Question 7.15: Calculate the input-referred thermal noise voltage of the am...
Calculate the input-referred thermal noise voltage of the amplifier shown in Fig. 7.42(a), assuming both transistors are in saturation. Also, determine the total output thermal noise if the circuit drives a load capacitance C_L . What is the output signal-to-noise ratio if a low-frequency sinusoid of peak amplitude V_m is applied to the input?
Representing the thermal noise of M_1 and M_2 by current sources [Fig. 7.42(b)] and noting that they are uncorrelated, we write
\overline{V_{n, \text { out }}^2}=4 k T\left(\gamma g_{m 1}+\gamma g_{m 2}\right)\left(r_{O 1} \| r_{O 2}\right)^2 (7.76)
(In reality, γ may not be the same for NMOS and PMOS devices.) Since the voltage gain is equal to g_{m 1}\left(r_{O 1} \| r_{O 2}\right) , the total noise voltage referred to the gate of M_1 is
\begin{aligned} \overline{V_{n, i n}^2} &=4 k T\left(\gamma g_{m 1}+\gamma g_{m 2}\right) \frac{1}{g_{m 1}^2} &(7.77)\\ &=4 k T \gamma\left(\frac{1}{g_{m 1}}+\frac{g_{m 2}}{g_{m 1}^2}\right) &(7.78) \end{aligned}Equation (7.78) reveals the dependence of \overline{V_{n, i n}^2} upon g_{m 1} and g_{m 2}, confirming that g_{m 2} must be minimized because M_2 serves as a current source rather than a transconductor. ^{11}
The reader may wonder why M_1 and M_2 in Fig. 7.42 exhibit different noise effects. After all, if the noise currents of both transistors flow through r_{O 1} \| r_{O 2}, why should g_{m 1} be maximized and g_{m 2} minimized? This is simply because, as g_{m 1} increases, the output noise voltage rises in proportion to \sqrt{g_{m 1}} whereas the voltage gain of the stage increases in proportion to g_{m 1}. As a result, the input-referred noise voltage decreases. Such a trend does not apply to M_2.
To compute the total output noise, we integrate (7.76) across the band:
\overline{V_{n, \text { out }, \text { tot }}^2}=\int_0^{\infty} 4 k T \gamma\left(g_{m 1}+g_{m 2}\right)\left(r_{O 1} \| r_{O 2}\right)^2 \frac{d f}{1+\left(r_{O 1} \| r_{O 2}\right)^2 C_L^2(2 \pi f)^2} (7.79)
Using the results of Example 7.3, we have
\overline{V_{n, u u t, t o t}^2}=\gamma\left(g_{m 1}+g_{m 2}\right)\left(r_{O 1} \| r_{O 2}\right) \frac{k T}{C_L} (7.80)
A low-frequency input sinusoid of amplitude V_m yields an output amplitude equal to g_{m 1}\left(r_{O 1} \| r_{O 2}\right) V_m . The output SNR is equal to the ratio of the signal power and the noise power:
We note that to maximize the output SNR, C_L must be maximized, i.e., the bandwidth must be minimized. Of course, the bandwidth is also dictated by the input signal spectrum. This example indicates that it becomes exceedingly difficult to design broadband circuits while maintaining low noise.
^{11}A device or a circuit that converts a voltage to a current is called a transconductor or a V/I converter