Review complex numbers, especially Euler's formula $e^{i\theta }=\cos \theta +i\sin \theta$; the real and imaginary parts of a complex number, as in z = x+iy, with x = Re z and y = Im z; the modulus (magnitude) and phase of a complex number, as in $z=\left\vert z\right\vert e^{i\theta }$, with

$$\left\vert z\right\vert =\sqrt{x^{2}+y^{2}}\quad \quad \quad \quad \quad \tan \theta =y/x;$$

the notion of complex conjugate $z^{*}=x-iy=\left\vert z\right\vert e^{-i\theta }$;the product of complex numbers and in particular $zz^{*}=x^{2}+y^{2}=\left\vert z\right\vert ^{2};$ and the inverse of a complex number, as in

$$\frac{1}{z}=\frac{1}{x+iy}=\frac{x-iy}{x^{2}+y^{2}}=\frac{z^... ...right\vert ^{2}}=\frac{e^{-i\theta }}{\left\vert z\right\vert }$$

These formulas are very convenient to find current- voltage relations in a circuit (or part of it), with $V=V_{0}e^{i\omega t},$ $I=I_{0}e^{i\omega t},$ and impedance

$$Z=R+i\omega L+\frac{1}{i\omega C}$$

(Note: in discussing the harmonic oscillator, we used $e^{-i\omega t}$ for consistency with quantum theory, because in quantum theory the time evolution of an eigenstate is given by $e^{-i\omega t}$, with $\omega =E/\hbar$. However, in circuit theory it is customary to use $e^{i\omega t}$, and here we follow this convention, as in Feynman, Chapter 22, volume II. Tipler uses phasors rotating counterclockwise, which are equivalent to using $e^{i\omega t}$.)

All the formulas developed for d.c. circuits can be directly extended to a.c. circuits. Simply, V = ZI and $I=V/Z=VZ^{*}/\left\vert Z\right\vert ^{2},$ with

$$\left\vert Z\right\vert ^{2}=R^{2}+\left( \omega L-\frac{1}{... ...{2}}{\omega ^{2}}\left( \omega ^{2}-\omega _{0}^{2}\right) ^{2}$$

where $\omega _{0}^{2}=1/LC$ is the resonant frequency of the circuit. Also, in the end we are usually interested in time-averaged values, which can be computed as shown below for the dissipated power. Use this to solve problems 2 and 3 in Assignment 8.

The time average of the dissipated power is

$$P_{av}=\frac{1}{2}\,R\left\vert I\right\vert ^{2}=\frac{1}{2... ...ac{R\left\vert V\right\vert ^{2}}{\left\vert Z\right\vert ^{2}}$$

Defining V_rms by $\left( V_{rms}\right) ^{2}=\frac{1}{2}\left\vert V\right\vert ^{2}$ and I_rms by $\left( I_{rms}\right) ^{2}=\frac{1}{2}\left\vert I\right\vert ^{2}$, we can also write

$$P_{av}=R\left( I_{rms}\right) ^{2}=\,\frac{R\left( V_{rms}\right) ^{2}}{\left\vert Z\right\vert ^{2}}$$

(Note: Tipler's Z is the modulus of the complex impedance, given by $\left\vert Z\right\vert$ in our notation here.)

To obtain the formula for P_av, recall that the real potential and current are obtained by taking the real part of V and I according to the general formula Re z = $\frac{1}{2}\left( z+z^{*}\right)$. The instantaneous power is then 1/4 of

(I+I^*)(V+V^*) = (I+I^*)(ZI+Z^*I^*) = ZI² + Z^*I^{* 2} + (Z+Z^*)II^*.

The average of $I^{2}=I_{0}^{2}e^{2i\omega t}$is 0, and so is the average of I^*2, while Z+Z^*=2R.