An op-amp has differential gain `A_d`=10 .common mode voltage applied to it is 2 millivolts and corresponding output voltage is 0.2 microvolt.Then the value of CMRR(dB) is ?

Explanation

To solve the problem, we need to calculate the Common Mode Rejection Ratio (CMRR) using the given data. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the CMRR Formula
The Common Mode Rejection Ratio (CMRR) is defined as:
\[ \text{CMRR} = 20 \log_{10} \left( \frac{A_d}{A_{cm}} \right) \]
where \( A_d \) is the differential gain and \( A_{cm} \) is the common mode gain.

### Step 2: Identify Given Values
From the question, we have:
- Differential gain \( A_d = 10 \)
- Common mode voltage (input) \( V_{cm} = 2 \, \text{mV} = 2 \times 10^{-3} \, \text{V} \)
- Corresponding output voltage \( V_{out} = 0.2 \, \mu\text{V} = 0.2 \times 10^{-6} \, \text{V} \)

### Step 3: Calculate the Common Mode Gain \( A_{cm} \)
The common mode gain \( A_{cm} \) can be calculated using the formula:
\[ A_{cm} = \frac{V_{out}}{V_{cm}} \]
Substituting the values:
\[ A_{cm} = \frac{0.2 \times 10^{-6}}{2 \times 10^{-3}} = \frac{0.2 \times 10^{-6}}{2 \times 10^{-3}} = 0.1 \times 10^{-3} = 10^{-4} \]

### Step 4: Substitute Values into the CMRR Formula
Now we can substitute \( A_d \) and \( A_{cm} \) into the CMRR formula:
\[ \text{CMRR} = 20 \log_{10} \left( \frac{10}{10^{-4}} \right) \]
This simplifies to:
\[ \text{CMRR} = 20 \log_{10} (10^5) = 20 \times 5 \log_{10} (10) \]

### Step 5: Simplify the Expression
Since \( \log_{10} (10) = 1 \):
\[ \text{CMRR} = 20 \times 5 \times 1 = 100 \]

### Final Answer
Thus, the value of CMRR is:
\[ \text{CMRR} = 100 \, \text{dB} \]