https://file.aiquickdraw.com/mathbot/file/1728828981949w7k22oho.jpg

To solve this circuit problem, let's first break down the information and approach required.

### Given Information:
1. **Transistors (W/L)\(_{1,2}\) = (25/0.5)**: The width-to-length ratio for both transistors M1 and M2.
2. **\( \mu_n C_{ox} = 50 \, \mu A/V^2 \)**: Process transconductance parameter.
3. **\( V_{TH} = 0.6 \, V \)**: Threshold voltage of both transistors.
4. **Voltage across \( R_{SS} = 0.5 \, V \)**: The voltage across the source resistance \( R_{SS} \).
5. **\( A_v = 5 \)**: Voltage gain.
6. **\( V_{DD} = 3 \, V \)**: Supply voltage.
7. **\( I_D = 1 \, mA \)**: Drain current through each transistor.

### Approach:
We need to calculate the **input and output common-mode voltages** for this differential pair circuit.

#### Step 1: Calculate \( R_{SS} \)
The total current through the tail current source is \( I_{SS} = 1 \, mA \), and the voltage across \( R_{SS} \) is given as 0.5 V. Using Ohm's law, we can calculate \( R_{SS} \) as:
\[
R_{SS} = \frac{V_{R_{SS}}}{I_{SS}} = \frac{0.5 \, V}{1 \, mA} = 500 \, \Omega
\]

#### Step 2: Calculate \( V_{GS} \)
The gate-source voltage \( V_{GS} \) can be calculated using the equation for the MOSFET in saturation:
\[
I_D = \frac{1}{2} \mu_n C_{ox} \left( \frac{W}{L} \right) (V_{GS} - V_{TH})^2
\]
Substituting the known values:
\[
1 \, mA = \frac{1}{2} \times 50 \, \mu A/V^2 \times \frac{25}{0.5} \times (V_{GS} - 0.6)^2
\]
Simplifying:
\[
1 \, mA = 1250 \, \mu A/V^2 \times (V_{GS} - 0.6)^2
\]
\[
(V_{GS} - 0.6)^2 = \frac{1 \, mA}{1250 \, \mu A/V^2} = 0.0008 \, V^2
\]
\[
V_{GS} - 0.6 = \sqrt{0.0008} \approx 0.0283 \, V
\]
\[
V_{GS} \approx 0.6283 \, V
\]

#### Step 3: Input Common-Mode Voltage \( V_{in,cm} \)
The input common-mode voltage \( V_{in,cm} \) is simply the gate-source voltage \( V_{GS} \):
\[
V_{in,cm} = V_{GS} = 0.6283 \, V
\]

#### Step 4: Output Common-Mode Voltage \( V_{out,cm} \)
The output common-mode voltage is taken as the voltage at the drains of the transistors, which can be calculated as:
\[
V_{out,cm} = V_{DD} - I_D R_D
\]
We are not given \( R_D \) directly, but we can use the voltage gain formula. Since the small-signal voltage gain \( A_v = 5 \), we know:
\[
A_v = \frac{g_m R_D}{2}
\]
Where \( g_m \) is the transconductance of the MOSFET. The transconductance \( g_m \) is given by:
\[
g_m = \sqrt{2 \mu_n C_{ox} \frac{W}{L} I_D}
\]
Substituting values:
\[
g_m = \sqrt{2 \times 50 \, \mu A/V^2 \times \frac{25}{0.5} \times 1 \, mA} = \sqrt{125 \, mA/V} = 11.18 \, mA/V
\]
Now, using the gain formula:
\[
5 = \frac{g_m R_D}{2} \implies R_D = \frac{5 \times 2}{g_m} = \frac{10}{11.18} \approx 0.894 \, k\Omega
\]
Now calculate \( V_{out,cm} \):
\[
V_{out,cm} = V_{DD} - I_D R_D = 3 \, V - 1 \, mA \times 0.894 \, k\Omega \approx 3 \, V - 0.894 \, V = 2.106 \, V
\]

### Final Answers:
1. **Input common-mode voltage \( V_{in,cm} \) = 0.628 V**
2. **Output common-mode voltage \( V_{out,cm} \) = 2.106 V**

Would you like further details or explanations?

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**Related Questions:**
1. What is the role of \( R_{SS} \) in this circuit, and how does it affect common-mode rejection?
2. How would the common-mode voltages change if the supply voltage \( V_{DD} \) were increased?
3. What is the significance of the small-signal gain \( A_v = 5 \) in determining the output?
4. How does the width-to-length ratio \( W/L \) influence the transistor’s behavior in this circuit?
5. How does varying the tail current \( I_{SS} \) affect the common-mode voltages?

**Tip:** In differential pairs, the input common-mode voltage is typically close to the gate-source voltage, which depends on the biasing of the transistors.

In the circuit shown below calculate the input and output common-mode voltages. Assume (W/L)1,2 = (25/0.5), μnCox = 50μA/V², VTH = 0.6 V, voltage across Rss = 0.5V, Av = 5, and VDD = 3V.

Explore how to calculate input and output common-mode voltages in a MOSFET differential amplifier circuit. This problem involves DC biasing, saturation current equations, and voltage gain calculations. Perfect for undergraduate students in electronics or electrical engineering.

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