Lesson 17 Intro to AC & Sinusoidal Waveforms

Learning Objectives Compare AC and DC voltage and current sources as

defined by voltage polarity, current direction and magnitude over time.

Define the basic sinusoidal wave equations and waveforms, and determine amplitude, peak to peak values, phase, period, frequency, and angular velocity.

Determine the instantaneous value of a sinusoidal waveform.

Graph sinusoidal wave equations as a function of time and angular velocity using degrees and radians.

Define effective / root mean squared values. Define phase shift and determine phase differences

between same frequency waveforms.

DC sources have fixed polarities and magnitudes. DC voltage and current sources are represented by capital E and I.

Direct Current (DC)REVIEW

Alternating Current (AC) A sinusoidal ac waveform starts at zero

Increases to a positive maximumDecreases to zeroChanges polarity Increases to a negative maximumReturns to zero

Variation is called a cycle

Alternating Current (AC) AC sources have a sinusoidal waveform. AC sources are represented by lowercase e(t) or i(t) AC Voltage polarity changes every cycle

Generating AC Voltage Rotating a coil in fixed magnetic field generates

sinusoidal voltage.

AC current changes direction each cycle with the source voltage.

Sinusoidal AC Current

Time Scales Horizontal scale can represent degrees or time.

Period Period of a waveform

Time it takes to complete one cycle Time is measured in seconds The period is the reciprocal of frequency

T = 1/f

Frequency Number of cycles per second of a waveform

FrequencyDenoted by f

Unit of frequency is hertz (Hz) 1 Hz = 1 cycle per second

Amplitude and Peak-to-Peak Value Amplitude of a sine wave

Distance from its average to its peak We use Em for amplitude Peak-to-peak voltage

Measured between minimum and maximum peaks We use Epp or Vpp

Example Problem 1

What is the waveform’s period, frequency, Vm and VPP?

The Basic Sine Wave Equation The equation for a sinusoidal source is given

where Em is peak coil voltage and is the angular position

The instantaneous value of the waveform can be determined by solving the equation for a specific value of

sin( ) Vme E

(37 ) 10sin(37 ) V = 6.01 Ve

Example Problem 2

A sine wave has a value of 50V at =150˚. What is the value of Em?

Radian Measure Conversion for radians to degrees.

2 radians = 360º

Angular Velocity The rate that the generator coil rotates is called its

angular velocity (). Angular position can be expressed in terms of

angular velocity and time. = t (radians)

Rewriting the sinusoidal equation:

e (t) = Em sin t (V)

Relationship between , T and f Conversion from frequency (f) in Hz to angular

velocity () in radians per second

= 2 f(rad/s)

In terms of the period (T)

22 (rad/s)f

T

Sinusoids as functions of time Voltages can be expressed as a function of time

in terms of angular velocity ()

e (t) = Em sin t (V)

Or in terms of the frequency (f)

e (t) = Em sin 2 f t (V) Or in terms of Period (T)

( ) sin 2m

te t E

T (V)

Instantaneous Value The instantaneous value is the value of the

voltage at a particular instant in time.

Example Problem 3

A waveform has a frequency of 100 Hz, and has an instantaneous value of 100V at 1.25 msec.

Determine the sine wave equation. What is the voltage at 2.5 msec?

A phase shift occurs when e(t) does not pass through zero at t = 0 sec

If e(t) is shifted left (leading), then

e = Em sin ( t + )

If e(t) is shifted right (lagging), then

e = Em sin ( t - )

Phase Shifts

Phase shift The angle by which the wave LEADS or LAGS

the zero point can be calculated based upon the Δt

The phase angle is written in DEGREES

10 s360 360 36

100 s

t

T

V and i are in phase.

PHASE RELATIONS

i leads v by 80°. i leads v by 110°.

Example Problem 4

Write the equations for the waveform below. Express the phase angle in degrees.

v = Vm sin ( t + )

Effective (RMS) Values

Effective values tell us about a waveform’s ability to do work.

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

waveform supplies in terms of its ability to produce the same average power

They are “Root Mean Squared” (RMS) values: The terms RMS and effective are synonymous.

0.7072

0.7072

mrms m

mrms m

VV V

II I