Question 4.14: Objective: Determine the small-signal voltage gain and outpu...
Objective: Determine the small-signal voltage gain and output resistance of the source-follower amplifier shown in Figure 4.47(a).
Assume the reference bias current is I_{Bias} = 0.20 mA and the bias voltage is V_{DD} = 3.3 V. Assume that all transistors are matched (identical) with parameters V_{T N} = 0.4 V, K_{n} = 0.20 mA/V^{2} , and λ = 0.01 V^{−1}.
We may note that since M_{3} and M_{2} are matched transistors and have the same gate-to-source voltages, the drain current in M_{1} is I_{D1} = I_{Bias} = 0.2 mA.
(voltage gain): From Figure 4.47(c), we find the small-signal output voltage to be
V_{o} = g_{m1} V_{gs} (r_{o1} || r_{o2})A KVL equation around the outside loop produces
V_{i} = V_{gs} + V_{o} = V_{gs} + g_{m1} V_{gs} (r_{o1} || r_{o2})or
V_{gs} = \frac{V_{i}}{1 + g_{m1} (r_{o1} || r_{o2})}Substituting this equation for V_{gs} into the output voltage expression, we obtain the small-signal voltage gain as
A_{v} = \frac{V_{o}}{V_{i}} = \frac{g_{m1} (r_{o1} || r_{o2})}{1 + g_{m1} (r_{o1} || r_{o2})}The small-signal equivalent circuit parameters are determined to be
g_{m1} = 2 \sqrt{K_{n} I_{D1}} = 2 \sqrt{(0.20)(0.20)} = 0.40 mA/Vand
r_{o1} = r_{o2} = \frac{1}{\lambda I_{D}} = \frac{1}{(0.01)(0.20)} = 500 k \OmegaThe small-signal voltage gain is then
A_{v} = \frac{(0.40)(500 || 500)}{1 + (0.40)(500 || 500)}or
A_{v} = 0.99(output resistance): The output resistance can be determined from the
equivalent circuit shown in Figure 4.47(d). The independent source V_{i} is set equal to zero and a test voltage V_{x} is applied to the output.
Summing currents at the output node, we find
From the circuit, we see that
V_{gs} = – V_{x} We then have
I_{x} = V_{x} \left(g_{m1} + \frac{1}{r_{o2}} + \frac{1}{r_{o1}} \right)The output resistance is then given as
R_{o} = \frac{V_{x}}{I_{x}} = \frac{1}{g_{m1}} || r_{o2} || r_{o1}We find
R_{o} = \frac{1}{0.40} ||500 ||500or
R_{o} = 2.48 k \OmegaComment: A voltage gain of A_{v} = 0.99 is typical of a source-follower circuit. An output resistance of R_{o} = 2.48 k \Omega is relatively small for a MOSFET circuit and is also a characteristic of a source-follower circuit