## Chapter 4

## Q. 4.11

Objective: Design an NMOS amplifier with an enhancement load to meet a set of specifications.

Specifications: An NMOS amplifier with the configuration shown in Figure 4.39(a) is to be designed to provide a small-signal voltage gain of |A_{v}| = 10. The Q-point is to be in the center of the saturation region. The circuit is to be biased at V_{D D} = 5 V.

Choices: NMOS transistors with parameters V_{T N} = 1 V, k´_{n} = 60 μA/V^{2} , and λ = 0 are available. The minimum width-to-length ratio is (W/L)_{min} = 1. Tolerances of ±5 percent in the k´_{n} and V_{TN} parameters must be considered.

## Step-by-Step

## Verified Solution

(ac design): From Equation (4.50), we have

|A_{v}| = 10 = \sqrt{\frac{(W/L)_{D}}{(W/L)_{L}}}

which can be written as

\left( \frac{W}{L} \right)_{D} = 100 \left( \frac{W}{L} \right)_{L}

If we set (W/L)_{L} = 1, then (W/L)_{D} = 100 .

(dc design): Setting the currents in the two transistors equal to each other (both transistors biased in saturation region), we have

i_{D D} = K_{n D} (v_{G S D} − V_{T N D})^{2} = i_{DL} = K_{nL} (v_{G SL} − V_{T N L})^{2}

From Figure 4.39(a), we see that v_{G SL} = V_{D D} − v_{O} . Substituting, we have

K_{n D} (v_{G S D} − V_{T N D})^{2} = K_{nL} (V_{D D} − v_{O} − V_{T N L} )^{2}

Solving for v_{O}, we have

v_{O} = (V_{D D} − V_{T N L} ) − \sqrt{\frac{K_{n D}}{K_{nL}} } (v_{G S D} − V_{T N D} )

At the transition point,

v_{Ot} = v_{DS D} (sat) = v_{G S Dt} − V_{T N D}

where v_{GSDt} is the gate-to-source voltage of the driver at the transition point. Then

v_{GSDt} − V_{T N D} = (V_{D D} − V_{T N L} ) − \sqrt{\frac{K_{n D}}{K_{nL}}} (v_{G S Dt} − V_{T N D} )

Solving for v_{GSDt} , we obtain

v_{GSDt} = \frac{(V_{D D} − V_{T N L} ) + V_{T N D} \left( 1 + \sqrt{\frac{K_{n D}}{K_{nL}}} \right)}{1 + \sqrt{\frac{K_{n D}}{K_{nL}}}}

Noting that

\sqrt{\frac{K_{n D}}{K_{nL}}} = \sqrt{\frac{(W/L)_{D}}{(W/L)_{L}}} = 10

we find

v_{G S Dt} = \frac{(5 − 1) + (1)(1 + 10)}{1 + 10} = 1.36 V

and

v_{Ot} = v_{DS Dt} = v_{G S Dt} − V_{T N D} = 1.36 − 1 = 0.36 V

Considering the transfer characteristics shown in Figure 4.41, we see that the center of the saturation region is halfway between the cutoff point (v_{G S D} = V_{T N D} = 1 V) and the transition point (v_{G Sdt} = 1.36 V), or

V_{G S Q} = \frac{1.36 − 1.0}{2} + 1.0 = 1.18 V

Also

V_{DS D Q} = \frac{4 − 0.36}{2} + 0.36 = 2.18 V

Trade-offs: Considering the tolerances in the k´_{n} parameter, we find the range in the small-signal voltage gain to be

|A_{v}|_{max} = \sqrt{\frac{k´_{n D}}{k´_{nL}} \cdot \frac{(W/L)_{D}}{(W/L)_{L}}} = \sqrt{\frac{1.05}{0.95} \cdot (100)} = 10.5

and

|A_{v}|_{min} = \sqrt{\frac{k´_{n D}}{k´_{nL}} \cdot \frac{(W/L)_{D}}{(W/L)_{L}}} = \sqrt{\frac{0.95}{1.05} \cdot (100)} = 9.51

The tolerances in the k´_{n} and V_{T N} parameters will also affect the Q-point. This analysis is left as an end-of-chapter problem.

Comment: These results show that a very large difference is required in the sizes of the two transistors to produce a gain of 10. In fact, a gain of 10 is about the largest practical gain that can be produced by an enhancement load device. A larger small-signal gain can be obtained by using a depletion-mode MOSFET as a load device, as shown in the next section.

Design Pointer: The body effect of the load transistor was neglected in this analysis. The body effect will actually lower the small-signal voltage gain from that determined in the example