Principles & Applications Electricity Principles & Applications Sixth Edition Richard J. Fowler Chapter 9 Power in AC Circuits ©2003 Glencoe/McGraw-Hill

INTRODUCTION Phase Shift True Power Apparent Power Angle Theta Power Factor

Ac-Power Facts Resistance doesn’t cause phase shift. Reactance causes 90° of phase shift. Theta is the angle by which the current leads or lags the voltage. Trigonometric functions can be used to find the resistive and reactive parts of a phasor. Apparent power calculations ignore phase shift. It is advantageous for a system to operate with a power factor close to one.

Power in a Resistive Ac Circuit Note that both power pulses are positive. There are only two pulses per cycle. Voltage and current are in-phase. 5V V 2 A I P = IV in an ac circuit containing only resistance. P = 2 A x 0.707 x 5V x 0.707 = 5 W

Resistive-Power Quiz Power equals current times voltage in an ac circuit when the only load is ____. resistance Use ____ values of current and voltage when calculating power. rms Without phase shift, there are ____ power pulses for each cycle. two There are no ____ power pulses when current and voltage are in phase. negative An ac circuit with a 30-VP-P source and a 10-W resistor load uses ____ watts of power. 11.25

Reactance Causes Phase Shift V I 50° A combination circuit with capacitance causes I to lead V.

Reactance Causes Phase Shift 60° V I A combination circuit with inductance causes V to lead I.

= Working with Right Triangles opposite adjacent tan q = opposite hypotenuse opposite hypotenuse opposite side sin q = q adjacent side adjacent hypotenuse cos q = Iresistive Ireactive Isource Vsource q Iresistive Ireactive Isource q = The hypotenuse can be either the total voltage or the total current. Of course, the sides must be the same quantity as the hypotenuse.

Combination-Loads Quiz Capacitance causes current to ____ voltage. lead Inductance causes current to ____ voltage. lag The ____ of a triangle of current phasors represents the total current. hypotenuse The resistive current is ____ amperes when theta is 40° and the total current is 12 amperes. 9.2 The total voltage is ____ volts when theta is 25° and the reactive voltage is 40 volts. 94.6

Power in a Phase-Shifted Circuit cos 30° = 0.866 30° 3V 2 A I V The power pulse is negative when I and V have opposite signs. P = I V cos q = 2 A x 0.707 x 3V x 0.707 x 0.866 = 2.6 W

Power in a Pure Reactance Circuit The negative power pulses cancel the positive pulses. There is no true power. I V 90° This circuit has capacitance because I leads V.

Two Types of Power in Ac Circuits V 25 V Combination of resistance and reactance or calculated using the formula Power (true power) is measured with a wattmeter P = IV cos q. Measured P = 80 W. Apparent power is determined by measuring current and voltage and then using the formula Papp = IV. Papp = 4 A x 25 V = 100 VA When the load is all resistance, power (P) = apparent power (Papp).

Determining Power Factor (PF) & Angle Theta (q) V 30 V Combination of resistance and reactance The measured power, current, and voltage can be used to determine power factor and angle theta. Papp = 3 A x 30 V = 90 VA PF = cos q = P / Papp = 75.5 W / 90 VA = 0.839 q = arccos 0.839 = 33° (Arccos means the angle that has this cos.)

Calculating Currents Determine the resistive and reactive currents when the source current is 43 A and the power factor is 0.92. cos q = Iresistive / IT Therefore, Iresistive = IT x cos q Iresistive = 43 A x 0.92 = 39.56 A arccos 0.92 = 23.1° ( Arccos means the angle that has this cos.) And, tan 23.1° = 0.426 tan q = Ireactive / Iresistive Therefore, Ireactive = Iresistive x tan q Ireactive = 39.56 A x 0.426 = 16.85 A

Combination-Load-Power Quiz There are ____ negative power pulses per cycle when current and voltage are out of phase. two With combination loads, P = IV cos q will yield the ____ power. true With combination loads, P = IV will yield the ____ power. apparent The unit for apparent power is the _____. VA The power is ____ watts when I = 16 A, V = 240 V, and q = 20°. 3608.4 The power factor is ____ when P = 860 W, I = 10 A, and V = 120 V. 0.717

REVIEW Phase Shift True Power Apparent Power Angle Theta Power Factor