Complex Numbers in a Circuit. The voltage across a circuit element in an ac circuit is not necessarily in phase with the current through that circuit element. Therefore the voltage amplitudes across the circuit elements in a branch in an ac circuit do not add algebraically. A method that is commonly employed to simplify the analysis of an ac circuit driven by a sinusoidal source is to represent the impedance $Z$ as a complex number. The resistance $R$ is taken to be the real part of the impedance, and the reactance $X=X_{L}-X_{C}$ is taken to be the imaginary part. Thus, for a branch containing a resistor, inductor, and capacitor in series, the complex impedance is $Z_{\text { cpx }}=R+i X,$ where $i^{2}=-1 .$ If the voltage amplitude across the branch is $V_{\mathrm{cpx}},$ we define a complex current amplitude by $I_{\mathrm{cpx}}=V_{\mathrm{cpx}} / Z_{\mathrm{cpx}}$ the actual current amplitude is the absolute value of the complex current amplitude; that is, $I=\left(I_{\mathrm{cpx}} * I_{\mathrm{cpx}}\right)^{1 / 2}$ value of the complex current amplitude; that is, $I=\left(I_{\text { cpx }} * I_{\text { cpx }}\right)^{1 / 2}$ The phase angle $\phi$ of the current with respect to the source voltage is given by $\tan \phi=\operatorname{Im}\left(I_{\mathrm{cpx}}\right) / \operatorname{Re}\left(I_{\mathrm{cpx}}\right) .$ The voltage amplitudes $V_{R-\mathrm{cpx}}, V_{L-\mathrm{cpx}},$ and $V_{C \text { cpx across the resistance, inductance, and }}$ capacitance, respectively, are found by multiplying $I_{\mathrm{cpx}}$ by $R, i X_{L}$ and $-i X_{C},$ respectively. From the complex representation for the voltage amplitudes, the voltage across a branch is just the algebraic sum of the voltages across each circuit element: $V_{\text { cpx }}=V_{R \text { -cpx }}+$ $V_{L \mathrm{cpx}}+V_{C-\mathrm{cp} x}$ . The actual value of any current amplitude or voltage amplitude is the absolute value of the corresponding complex quantity. Consider the $L_{-} R-C$ series circuit shown in Fig. $\mathrm{P} 31.75 .$ The values of the circuit elements, the source voltage amplitude, and the source angular frequency are as shown. Use the phasor diagram techniques presented in Section 31.1 to solve for (a) the current amplitude and (b) the phase angle $\phi$ of the current with respect to the source voltage. (Note that this angle is the negative of the phase angle defined in Fig. $31.13 .$ ) Now analyze the same circuit using the complex-number approach. (c) Determine the complex impedance of the circuit, $Z_{\text { cpx. }}$ Take the absolute value to obtain $Z,$ the actual impedance of the circuit. (d) Take the voltage amplitude of the source, $V_{\text { cpx }},$ to be real, and find the complex current amplitude $I_{\text { cpx }} .$ Find the actual current amplitude by taking the absolute value of $I_{\mathrm{cpx}}$ . (e) Find the phase angle $\phi$ of the current with respect to the source voltage by using the real and imaginary parts of $I_{\mathrm{cp}}$ , as explained above.(f) Find the complex representations of the voltages across the resistance, the inductance, and the capacitance. (g) Adding the answers found in part (f), verify that the sum of these complex numbers is real and equal to $200 \mathrm{V},$ the voltage of the source.