COVERAGE TOPICSCOVERAGE TOPICS
1. AC Fundamentals• AC sinusoids• AC response (reactance, impedance)• Phasors and complex numbers
2. AC Analysis• RL, RC, RLC circuit analysis• Mesh and Nodal analysis
3. AC power• Average power, Reactive power, Complex
power• Power triangle
4. Three phase circuit• Y and Delta connection• Line and Phase voltages
AC SINUSOIDSAC SINUSOIDS
SCOPESCOPE Explain the difference between AC and DC Express angular measure in both degrees and radians. Compute the peak, peak-peak, and instantaneous values
of a waveform. Define and solve for the RMS value Define cycle, period, and frequency Given the analytical expression, sketch and explain the
graph of a sinusoid. Determine the relative phase of a sinusoidal waveform.
OBJECTIVESOBJECTIVES (cont) (cont)
Determine the total voltages and currents that have DC and AC components.
Apply Ohm’s Law, KCL, and KVL to analyze a simple AC circuit.
Write the time domain equation for any sinusoidal waveform with a DC component.
SINE WAVESSINE WAVES
Voltage can be produced such that, over time, it follows the shape of a sine wave
The magnitude of the voltage continually changes.
Polarity may or may not change. When it does not change, the current does not change
direction. When polarity does change, the current changes direction. When graphing a sinusoidal voltage, the polarity changes
only when the magnitude alternates between “+” and “-” values.
AC SINEWAVEAC SINEWAVE
1 cycle1 cycle
Voltage is positive
Voltage is positive
Polarity change
t
voltage
+
-
0
voltage
+
-
0
Voltage is positive
Polarity change
t
voltage
+
-
0
voltage
+
-
0
OTHER ACsOTHER ACs
SINE WAVE
TRIANGLE WAVE
SQUARE WAVE
HOW IS A SINE WAVE HOW IS A SINE WAVE GENERATED ?GENERATED ?
Electromagnetic Induction. (Ship AC generators produce sine wave voltages through electromagnetic induction): magnetic field conductor relative motion between the two.
Electronic Signal Generators Function Generators: multi-waveforms.
GENERATING AC GENERATING AC VOLTAGESVOLTAGES
One way to generate ac voltage is to rotate a coil of wire at constant angular velocity in a fixed magnetic field
FARADAY’S LAWFARADAY’S LAW
“ “ Voltage is induced in a circuit Voltage is induced in a circuit whenever the flux linking (i.e. passing whenever the flux linking (i.e. passing through) the circuit is changing and through) the circuit is changing and that the magnitude of the voltage is that the magnitude of the voltage is proportional to the rate of change of proportional to the rate of change of the flux linkages”the flux linkages”
DC vs ACDC vs AC
DC Source: voltage POLARITY of the source and current DIRECTION do not change over time.
V1 ohm
IVoltage
time
AC SOURCEAC SOURCE
AC source: Voltage polarity changes therefore the current changes direction.
V(1.25s)= +2v
1 ohm
I
0 time(sec)
2v
-2v
1 2 3 4
V(3.75s)= -2v
1 ohmI
PERIOD AND FREQUENCYPERIOD AND FREQUENCY
Period: Time to complete one complete cycle Symbol: T
Frequency: Number of cycles in one second Symbol: f Measured in hertz (Hz)
Tf
1t
V
FREQUENCYFREQUENCY
Definition: the number of cycles per second of a waveform
Denoted by the lower case letter f Its unit is the hertz (Hz)
secondper cycle 1 hertz 1
1 cycle
1 second
f=1 Hz
Ex. Ex.
f=2 Hz
Ex. Ex.
1 cycle
1 second
1 cycle
60 cycles1 cycle
1 second
?
Ex. Ex.
PERIODPERIOD
Definition: the duration of one cycle. It is the inverse of frequency. Denoted by the upper case letter T Measured in second, s
Hz)(T
1 f and )s(
f
1T
The period of a waveform can be measured between any two corresponding point.
Often it is measured between zero points because they are easy to establish on an oscilloscope trace
T(between peaks)
T (between zero points)
T (Any two identical points)
t
Ex. Ex.
Figure shows an oscilloscope trace of a square wave. Each horizontal division represents 50 μs. Determine the frequency.
SolutionSolution
Since the wave repeats itself every 200 μs, its period (T) is 200 μs and,
kHz 510200
16
sf
Ex. Ex.
Determine the period and frequency of the waveform of the figure above.
