ALTERNATING CURRENT A'C' CIRCUITS

1
ALTERNATING CURRENT (A.C.) CIRCUITS

• In practice the Generators of electrical energy
  and connected with them Loads are usually far
  away from each other, therefore the electrical
  energy is usually transmitted over long
  distances.
• The main condition for long-distance electric
  power transmission and distribution is the
  possibility of voltage transformation.
• This can only be done by sinusoidal alternating
  current (A.C.) device known as a Transformer,
  providing minimum power losses.
• The higher is distance, the higher is voltage of
  transmission lines.

2
Sinusoidal Current

• Advantages of Sinusoidal Current
• The simple construction of sinusoidal current
  generators
• Sinusoidal Current passing through elements of
  circuit doesnt change its sinusoidal form
• Simplicity of circuit analysis

3
Construction of Synchronous Generator
4
Sinusoidal Emf, Current, Voltage

• e, i, v - instantaneous values of the sinusoidal
  emf, current and voltage at the instant of time t
• Em, Im, Vm - their amplitudes (peak values)
• - initial phase of sinusoidal emf (rad)
• - angular frequency (s-1)
• f - frequency (Hz) f 50Hz (in US and
  Canada f 60Hz)
• T 1/f period (s)

5
Graphical Symbols for A.C. Generators
Sinusoidal voltage generator
Sinusoidal current generator
v
? j
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The Average Value of Sinusoidal Quantities

• The average value of a sinusoidal quantity is the
  average
• value of sinusoidal function over the positive
  half period

• The average value of a sinusoidal current

• The average value of a sinusoidal emf

• The average value of a sinusoidal voltage

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The RMS Value of Sinusoidal Quantities
The Root-Mean-Square (RMS) value (or
Effective value) of a sinusoidal current is equal
to the value of direct current, which passing
through some resistance R during some period of
time, produces the same amount of heat energy as
sinusoidal current passing through the same
resistance R during the same period of time, t
T
QD.C. I2RT - Heat energy produced by D.C.
- Heat energy produced by A.C.
?
QD.C. QA.C.
?
The effective value of a sinusoidal current is
defined as its root-mean-square (RMS) value over
a period.
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The RMS Value of Sinusoidal Current, EMF and
Voltage

• The RMS value of a sinusoidal current

• The RMS value of a sinusoidal emf

• The RMS value of a sinusoidal voltage

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Representation of Sinusoidal Quantities as
Rotating Vectors (Phasors)
y y1 y2 i1 i2 i
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Phasor Diagram of A.C.Circuit

• The group of phasors presenting currents or/and
  voltages of given circuit is called the Phasor
  Diagram
• To construct the phasor diagram of the circuit,
  one of the phasors, called Reference Phasor, is
  directed arbitrary (reference phasors and
  are directed horisontally)

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Representation of Sinusoidal Quantities as
Complex Numbers
j
j y
x
0
1
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Representation of Sinusoidal Current, (Voltage,
EMF) as Complex Numbers
In Exponential form
In Trigonometric form
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Elements of A.C. Circuits 1

• The alternating current passing through
  electrical circuit may initiate the following
  phenomena
• Conversion electrical energy into non-electrical
  energy,
• The alternating magnetic field and associated
  with it self-induction emf,
• The alternating electric field and associated
  with it charging-discharging periodical process.

