Question 12.51: Figure 12.87 shows a model used by one manufacturer for ai...
Figure 12.87 shows a model used by one manufacturer for air-core rf coils. The resistance r_L accounts for skin effect and is given by r_L=k \sqrt{f}, where k is an experimentally determined parameter.
In Fig. 12.87, let R_C=6 \Omega, R_S=1 m \Omega, C=68 fF =68 \times 10^{-15} F , L=1 nH , and k=3.4 \times 10^{-6} \Omega Hz ^{-1 / 2}. Construct a graph of the magnitude in dB of the equivalent impedance Z_{ab} at the terminals a-b versus frequency f , for 100 MHz \leq f \leq 100 GHz. Use a logarithmic scale for frequency and normalize the impedance to the impedance at the self-resonant frequency. Show also a graph of the magnitude in dB of the impedance of an ideal inductor having inductance L , also normalized to the impedance at the self-resonant frequency.
From (12.88), the self-resonant frequency of the circuit is given by
\omega_r=\frac{1}{\sqrt{L C}} \sqrt{\frac{R_L^2 C-L}{R_C^2 C-L}} (12.88)
\begin{aligned} f_r &=\frac{1}{2 \pi \sqrt{L C}} \sqrt{\frac{r_L^2\left(f_r\right) C-L}{R_C^2 C-L}} \\\\ &=\frac{1}{2 \pi \sqrt{L C}} \sqrt{\frac{k^2 f_r C-L}{R_C^2 C-L}} \end{aligned}
which yields
\begin{aligned} f_r &=\frac{-k^2 C+\sqrt{k^4 C^2-16 \pi^2 L^2 C\left(R_C^2 C-L\right)}}{8 \pi^2 L C\left(L-R_C^2 C\right)} \\\\ & \cong 19.3 GHz . \end{aligned}
From Fig. 12.87, the equivalent impedance at the terminals a-b is given by
\begin{aligned} Z &=R_S+\frac{1}{\left[R_C+(j 2 \pi f C)^{-1}\right]^{-1}+[k \sqrt{f}+j 2 \pi f L]^{-1}} \\ &=R_S+\frac{1}{\left[R_C+(j \omega C)^{-1}\right]^{-1}+[k \sqrt{\omega /(2 \pi)}+j \omega L]^{-1}} . \end{aligned} (12.121)
The equivalent impedance at the selfresonant frequency f_r is
Z_r=Z\left(j \omega_r\right)=2.27 k \Omega
where \omega_r=2 \pi f_r . Figure 12.88 shows graphs of the magnitude of the normalized impedance
of the model in Fig. 12.87 (solid line) and of the normalized impedance of an ideal inductor having inductance L (dashed line), where Z is given by (12.121). The normalized impedance (in dB ) of the model and that of the associated ideal inductor are given by
\begin{aligned} |Z|( dB ) &=20 \log \left|\frac{Z(j \omega)}{Z\left(j \omega_r\right)}\right| \\\\ \left|Z_L\right|( dB ) &=20 \log \left|\frac{j \omega L}{Z\left(j \omega_r\right)}\right| \end{aligned}
respectively. Figure 12.88 indicates that the physical inductor approximates an ideal inductor for 10 MHz < f <10 GHz . Below 10 MHz , the series resistance R_S becomes significant and above 10 GHz , the shunt capacitance is significant.
