AC Circuits Download: http://skillcruise.com/academics/electrical-engineering/

1. AC CircuitsAC CircuitsAC CircuitsAC Circuits

2. ALTERNATING CURRENT
ALTERNATING CURRENT
 A current that is constantly changing in amplitude and direction.
Advantages of AC:
 Magnitude can easily be changed (stepped-up or stepped down)
with the use of a transformer
 Can be produced either single phase for light loads, two phase for
control motors, three phase for power distribution and large motor
loads or six phase for large scale AC to DC conversion.

3. AC WAVEFORMS

4. AC WAVEFORMS
• Period (T) – the time of one complete cycle in seconds
• Frequency (f) – the number of cycles per second (Hertz)
• 1 cycle/second (cps) = 1 Hertz (Hz)
• Proper operation of electrical equipmnent requires specific frequency
• Frequencies lower than 60 Hz would cause flicker when used in lightning
• Wavelength (λ) – the length of one complete cycle
• Propagation Velocity (v) – the speed of the signal
• Phase (Φ) – an angilar measurement that specifies the position of a sine wave relative
to reference
Parameters of Alternating Signal

5. AC WAVEFORMS
THE SINUSOIDAL WAVE
 Is the most common AC waveform that is practically generated by
generators used in household and industries
 General equation for sine wave:
Where
a(t) – instantaneous amplitude of voltage or current at a given time (t)
Am – maximum voltage or current amplitude of the signal
ω – angular velocity in rad/sec; ω = 2πf
t – time (sec)
Φ – phase shift ( + or – in degrees)
A(t) = Am sin (ωt + Φ)

6. AC WAVEFORMS
• AMPLITUDE
It is the height of an AC waveform as depicted on a graph over time (peak,
peak-to-peak, average, or RMS quantity)
• PEAK AMPLITUDE
the height of an AC waveform as measured from the zero mark to the highest
positive or lowest negative point on a graph. Also known as the crest
amplitude of a wave.
Measurements of AC Magnitude

7. AC WAVEFORMS
• PEAK-TO-PEAK AMPLITUDE
the total height of an AC waveform as measured from maximum
positive to maximum negative peaks on a graph. Often
abbreviated as “P-P”

8. AC WAVEFORMS
• AVERAGE AMPLITUDE
the mathematical “mean” of all a waveform’s points over the period of one
cycle. Technically, the average amplitude of any waveform with equal-
area portions above and below the “zero” line on a graph is zero.
 For a sine wave, the average value so calculated is approximately 0.637
of its peak value.

9. AC WAVEFORMS
• RMS AMPLITUDE
“RMS” stands for Root Mean Square, and is a way of expressing an AC
quantity of voltage or current in terms functionally equivalent to DC.
Also known as the “equivalent” or “DC equivalent” value of an AC
voltage or current.
• For a sine wave, the RMS value is approximately 0.707 of its peak
value.

10. AC WAVEFORMS
• The crest factor of anThe crest factor of an
AC waveform is theAC waveform is the
ratio of its peakratio of its peak
(crest) to its RMS(crest) to its RMS
value.value.
• The form factor of anThe form factor of an
AC waveform is theAC waveform is the
ratio of its RMS valueratio of its RMS value
to its average value.to its average value.

11. AC QUANTITIES
RESISTANCE (R)
• Opposes the AC current similar to DC circuits
• Opposition offered by resistors
REACTANCE (X)
• Depends on the AC frequency of the AC source which is the
opposition to current due to inductance and capacitance

12. AC QUANTITIES
Inductive Reactance (XL)
• The property of the inductor to oppose the alternating current
Inductive Susceptance (BL)
• Reciprocal of inductive reactance
Capacitive Reactance (XC)
• The property of a capacitor to oppose alternating current
Capacitive Susceptance (BC)
• Reciprocal of capacitive reactance
XL = 2πfL

13. AC QUANTITIES
IMPEDANCE (Z)
 Total opposition to the flow of Alternating current
 Combination of the resistance in a circuit and the reactances involved
Phasor Diagram of Impedance
Z = R + jXeq Z = |Z| ∠φ

14. AC QUANTITIES
• If I = Im ∠β is the resulting current drawn by a passive, linear
RLC circuit from a source voltage V = Vm ∠θ, then
Where: Z = Vm = √ R2
+ X2
= magnitude of the impedance
Im
φ = θ – β = tan-1
X = phase angle of the impedance
R
R = Zcos φ = active or real component of the impedance
X = Zsin φ = reactive or quadrature component of impedance
Z = V = Vm ∠θ = Z ∠φ
I Im ∠β
Z cosφ + jZsin φ = R + jX = √ R2
+ X2
∠ tan-1
X
R

