Chapter 5, Problem 23P

An electric circuit consists of a resistor R, an inductor L, and a capacitor C. connected as shown in Fig. P5.23. The impedance of the resistor is $Z_R=R\Omega$, the impedance of the inductor is $Z_L=j\omega L$ and the impedance of the capacitor is $Z_C=\frac{1}{j\omega C}\Omega$ where $j=\sqrt{-1}$.

Suppose $R=9\Omega ,L=3mH,C=250\mu F$ and $\omega =1000rad/s.$

Figure P5.23 Series-parallel combination of R, L, and C for problem P5-23.

1. Write $Z_R$, $Z_L$, and $Z_C$
2. as complex numbers in both rectangular and polar form.
3. Calculate the total impedance of the inductor-capacitor parallel combination, which is given as $Z_{LC}=\frac{Z_L{Z}_C}{Z_L+Z_C}$. Express the result in both rectangular and polar forms.
4. Now suppose that the total impedance of the circuit is $Z=Z_R+Z_L{}_C$. Determine the total impedance in both rectangular and polar forms.
5. Determine the complex conjugate of $Z$ and compute the product $Z_{}{Z}_{}{}^{*}$.