Chapter 7
Q. 7.4
\text { Calculate the equivalent noise voltage of two parallel resistors } R_1 \text { and } R_2 \text { [Fig. 7.18(a)]. }
Step-by-Step
Verified Solution
As shown in Fig. 7.18(b), each resistor exhibits an equivalent noise current with the spectral density 4 k T / R. Since the two noise sources are uncorrelated, we add the powers:
\begin{aligned} \overline{I_{n, t o t}^2} &=\overline{I_{n 1}^2}+\overline{I_{m 2}^2} (7.24) \\ &=4 k T\left(\frac{1}{R_1}+\frac{1}{R_2}\right) \end{aligned} (7.25)Thus, the equivalent noise voltage is given by
\begin{aligned} \overline{V_{n, t o t}^2} &=\overline{I_{n, t o t}^2}\left(R_1 \| R_2\right)^2 (7.26)\\ &=4 k T\left(R_1 \| R_2\right) \end{aligned} (7.27)as intuitively expected. Note that our notation assumes a 1-Hz bandwidth.