7.4 The MOS amplifier of Fig. 7.4(a), when operated with VDD = 2 V, has a maximum small-signal voltage gain
magnitude of 14 V/V. Find VOV and VDS for bias point Q at which a voltage gain of −12 V/V is obtained.


To solve this problem, let's break down the information and apply the relevant formulas.

Given:
- \( V_{DD} = 2 \, \text{V} \)
- Maximum small-signal voltage gain: \( |A_v|_{\text{max}} = 14 \, \text{V/V} \)
- Required small-signal voltage gain: \( A_v = -12 \, \text{V/V} \)

To find:
- \( V_{OV} \) (overdrive voltage)
- \( V_{DS} \) (drain-source voltage)

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### Solution

The small-signal voltage gain \( A_v \) for a common-source MOS amplifier with a drain resistor \( R_D \) is given by:
\[
A_v = -g_m R_D
\]
where \( g_m \) is the transconductance.

Since the maximum voltage gain is given as \( |A_v|_{\text{max}} = 14 \, \text{V/V} \), we can write:
\[
g_m R_D = 14
\]

Now, we need to determine the value of \( g_m \) when \( A_v = -12 \, \text{V/V} \):
\[
g_m R_D = 12
\]

From the above equations, we have two conditions:
1. \( g_{m, \text{max}} R_D = 14 \)
2. \( g_{m, Q} R_D = 12 \)

Thus, the ratio of transconductances at the operating points is:
\[
\frac{g_{m, Q}}{g_{m, \text{max}}} = \frac{12}{14} = \frac{6}{7}
\]

### Step 2: Express \( g_m \) in Terms of \( V_{OV} \)

The transconductance \( g_m \) of a MOSFET in saturation is given by:
\[
g_m = \frac{2I_D}{V_{OV}}
\]

Since \( g_m \propto \frac{1}{V_{OV}} \), we have:
\[
\frac{V_{OV, \text{max}}}{V_{OV, Q}} = \frac{g_{m, \text{max}}}{g_{m, Q}} = \frac{7}{6}
\]

So,
\[
V_{OV, Q} = \frac{6}{7} V_{OV, \text{max}}
\]

To find \( V_{OV, Q} \) and \( V_{DS} \) specifically, additional details such as the value of \( V_{OV, \text{max}} \) or specific information about the drain current \( I_D \) would be required. If you have further details, please provide them, and I’ll continue the calculations.

Would you like to go deeper into any step?

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1. What is the significance of the overdrive voltage \( V_{OV} \) in MOSFET operation?
2. How does the transconductance \( g_m \) influence the gain of the MOS amplifier?
3. Why does \( V_{OV} \) need to be adjusted for different gains?
4. How can \( V_{DS} \) affect the MOSFET’s operation point?
5. What happens to gain if \( V_{OV} \) is minimized?

**Tip:** Keep in mind that MOSFET small-signal parameters are tied to its operating region. Ensuring the MOSFET remains in saturation is key to maintaining a stable gain

The MOS amplifier of Fig. 7.4(a), when operated with VDD = 2 V, has a maximum small-signal voltage gain magnitude of 14 V/V. Find VOV and VDS for bias point Q at which a voltage gain of −12 V/V is obtained.

Find the overdrive voltage (VOV) and drain-source voltage (VDS) for a MOS amplifier with a required voltage gain of -12 V/V, given a VDD of 2V and maximum gain magnitude of 14 V/V. This solution involves MOSFET small-signal analysis and transconductance calculations, relevant to electronics students.

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