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## Q. 7.4

$\text { Calculate the equivalent noise voltage of two parallel resistors } R_1 \text { and } R_2 \text { [Fig. 7.18(a)]. }$ ## 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.