Chapter 15

AC Fundamentals

Alternating Current

• Voltages of ac sources alternate in polarity and vary in magnitude over time.

• These voltages produce currents which vary in magnitude and alternate in direction.

• A sinusoidal ac waveform starts at zero, increases to a positive maximum, decreases to zero, changes polarity, increases to a negative maximum, then returns to zero.

Generating AC Voltages

Generating AC Voltages

Voltage and Current Conventions for AC

• Assign a reference polarity(+) for the source.• When the voltage e has a positive value, its

actual polarity is the same as the reference polarity(+).

• When e is negative, its actual polarity is opposite that of the reference polarity.

• When i has a positive value, its actual direction is the same as the reference arrow.

• If i is negative, its actual direction is opposite that of the reference.

Frequency

• The number of cycles per second of a waveform is called its frequency.

• Frequency is denoted f.• The unit of frequency is the hertz.• 1 Hz = 1 cycle per second

Period

• The period of a waveform is the duration of one cycle.

• It is measured in units of time.• It is the inverse of frequency.• T = 1/f

Amplitude and Peak-to-Peak Value

• The amplitude of a sine wave is the distance from its average to its peak.

• We use Em for amplitude.

• Peak-to-peak voltage is measured between the minimum and maximum peaks.

• We use Ep-p or Vp-p.

Peak Value

• The peak value of a voltage or current is its maximum value with respect to zero.

• If a sine wave rides on top of a dc value, the peak is the sum of the dc voltage and the ac waveform amplitude.

The Basic Sine Wave Equation

• The voltage produced by a generator is

e = Emsin .

• Em is the maximum voltage and is the instantaneous angular position of the rotating coil of the generator.

• The voltage at any point on the sine wave may be found by multiplying Em times the sine of angle at that point.

Angular Velocity

• The rate at which the generator coil rotates is called its angular velocity, .

• The units for are revolutions/second, degrees/sec, or radians/sec.

tt

Radian Measure

is usually expressed in radians.• 2 radians = 360°• To convert from degrees to radians, multiply

by /180.• To convert from radians to degrees, multiply

by 180/.

Relationship between ,T, and f

• One cycle of a sine wave may be represented by = 2 rads or t = T sec.

fT

T

t

2

2

2

Voltages and Currents as Functions of Time

• Since = t, the equation e = Emsin becomes e = Emsin t.

• Also v = Vmsin t and i = Imsin t.

• These equations may be used to compute voltages and currents at any instant of time.

Voltages and Currents with Phase Shifts

• If a sine wave does not pass through zero at

t = 0, it has a phase shift.• For a waveform shifted left,

v = Vmsin(t + ).

• For a waveform shifted right,

v = Vmsin(t - ).

Phasors• A phasor is a rotating line whose projection

on a vertical axis can be used to represent sinusoidally varying quantities.

• A sinusoidal waveform can be created by plotting the vertical projection of a phasor that rotates in the counterclockwise direction at a constant angular velocity .

• Phasors apply only to sinusoidal waveforms.

Shifted Sine Waves

• Phasors may be used to represent shifted waveforms.

• The angle is the position of the phasor at t = 0 seconds.

Phase Difference

• Phase difference is the angular displacement between waveforms at the same frequency.

• If the angular displacement is 0°, the waveforms are in phase; otherwise they are out of phase.

• If v1 = 5 sin(100t) and v2 = 3 sin(100t - 30°), v1 leads v2 by 30°.

Average Value• To find an average value of a waveform,

divide the area under the waveform by the length of its base.

• Areas above the axis are positive, areas below the axis are negative.

• Average values are also called dc values because dc meters indicate average values rather than instantaneous values.

Sine Wave Averages

• The average value of a sine wave over a complete cycle is zero.

• The average over a half cycle is not zero.• The full-wave average is 0.637 times the

maximum value.• The half-wave average is 0.318 times the

maximum value.

Effective Values

• An effective value is an equivalent dc value.• It tells how many volts or amps of dc that an

ac waveform is equal to in terms of its ability to produce the same average power.

• In North America, house voltage is 120 V(ac). This means that the voltage is capable of producing the same average power as 120 V(dc).

Effective Values• To determine the effective power, we set

Power(dc) = Power(ac).

Pdc = pac

I2R = i2R where i = Imsin t

• By applying a trigonometric identity, we are able to solve for I in terms of Im.

• Ieff = .707Im

• Veff = .707Vm

• The effective value is also known as the RMS value.