Questions

Design an op-amp integrator with \(\text{R} = 100\,\text{k}\Omega\), \(\text{C} = 0.1 \mu F\). To avoid drift at low frequencies, a resistor \(R_{\text{f}}\) is placed in parallel with the capacitor. Determine the low-frequency gain limit, If \(R_{\text{f}} = 10\,\text{M}\Omega\).
An op-amp differentiator has \(\text{R} = 10\,\text{k}\Omega\), \(\text{C} = 0.01 \mu \text{F}\). To limit high-frequency noise, a resistor \(R_{\text{f}} = 100\,\text{k}\Omega\) is added in series with the capacitor. Derive the modified transfer function.
A log amplifier uses a diode with \(V_T = 26 \,\text{mV}\). Calculate the output voltage range, If input varies from \(0.1 \,\text{V}\) to \(100 \,\text{V}\).
An antilog amplifier has input \(–60 \,\text{mV}\). Calculate percentage error in output, If temperature increases so that \(V_T\) rises from \(26 \,\text{mV}\) to \(30 \,\text{mV}\), .
Design a summing amplifier that computes \(V_{\text{out}} = -(2\,V_1 + 0.5\,V_2)\). Choose resistor values with \(R_{\text{f}} = 20\,\text{k}\Omega\).
Design a comparator with hysteresis using op-amp. Reference = 2 V, feedback resistor ratio chosen so hysteresis = \(\pm 0.5 \,\text{V}\). Find switching thresholds.
A zero-crossing detector has a small offset voltage of \(10 \,\text{mV}\) at the op-amp input. Input sine wave amplitude \(= 1 \,\text{V}\). Determine effect on output.
Explain how log and antilog amplifiers can be combined to compute RMS value of a signal.

Answers

\(V_{\text{out}}(t) = -\frac{1}{\text{RC}} \int_0^t V_{in}(\tau) \, d\tau + V_{out}(0)\)
\(V_{\text{out}}(t) = -100 \int_0^t V_{in}(\tau) \, d\tau \)
\(V_{\text{out}}(t) = -100 \int_0^t 1 \, \text{d}\tau = -100 \cdot t\)
\(V_{\text{out}}(t) = -\text{RC} \, \frac{\text{d}V_{\text{in}}(t)}{\text{dt}} = -0.2 \cos(1000\,t) \, V\)
\(V_{\text{out}} = -V_T \ln\left(\frac{V_{\text{in}}}{I_S R}\right)\); Assume \(I_S R = 1\);
For \(V_{\text{in}} = 0.1\,\text{V}\): \(V_{\text{out}} \approx +60\,\text{mV}\);
For \(V_{\text{in}} = 100\,\text{V}\): \(V_{\text{out}} \approx -120\,\text{mV}\)
\(V_{\text{out}} = e^{-V_{\text{in}}/V_T}\);
At \(V_T = 26\,\text{mV}\): \(\approx 10.1\);
At \(V_T = 30\,\text{mV}\): \(\approx 7.39\); Error = \(\frac{10.1 - 7.39}{10.1} \times 100 \approx 27\%\)
\(V_{\text{out}} = -R_{\text{f}}\left(\frac{V_1}{R_1} + \frac{V_2}{R_2}\right)\); \(R_1 = 10\,\text{k}\Omega\), \(R_2 = 40\,\text{k}\Omega\), \(R_{\text{f}} = 20\,\text{k}\Omega\).
Upper threshold \(= 2 \,\text{V} + 0.5 \,\text{V} = 2.5 \,\text{V}\);
Lower threshold \(= 2 \,\text{V} – 0.5 \,\text{V} = 1.5 \,\text{V}\)
\(\Delta \theta = \sin^{-1}\left(\frac{0.01}{1}\right) \approx 0.57^\circ\)
Op-amp circuits can approximate RMS computation by combining \(\text{log} \rightarrow \text{scaling} \rightarrow \text{averaging} \rightarrow \text{antilog}\). Log amplifier converts multiplication into addition: \(\ln(x^2) = 2\ln(x)\)