15. AC QUANTITIES
ADMITTANCE (Y)
 The reciprocal of impedance
 Expressed in siemens or mho (S)

16. AC RESISTOR CIRCUIT
With an AC circuit like this which is purely
resistive, the relationship of the voltage
and current is as shown:
• Voltage (e) is in phase with the current (i)
• Power is never a negative value. When
the current is positive (above the line), the
voltage is also positive, resulting in a
power (p=ie) of a positve value
• This consistent “polarity” of a power tell us
that the resistor is always dissipating
power, taking it from the source and
releasing it in the form of heat energy.
Whether the current is negative or
positive, a resistor still dissipated energy.
Impedance (Z) = R

17. AC INDUCTOR CIRCUIT
• The most distinguishing electrical characteristics of an L circuit is that current lags voltage by 90
electrical degrees
• Because the current and voltage waves arae 90o
out of phase, there sre times when one is positive
while the other is negative, resulting in equally frequent occurences of negative instantaneous power.
• Negative power means that the inductor is releasing power back to the circuit, while a positive power
means that it is absorbing power from the circuit.
• The inductor releases just as much power back to the circuit as it absorbs over the span of a complete
cycle.
Impedance (Z) = jXL

18. AC INDUCTOR CIRCUIT
• Inductive reactance is the opposition that an inductor offers to
alternating current due to its phase-shifted storage and release of
energy in its magnetic field. Reactance is symbolized by the
capital letter “X” and is measured in ohms just like resistance (R).
• Inductive reactance can be calculated using this formula: XL =
2πfL
• The angular velocity of an AC circuit is another way of expressing
its frequency, in units of electrical radians per second instead of
cycles per second. It is symbolized by the lowercase Greek letter
“omega,” or ω.
• Inductive reactance increases with increasing frequency. In other
words, the higher the frequency, the more it opposes the AC flow
of electrons.

19. AC CAPACITOR CIRCUIT
• The most distinguishing electrical characteristics of an C circuit is that leads the voltage by 90 electrical degrees
• The current through a capacitor is a reaction against the change in voltage across it
• A capacitor’s opposition to change in voltage translates to an opposition to alternating voltage in general,
which is by definition always changing in instantaneous magnitude and direction. For any given magnitude of
AC voltage at a given frequency, a capacitor of given size will “conduct” a certain magnitude of AC current.
• The phase angle of a capacitor’s opposition to current is -90o
,meaning that a capacitor’s opposition to current is
a negative imaginary quantity
Impedance (Z) = -jXC

20. AC CAPACITOR CIRCUIT
• Capacitive reactance is the opposition that a capacitor offers to
alternating current due to its phase-shifted storage and release of
energy in its electric field. Reactance is symbolized by the capital
letter “X” and is measured in ohms just like resistance (R).
• Capacitive reactance can be calculated using this formula:
XC = 1/(2πfC)
• Capacitive reactance decreases with increasing frequency. In
other words, the higher the frequency, the less it opposes (the
more it “conducts”) the AC flow of electrons.

21. SERIES RESITOR-INDCUTOR CIRCUIT
 For a series resistor-inductor circuit, the voltage and current
relation is determined in its phase shift. Thus, current lags
voltage by a phase shift (θ)
Impedance (Z) = R + jXL
Admittance (Y) = 1 = R – jXL
R + jXL R2
+ jXL
2

22. SERIES RESITOR-INDCUTOR CIRCUIT
• When resistors and inductors are mixed together in circuits, the
total impedance will have a phase angle somewhere between 0o
and +90o
. The circuit current will have a phase angle somewhere
between 0o
and -90o
. Series AC circuits exhibit the same
fundamental properties as series DC circuits: current is uniform
throughout.
Phase shift (θ) = Arctan ( XL ) |Z| = √ R2
+ jXL
2
= e
R i

23. SERIES RESISTOR-CAPACITOR CIRCUIT
 For a series resistor – capacitor circuit, the voltage and current
relation is determined by the phase shift. Thus the current leads the
voltage by an angle less than 90 degrees but greater than 0
degrees.
Impedance (Z) = R – jXC
Admittance (Y) = 1 = R + jXC
R – jXC R2
+ jXC
2

24. SERIES RESISTOR-CAPACITOR CIRCUIT
Phase shift (θ) = Arctan ( XC ) |Z| = √ R2
+ jXC
2
= e
R i

25. PARALLEL RESISTOR-INDUCTOR
Y = G – jβL where: G – conductance = 1/R
βL – susceptance = 1/XL
Z = E , by Ohm’s Law
I
• The basic approachwith regarda to parallel circuits is using
admittance because it is additive

26. PARALLEL RESISTOR-INDUCTOR
• When resistors and inductors are mixed together in parallel circuits
(just like in series cicuits), the total impedance will have a phase
angle somewhere between 0o
and +90o
. The circuit current will
have a phase angle somewhere between 0o
and -90o
.
• Parallel AC circuits exhibit the same fundamental properties as
parallel DC circuits: voltage is uniform throughour the circuit,
brach currents add to form the total current, and impedances
diminish (through the reciprocal formula) to form the total
impedance.

