Lesson 14: Introduction to AC and Sinusoids

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.

Direct Current (DC) REVIEW DC sources have fixed polarities and magnitudes. DC voltage and current sources are represented by capital E and I.

Alternating Current (AC) A sinusoidal AC waveform starts at zero then: Increases to a positive maximum… Decreases to zero… Changes polarity… Increases to a negative maximum… Returns to zero. Variation is called a cycle. AC sources have a sinusoidal waveform. AC sources are represented by lowercase e(t) or i(t).

Generating AC Voltage Rotating a coil in fixed magnetic field generates sinusoidal voltage.

Sinusoidal AC Current AC current changes direction each cycle with the source voltage.

Time Scales Horizontal scale can represent degrees or time.

Frequency Frequency (f) is the number of cycles per second of a waveform. Unit of frequency is hertz (Hz). 1 Hz = 1 cycle per second.

Period Period of a waveform: Time is measured in seconds. Time it takes to complete one cycle. Time is measured in seconds. The period (T) is the reciprocal of frequency:

Amplitude and Peak-to-Peak Value 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 minimum and maximum peaks We use Epp or Vpp Amplitude Peak-to-Peak

Example Problem 1 What is the waveform’s period, frequency, Vm and VPP? Amplitude Peak-to-Peak T = 0.4s

The Basic Sine Wave Equation The equation for a sinusoidal source is given: where Em is peak coil voltage and  is the angular position. 

Instantaneous Value The instantaneous value is the value of the voltage at a particular instant in time.  The instantaneous value of the waveform can be determined by solving the equation for a specific value of . For example, if  =37⁰ and amplitude were 10V, then the instantaneous value at that point would be:

Example Problem 2 A sine wave has a value of 50V at  = 150˚. What is the value of Em (the amplitude)? 

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 (NOT WTF!!) Conversion from frequency (f) in Hz to angular velocity () in radians per second  = 2 f (rad/s) In terms of the period (T)

Sinusoids as Functions of Time Voltages can be expressed as a function of time in terms of angular velocity (): Or in terms of the frequency (f): Or in terms of Period (T):

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? Now, calculate the voltage at 2.5 msec:

Phase Shifts 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 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.

Phase Relationships i leads v by 80° i leads v by 110° V and i are in phase

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.

Example Problem 4 Tie it all together: The 120VDC source shown delivers 3.6 W to the load. Determine the peak values of the sinusoidal voltage and current (Em and IM) such that the AC source delivers the same power to the load.

QUESTIONS?