T2 = 10 ms
T1 = 8 ms
SolutionSolution
Time interval T1 does not represent a period as it is not measured between corresponding points. Interval T2, however, is. Thus, T = 10 ms and,
Hz1001010
13
sf
PEAK VALUES (VPEAK VALUES (VPP, I, IPP))
Max Voltage (Current) Symbol VM ( IM ) The maximum value of V (I) measured from
the point of inflection (“baseline or DC offset”)
From the graph: VM - VDC
Also called “Amplitude”
baselinebaseline
VVMM oror Amplitude Amplitude
VVDCDC
t
V
PEAK TO PEAK VALUES PEAK TO PEAK VALUES (V(VPP, PP, IIPPPP))
Peak to Peak Voltage (Current) Symbol VPP ( IPP ) The difference between the maximum value of
V (I) and the minimum value of V (I) From the graph: VMAX – VMIN
Equals twice peak value VPP = 2VP
VVPPPP
VVMINMIN
VVMAXMAX
t
V
ROOT MEAN SQUARE ROOT MEAN SQUARE (V(VRMSRMS, I, IRMS RMS ))
Named for the mathematical process by which the value is calculated. “Effective Voltage (VEFF)”
The RMS value of a sine wave is equal to the value of an equivalent DC circuit that would produce the same heating effect or power in a load as the given sine wave.”
Most meters read in RMS (lab DMM)
The voltage accessed at electrical wall sockets is RMS.
ROOT-MEAN-SQUARE ROOT-MEAN-SQUARE (V(VRMSRMS, I, IRMS RMS ))
PPRMS V0.707V2
2V
COMPATIBILITY OF VALUESCOMPATIBILITY OF VALUES
When Peak voltages are used as source values, current calculations will also be in Peak values.
Likewise, an RMS source produces answers in RMS.
When solving a problem make sure all values are expressed ONE way (peak, peak to peak, or RMS)!
VVMMVVrmsrms
Vpp
VOLTAGE & CURRENT VOLTAGE & CURRENT VALUESVALUES
Ohm’s Law still applies: V=IR If current changes with time and R is a constant,
voltage will also change with time Voltage will be proportional to current
VOLTAGE & CURRENT VOLTAGE & CURRENT VALUESVALUES
A graph of current and voltage in a resistor produces identical waveforms:Peak at the same timeCross the same baseline, at the
same time Differ only in amplitude:
IP is 1/R of VP
INSTANTANEOUS INSTANTANEOUS VALUESVALUES
Instantaneous Values ( v, i ) value of voltage and current at any:
instant in time or at at any angle
Mathematically expressed 2 ways:
sin 2
sin
M
M
v(t) V ( ft )
v( ) V ( )
ANGULAR DOMAINANGULAR DOMAIN
We can identify points on the sine wave in terms of an angular measurement (degrees or radians). The instantaneous value of the sine wave can
be related to the angular rotation of the generator, (1 rotation = 360°=21 rotation = 360°=2 radians radians)
deg180
rad rad
180
deg
Sine Wave Angles: Degrees & Radians 2 radians = 360o 1 radian = 57.3o
TIME DOMAINTIME DOMAIN
Because the time to complete a cycle is frequency dependent, we can also identify points on the sine wave in terms of time.
To convert between the time domain and angular domain remember:
sin 2Mv(t) V ( ft )
2 ft t
PHASE ANGLEPHASE ANGLE
Symbol is (theta). It is expressed as an angle
Phase angle specifies the lateral shift in the position of a sine wave from a reference wave.
Examine the same event, on each wave: Two events occurring at the same angle or at
the same time are in phase. Events occurring at different angles or at
different times are out of phase.
PHASE ANGLE PHASE ANGLE (angular domain)(angular domain)
Wave A is the reference wave: Wave B is 90° out of phase.
PHASE ANGLE PHASE ANGLE (Time domain)(Time domain)
Wave A is the reference wave. Compare the positive peak events: Wave A peaks at 30ms; Wave B at 60ms T=120ms /360º = t/T = (60ms-30ms)/120ms. = 90º
LEADING & LAGGINGLEADING & LAGGING
Since wave B peaked after the reference wave peaked, we say it LAGS the reference wave by 90º ; = - 90º
If wave B was the reference, wave A would peak before the reference wave (B). We would say it LEADS the reference wave;
= + 90º Note: Because it is the reference wave, for ANY reference wave is 0 º
Ex:Ex:
Compute the phase angle if:V1(t) is the reference wave
V2 (t) is the reference wave
V 1(t)
V 2(t)
t = 1 ms/divt = 1 ms/div
Ex:Ex:
V 1(t)
V 2(t)
t = 1 ms/divt = 1 ms/div
V2 is the reference. Write the equations.
sin 2
sin
dc M
dc M
v(t) V V ( ft )
v( ) V V ( )
SUPERIMPOSED DC & ACSUPERIMPOSED DC & AC
A circuit can have both a DC voltage source and an AC
We say that the “AC rides on the DC” The graph of the voltage is displaced
vertically from 0, to the DC voltage level. Algebraically:
REVIEW QUIZREVIEW QUIZ
The difference between DC and AC ? 3 items required for electromagnetic
induction. Frequency is equal to ? Name 3 different Sine wave values. How many radians in 360 degrees ? If the peak value is 170 V, the RMS value = ? What type of shift does a phase angle
represent?