QB_PH_SEC8_030: Operational Amplifiers & Analog Circuits

An operational amplifier has a common-mode rejection ratio of $\text{CMRR} = 80\text{ dB}$. If the differential voltage gain is $A_d = 20000$, what is the magnitude of the common-mode gain $A_{\text{cm}}$?

Correct Answer

### Step-by-Step Derivation & Verification: 1. **Decibel Definition of CMRR**: $\text{CMRR}_{\text{dB}} = 20 \log_{10}\left(\frac{A_d}{A_{\text{cm}}}\right)$ 2. **Solving for Linear Ratio**: $\frac{\text{CMRR}_{\text{dB}}}{20} = \frac{80}{20} = 4$ $\frac{A_d}{A_{\text{cm}}} = 10^4 = 10000$ 3. **Evaluation of Common-Mode Gain**: $A_{\text{cm}} = \frac{A_d}{10000} = \frac{20000}{10000} = 2.0$ 4. **Official Answer**: 2.