor $L_a$, armature current $i_a$, back emf $e_b$, motor current $i_p$, motor torque $\theta$, and relation $$e_b = k_b \frac{d\theta}{dt}$$ Step 1: Write electrical equation: $$e_a = R_a i_a + L_a \frac{di_a}{dt} + e_b$$ Step 2: Mechanical equation relates torque and angular velocity: $$e_b = k_b \frac{d\theta}{dt}$$ Step 3: Use Laplace transform to find transfer function from input voltage $e_a$ to angular position $\theta$ or speed $\frac{d\theta}{dt}$. Step 4: The transfer function is: $$\frac{\Theta(s)}{E_a(s)} = \frac{k_t}{(L_a s + R_a)(J s + B) + k_t k_b}$$ where $k_t$ is torque constant, $J$ is moment of inertia, and $B$ is damping coefficient (assumed or given). Final answers: (a) $$\frac{e_o}{e_i} = -\frac{\left(\frac{1}{R_2} + \frac{1}{R_3} + j\omega C_2\right)^{-1} + \frac{1}{j\omega C_1}}{R_1}$$ (b) $$\frac{v_o}{v_i} = -\frac{C}{C_i (1 + 2R j\omega C)}$$ (c) $$\frac{v_o}{v_i} = -\frac{\left( \frac{1}{R} + j\omega 2C \right)^{-1}}{2R + \frac{1}{j\omega 2C}}$$ (d) $$\frac{\Theta(s)}{E_a(s)} = \frac{k_t}{(L_a s + R_a)(J s + B) + k_t k_b}$$