1. **Problem Statement:** Find the output voltage $v_o$ and output current $i_o$ of the op amp circuit given $v_s = 12 \cos 5000t$ V. 2. **Circuit Elements and Configuration:** - Input voltage source: $v_s = 12 \cos 5000t$ - Resistors: $R_1 = 10\text{k}\Omega$, $R_2 = 20\text{k}\Omega$ - Capacitors: $C_1 = 10\text{nF}$, $C_2 = 20\text{nF}$ 3. **Approach:** We analyze the circuit in the frequency domain using phasors. The angular frequency is $\omega = 5000$ rad/s. 4. **Impedances:** - Capacitive reactance: $Z_C = \frac{1}{j\omega C}$ - Calculate $Z_{C1}$ and $Z_{C2}$: $$Z_{C1} = \frac{1}{j5000 \times 10 \times 10^{-9}} = \frac{1}{j5 \times 10^{-4}} = -j2000\ \Omega$$ $$Z_{C2} = \frac{1}{j5000 \times 20 \times 10^{-9}} = \frac{1}{j10^{-3}} = -j1000\ \Omega$$ 5. **Node Analysis:** Let the node between $R_1$ and $C_1$ be $V_1$, and the node between $R_2$ and $C_2$ be $V_2$ (also the non-inverting input of the op amp). 6. **Op Amp Assumptions:** Ideal op amp with infinite input impedance and zero input current, so current into the input terminals is zero. 7. **Write node equations:** At node $V_1$: $$\frac{v_s - V_1}{R_1} = \frac{V_1 - V_2}{Z_{C1}}$$ At node $V_2$ (non-inverting input): $$\frac{V_1 - V_2}{Z_{C1}} = \frac{V_2}{R_2} + \frac{V_2 - v_o}{Z_{C2}}$$ 8. **Op amp output relation:** Since the op amp is in a voltage follower configuration at the non-inverting input, output voltage $v_o$ is related to $V_2$ by the feedback network. Here, the output current $i_o$ flows out of the op amp into the capacitor $C_2$ and resistor $R_2$. 9. **Simplify and solve for $V_2$ and $v_o$:** From the ideal op amp, $v_o = V_2$ (since no feedback resistor is shown, assume voltage follower). 10. **Calculate $V_1$ from first equation:** $$\frac{v_s - V_1}{R_1} = \frac{V_1 - V_2}{Z_{C1}} \Rightarrow (v_s - V_1) \frac{1}{R_1} = (V_1 - V_2) \frac{1}{Z_{C1}}$$ Rearranged: $$v_s \frac{1}{R_1} = V_1 \left( \frac{1}{R_1} + \frac{1}{Z_{C1}} \right) - V_2 \frac{1}{Z_{C1}}$$ 11. **Calculate $V_2$ from second equation:** $$\frac{V_1 - V_2}{Z_{C1}} = V_2 \left( \frac{1}{R_2} + \frac{1}{Z_{C2}} \right) - \frac{v_o}{Z_{C2}}$$ Since $v_o = V_2$, the right side simplifies to: $$V_2 \left( \frac{1}{R_2} + \frac{1}{Z_{C2}} - \frac{1}{Z_{C2}} \right) = V_2 \frac{1}{R_2}$$ So: $$\frac{V_1 - V_2}{Z_{C1}} = V_2 \frac{1}{R_2}$$ 12. **From step 11, express $V_1$ in terms of $V_2$:** $$V_1 = V_2 + Z_{C1} \frac{V_2}{R_2} = V_2 \left(1 + \frac{Z_{C1}}{R_2} \right)$$ 13. **Substitute $V_1$ into step 10:** $$v_s \frac{1}{R_1} = V_2 \left(1 + \frac{Z_{C1}}{R_2} \right) \left( \frac{1}{R_1} + \frac{1}{Z_{C1}} \right) - V_2 \frac{1}{Z_{C1}}$$ Simplify: $$v_s \frac{1}{R_1} = V_2 \left[ \left(1 + \frac{Z_{C1}}{R_2} \right) \left( \frac{1}{R_1} + \frac{1}{Z_{C1}} \right) - \frac{1}{Z_{C1}} \right]$$ 14. **Calculate $V_2$ (and thus $v_o$):** Plug in values: $$R_1 = 10,000, R_2 = 20,000, Z_{C1} = -j2000$$ Calculate terms: $$1 + \frac{Z_{C1}}{R_2} = 1 + \frac{-j2000}{20000} = 1 - j0.1$$ $$\frac{1}{R_1} + \frac{1}{Z_{C1}} = \frac{1}{10000} + \frac{1}{-j2000} = 0.0001 + j0.0005$$ Multiply: $$(1 - j0.1)(0.0001 + j0.0005) = 0.0001 + j0.0005 - j0.00001 - 0.00005 = 0.00005 + j0.00049$$ Subtract $\frac{1}{Z_{C1}} = \frac{1}{-j2000} = j0.0005$: $$0.00005 + j0.00049 - j0.0005 = 0.00005 - j0.00001$$ 15. **Calculate $V_2$:** $$V_2 = \frac{v_s / R_1}{0.00005 - j0.00001} = \frac{12 \cos 5000t \times 0.0001}{0.00005 - j0.00001} = \frac{0.0012 \cos 5000t}{0.00005 - j0.00001}$$ Calculate magnitude and phase of denominator: $$|0.00005 - j0.00001| = \sqrt{(0.00005)^2 + (0.00001)^2} = 5.1 \times 10^{-5}$$ Phase: $$\theta = -\tan^{-1} \left( \frac{0.00001}{0.00005} \right) = -11.31^\circ$$ Magnitude of $V_2$ phasor: $$\frac{0.0012}{5.1 \times 10^{-5}} \approx 23.53$$ Phase shift: $+11.31^\circ$ 16. **Convert back to time domain:** $$v_o = V_2 = 23.53 \cos(5000t + 11.31^\circ)$$ 17. **Calculate output current $i_o$:** Output current flows through $C_2$ and $R_2$ in parallel: $$i_o = \frac{v_o}{R_2} + C_2 \frac{d v_o}{dt}$$ Calculate each term: $$\frac{v_o}{R_2} = \frac{23.53 \cos(5000t + 11.31^\circ)}{20000} = 0.0011765 \cos(5000t + 11.31^\circ)$$ $$\frac{d v_o}{dt} = -23.53 \times 5000 \sin(5000t + 11.31^\circ) = -117650 \sin(5000t + 11.31^\circ)$$ $$C_2 \frac{d v_o}{dt} = 20 \times 10^{-9} \times (-117650) \sin(5000t + 11.31^\circ) = -0.002353 \sin(5000t + 11.31^\circ)$$ 18. **Final expression for $i_o$:** $$i_o = 0.0011765 \cos(5000t + 11.31^\circ) - 0.002353 \sin(5000t + 11.31^\circ)$$ **Summary:** $$v_o = 23.53 \cos(5000t + 11.31^\circ)\ \text{V}$$ $$i_o = 0.0011765 \cos(5000t + 11.31^\circ) - 0.002353 \sin(5000t + 11.31^\circ)\ \text{A}$$