Question 12.49: Design a gyrator-based tuned circuit subject to the followin...
Design a gyrator-based tuned circuit subject to the following specifications:
resonant frequency =f_0=400 Hz ,
quality factor =Q=10,
impedance at resonance =Z_0=100 \Omega .
We assume a suitable op amp is available (which is almost always the case for frequencies at which the gyrator described above is useful). From (12.115) and the specification on the impedance at resonance, we have
Z_0=R_1+j \omega_0 R_1 R_2 C+\frac{1}{j \omega_0 C_1}=R_1, (12.115)
R_1=Z_0=100 \Omega . (12.117)
Based upon (12.111), we require
\omega_0 R_1 C \ll 1 (12.111)
\frac{1}{R_1 C}=100 \omega_0,
which yields
C=\frac{1}{100 \omega_0 R_1}=39.79 nF .
From (12.114)
Q=\frac{\operatorname{Im}\left[Z_G\left(\omega_0\right)\right]}{\operatorname{Re}\left[Z_G\left(\omega_0\right)\right]}=\omega_0 R_2 C (12.114)
R_2=\frac{Q}{\omega_0 C}=100 k \Omega .
Finally, from (12.110)
\omega_0=\frac{1}{\sqrt{L C_1}} (12.110)
C_1=\frac{1}{\omega_0^2 L}=397.89 nF .
The impedance of the gyrator-based circuit is given by
Z_{G} = \frac{R_1 (1+j\omega R_2C)}{1+j \omega R_1C}+\frac{1}{ j \omega C_1}
The impedance of an equivalent passive RLC circuit is given by
Z_{R L C}=Z_0\left[1+j Q\left(\frac{\omega}{\omega_0}-\frac{\omega_0}{\omega}\right)\right] .
We may express the magnitudes of the normalized impedances in dB as
\begin{gathered} \left|Z_G\right|_{ dB }=20 \log \left(\left|\frac{Z_G}{R_1}\right|\right), \\\\ \left|Z_{R L C}\right|_{ dB }=20 \log \left(\left|\frac{Z_{R L C}}{R_1}\right|\right) . \end{gathered}
Figure 12.78 shows graphs of the magnitudes of the impedances versus frequency, on the same axes. The graphs are almost indistinguishable for f < 20 kHz , and indicate that the design is successful. The resonant peak appears to be at 400 Hz , in good agreement with the specification.
