ECE 1100: Introduction to Electrical and Computer Engineering Notes 23 Power in AC Circuits and RMS Spring 2008 David R. Jackson Professor, ECE Dept.

AC Power R [  ] + - v ( t ) Goal: Find the average power absorbed by resistor: f = frequency [Hz] V p = peak voltage Note: The phase of the voltage wave is assumed to be zero here for convenience.

AC Power (cont.) T = 1/f [s] cos (  t ) T p = T / 2 = 0.5 / f [s] cos 2 (  t )

AC Power (cont.) Note: We obtain the same result if we integrate over T p or T.

AC Power (cont.) Consider the integral that needs to be evaluated:

AC Power (cont.) “The average value of cos 2 is 1/2.”

AC Power (cont.) Hence I c = T/2 so

R [  ] + - v ( t ) Summary

Effective Voltage V eff Define: Then we have: Note: V eff is used the same way we use V in a DC power calculation.

Effective Voltage V eff R [  ] + - v ( t ) R [  ] V DC AC same formula

Example In the U. S., 60 Hz line voltage has an effective voltage of 120 [ V ]. Describe the voltage waveform mathematically. V eff = 120 [ V ] so

Example 60 Hz line voltage is connected to a 144 [  ] resistor. Determine the average power being absorbed. R = 144 [  ] [V] (eff)

RMS (Root Mean Square) This is a general way to calculate the effective voltage for any periodic waveform (not necessarily sinusoidal). t v(t)v(t) T digital pulse waveform tptp Duty cycle: D = t p / T

RMS (cont.) Hence, Also, By definition,

RMS (cont.) Hence

RMS (cont.) Define V RMS is the root (square root) of the mean (average) of the square of the voltage waveform V eff = V RMS Comparing with the formula for V eff, we see that

RMS (cont.) For sinusoidal (AC) signals, For other periodic signals, there will be a different relationship between V RMS and V p. (See the example at the end of these notes.)

RMS Current R [  ] + - i ( t ) v ( t ) The concept of effective (RMS) current works the same as for voltage. Define:

RMS Current (cont.) RMS current can be easily related to RMS voltage. R [  ] + - i ( t ) v ( t ) where

Example 60 Hz line voltage is connected to a 144 [  ] resistor. Determine the RMS current and the average power absorbed (using the current formula). 120 [V] (RMS) R = 144 [  ] + - I RMS

RMS Voltage and Current Power can also be expressed in terms of both RMS voltage and current. R [  ] + - I RMS V RMS - +

Example 60 Hz line voltage is connected to a 144 [  ] resistor. Determine the average power (using the voltage-current formula). R = 144 [  ] [ V ] (RMS) I RMS

Summary of AC Power R [  ] + - V RMS I RMS - +

Example (non-sinusoidal) Find the RMS voltage of a sawtooth waveform: t v ( t ) T VpVp

Example (cont.) Hence

Example (sawtooth wave) t v ( t ) T VpVp R [  ] + - v ( t ) Given: V p = 10 [V] R = 100 [  ] Find the average power absorbed by the resistor.

Example (cont.) t v ( t ) T V p = 10 [V] 100 [  ] + - v ( t ) (for sawtooth)