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Elements of A.C. Circuits 2
Resistive Element R Resistance (?, k?, M?)
R
Inductive Element L Indutance (H, mH)
Capacitive Element C Capacitance (F, ?F, ?F)
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The Simplest A.C. Circuit with only Resistive
Element 1
i Im Rsin(?t ?i )
v iR Im Rsin(?t ?i ) Vm sin(?t ?v)
?v ?i, - Initial phase of the voltage
Vm ImR - Amplidude value of the voltage
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The Simplest A.C. Circuit with only Resistive
Element 2
The initial phases of voltage and current are the
same ?v ?i, which means that in resistive
element the current and the voltage sinusoids
vary in phase. Or, in other words, the current in
resistive element coincides in phase with the
voltage across the element.
? ?u -?i 0
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The Simplest A.C. Circuit with Only Inductive
Element 1
As an inductive element of A.C. circuit can be
considered the ideal magnetic coil with
inductance L and resistance R 0
i Im R sin(?t ?i )
eL - v ? v LIm?cos(? t ?i)
v Im?Lcos(?t ?i ?/2) Vmsin(?t ?v)
Vm Im?L - Amplidude value of the voltage (1)
?v ?i ?/2 - Initial phase of the voltage
(2)
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The Simplest A.C. Circuit with Only Inductive
Element 2
xL ?L 2 ? fL (?) Inductive Reactance
The voltage across inductive element leads the
current (or the current lags behind the voltage)
in phase by an angle ?/2 or 900
? ?v -?i ?/2 Phasor-Shift between voltage
and current
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The Simplest A.C. Circuit with only Capacitive
Element 1

• As a capacitive element can be considered the
  ideal capacitor with capacitance C and resistance
  R 0

i Im R sin(?t ?i )
- Amplidude value of the
voltage (1) ?v ?i - ?/2 - Initial phase of
the voltage (2)
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The Simplest A.C. Circuit with only Capacitive
Element 2
The voltage across capacitive element lags behind
the current (or the current leads the voltage) in
phase by an angle ?/2 or 900
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The Single Loop Sinusoidal A.C. Circuit 1
According to Kirchhoffs Voltage Law
VR I R VL I xL VC I xC
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The Single Loop Sinusoidal A.C. Circuit 2
(1)
R z cos ?, x z sin?
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The Single Loop Sinusoidal A.C. Circuit 3

• 3. xL xC, ? 0
• The circuit is in
• Voltage (Series)
• Resonance condition
• zres zmin R ?
• ? Ires V /Rmin Imax

2. xLlt xC , ? lt 0 The circuit is capacitive in
its effect and the current leads the voltage in
phase

• xLgt xC, ? gt 0
• The circuit is inductive
• in its effect and the
• current I lags behind
• the voltage V in phase

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The Parallel Connection of Resistive, Inductive
and Capacitive Elements 1
According to Kirchhoffs Current Law
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The Parallel Connection of Resistive, Inductive
and Capacitive Elements 2
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The Parallel Connection of Resistive, Inductive
and Capacitive Elements 2
3. bL bC, ? 0 The circuit is in Current
(Parallel) Resonance condition yres ymin G
? ? Ires V ymin Imin
1. bLgt bC, ? gt 0 The circuit is inductive in
its effect and the current I lags behind the
voltage V in phase
2. bLlt bC, ? lt 0 The circuit is inductive in
its effect and the current I leads the voltage V
in phase
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Relationship Between Impedance and Admittance 1
Transforming the series
circuit into a parallel circuit
?
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Relationship Between Impedance and Admittance 2
Transforming the parallel
circuit into a series circuit
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Power Relations in A.C. Circuits 1
i
v
P

• p vi VmIm sin(?t ?v) sin(?t ?i)
• cos(?v - ?i) cos(2?t
  ? v ?i)
• p VIcos? VIcos(2?t ?v ?i).

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Power Relations in A.C. Circuits 2

• - ?/2 lt ? lt ?/2 ? cos ? gt0, P gt 0

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Power Relations in A.C. Circuits 3

• Active Power

(W, kW)
Reactive Power
(VAr, kVAr)
(VA, kVA)
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Power Relations in A.C. Circuits 4
? 0, cos? 1, P S VI sin ? 0, Q
0
, cos? 0, P 0 sin ? ?1 ,
Q S ?VI
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Calculation of Branched A.C. Circuit with the Aid
of Complex Numbers 1

• Ohms Law in Complex Form

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Calculation of Branched A.C. Circuit with the Aid
of Complex Numbers 2
Kirchhoffs Current Law in Complex Form
Kirchhoffs Voltage Law in Complex Form
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Calculation of Branched A.C. Circuit with the Aid
of Complex Numbers 3
The Complex Apparent Power
The Complex Apparent Power of the circuit is
called the product of the complex value of the
voltage and the conjugate of complex
value of a current