QB_PH_SEC8_011: Operational Amplifiers & Analog Circuits

An ideal operational amplifier differentiator has input resistor replaced by capacitor $C = 0.1\text{ }\mu\text{F}$ and feedback resistor $R_f = 100\text{ k}\Omega$. If the input voltage increases linearly with time as $V_{\text{in}}(t) = 200 t\text{ Volts}$ (where $t$ is in seconds), what is the magnitude of the steady output voltage $|V_{\text{out}}|$ in Volts?

Correct Answer

### Step-by-Step Derivation & Verification: 1. **Ideal Differentiator Transfer Equation**: $i_{\text{in}} = C \frac{d V_{\text{in}}}{dt}$ Due to virtual ground ($V_- = 0$) and zero input bias current: $i_{\text{in}} + i_f = 0 \implies C \frac{d V_{\text{in}}}{dt} + \frac{V_{\text{out}}}{R_f} = 0$ $V_{\text{out}}(t) = -R_f C \frac{d V_{\text{in}}(t)}{dt}$ 2. **Derivative Computation**: $V_{\text{in}}(t) = 200 t \implies \frac{d V_{\text{in}}}{dt} = 200\text{ V/s}$ 3. **Numerical Substitution**: - $R_f = 100\text{ k}\Omega = 10^5\,\Omega$ - $C = 0.1\text{ }\mu\text{F} = 10^{-7}\text{ F}$ $R_f C = 10^5 \times 10^{-7} = 10^{-2}\text{ s} = 0.01\text{ s}$ $V_{\text{out}} = -(0.01\text{ s})(200\text{ V/s}) = -2.0\text{ V}$ 4. **Magnitude Request**: $|V_{\text{out}}| = 2.0\text{ V}$.