Exercise 11.2 What the wire?

In physical realizations of RF circuits, resistors with some kind of leads are used. The resistor value is $R$, the length of the lead wires is $l$ mm on each side of the resistor. Assume that the inductance per length is $L=2\mathit{nH}∕\mathit{mm}$.

a)

Derive an equation for the impedance of the resistor with wires, $Z_{\mathit{res}}(\mathit{j\omega })$, at (radian) frequency $\omega$.

b)

Assuming that $R=100\mathrm{\Omega }$ and lead wires of 2cm at each side of the resistor. Derive the maximum frequency to keep the impedance $|Z_{\mathit{res}}(\mathit{j\omega })|<\sqrt{2}R$.

c)

Derive the impedance of a resistor with a nominal $R=100\mathrm{\Omega }$ and lead wires of 2cm at each side at the upper limit of the audible audio range, at 20 kHz.

d)

Assuming that $R=100\mathrm{\Omega }$ and lead wires with length $l$ at each side of the resistor. Derive the lead wire length to get the impedance $|Z_{\mathit{res}}(\mathit{j\omega })|<\sqrt{2}R$ at 20 kHz.