27. PARALLEL RESISTOR-CAPACITOR
Y = G + jβC
Where G – conductance = 1/R
βC – susceptance = 1/XC
• When resistors and capacitors are mixxed together in circuits, the total
impedance will have a phase angle somewhere between 0o
and -90o
.

28. APPARENT POWER (S)
APPARENT POWER
• Represents the rate at which the total energy is supplied to the
system
• Measured in volt-amperes (VA)
• It has two components, the Real Power and the Capacitive or
Inductive Reactive Power
S = Vrms Irms = Irms
2
|Z|

29. APPARENT POWER (S)
Power Triangle
Complex Power
S = P ± jQ

30. REAL POWER (R)
REAL POWER
• The power consumed by the resistive component
• Also called True Power, Useful Power and Productive Power
• Measured in Watts (W)
• It is equal to the product of the apparent power and the power
factor
• Cosine of the power factor angle (θ)
• Measure of the power that is dissipated by the cicuit in relation to the
apparent power and is usually given as a decimal or percentage
Pf = cos θ
P = Scos θ
Power Factor

31. REAL POWER (R)
• Ratio of the Real Power to the Apparent Power ( P )
S
when:
Pf = 1.0 I is in phase with V; resistive system
Pf = lagging I lags V by θ; inductive system
Pf = leading I leads V by θ; capacitive system
Pf = 0.0 lag I lags V by 90o; purely inductive
Pf = 0.0 lead I leads V by 90o; purely capacitive
• The angle between the apparent power and the real poweer in the power
triangle
Let v(t) = Vm cos(ωt + θv) volts
V = Vrms ∠θv
i(t) = Im cos(ωt + θi) A
I = Irms ∠θi
Power factor Angle (θ)

32. REAL POWER (R)
Where
θ = phase shfit between v(t) and i(t) or the phase angle
of the equivalent impedance
Instantaneous Power (watts)
Average Power (watts)
P(t) = v(t) i(t)
P(t) = ½ VmIm cos (θv – θi) + ½ VmIm cos (2ωt + θv + θi)
P(t) = ½ VmIm cos (θv – θi) = VmIm cos θ

33. REACTIVE POWER (QL or QC)
REACTIVE POWER
• Represents the rate at which energy is stored or released in any of the
energy storing elements (the inductor or the capacitor)
• Also called the imaginary power, non-productive or wattless power
• Measured in volt-ampere reactive (Var)
• When the capacitor and inductor are both present, the reactive
power associated with them take opposite signs since they do not
store or release energy at the same time
• It is positive for inductive power (QL) and negative for capacitive
power (QC)
• Ratio of the Reactive Power to the Apparent Power
• Sine of the power factor angle (θ)
Q = VmIm sin θ
Reactive factor
Rf = sin θ

34. BALANCED THREE PHASE SYSTEMS
BALANCED 3-PHASE SYSTEM
 Comprises of three identical single-phase systems operating at a 120o
phase displacement from one another. This means that a balance
three-phase system provides three voltages(and currents) that are
equal in magnitude and separated by 120o
from each other
Three-Phase, 3-wire systems
 Provide only one type of voltage(line to line) both single
phase and three phase loads
Three-Phase, 4-wire systems
 Provide two types of voltages(line to line and line to
neutral) to both single phase and three phase loads
CLASSIFICATION

35. BALANCED THREE PHASE SYSTEMS
and
VLL and VLN are out of phase by 30o
BALANCED Y-system
VLL = √3 VLN IL = IP
and
IL and IP are out of phase by 30o
Where
VLL or VL - line to line or line voltage
VLN or VP - line to neutral or phase voltage
IL - line current
IP - phase current
BALANCED ∆-system
IL = √3 IP VLL = VLN

36. watts
vars
va
Note: for balanced 3-phase systems:
IA + IB + IC =
0
VAN + VBN + VCN = 0
VAB + VBC + VCA = 0
P = 3VPIPcos θ = √3 VLIL cos θ
Q = 3VPIPsin θ = √3 VLIL sin θ
S = 3VPIP = √3 VLIL
THREE-PHASE POWER

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