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Alternating Current
Alternating CurrentAsist. Prof. Dr. Aytaç Gören
Asist. Prof. Dr. Levent Çetin
30.10.2012
Alternating Current
Contents
Alternating Voltage
Phase
Phasor Representation of AC
Behaviors of Basic Circuit Components under AC
Resistance, Reactance and Impedance
2
Power in AC Circuits
Alternating Current
Alternating Voltage
If the direction of current and voltage value of a source change due to time, then it is called an AC voltage.
The grid uses AC, since the generation and converting to mechanical energy of it is easy and efficient, moreover the loss in tranfer is less than DC.
3
Alternating Current
Alternating Voltage
4
Alternating CurrentVoltage of the grid.
Frequency or oscillation of a signal is the value of repetition observed in a changing signal in unit time. In other words, frequency refers how often an evvalue of the fequency of the grid in Turkey is ent occurs.The 50 [Hertz]. [1/s ] is also used instead of [Hertz].The time needed to complete one cycle is a period. Period is 1/f, so the period of the grid is 0.02 [s] in Turkey.
The change of the voltage in grid is defined with a sine
function.
The parameters in this equation are;
a) f is oscillation (frequency)
b) Vmax is the maximum value of the voltage
(amplitude).
5
Alternating Voltage
)2sin()( max ftVtVAV
Alternating Current7
T
RMS dttfT
tf0
21)()(
maxmax . VV
VRMS 70702
The maximum value or the amplitude of the alternating voltage is themaximum value of the sine wave during one period. This value is approximately311 [V] for the grid. But, this value is not used as nominal value. Instead, theRMS value of this sine wave is used. The RMS value may be said as theequivalent value of an alternating voltage to direct voltage. The effective valueof a signal is:
For the electrical grid, the effective value is app. 0.707 times ofthe maximum voltage value of the grid and it is 220 [v].
Alternating Voltage
Alternating Current807.10.2011
Alternatif Gerilim
Alternating Current9
Phase
An important point in operations of two time dependent signals iswhether they are synchronized or not. For electrical definitions, twovoltage signals or two corrent signals or one voltage with one currentsignals can be either synchronized or with phase diference.If two signals are synchronized, they both pass the zero points and themaximum value points at the same time.
Alternating Current10
PhaseIf two signals pass the zero points and maximum value points in differentmoments then a phase shift occurs. Phase or phase shift is the timedifference between two signals. The phase shift or phase is denoted withdegree in sine functions. If one of the two sine functions is accepted asreference signal; the value of angle of the signal that is not the referencesignal when the reference signal reaches zero is the phase value. Accordingly,+45 +90 +180 and 0 degrees of phase shifts are shown in the figure below.
Alternating Current11
Phase
Alternating Current12
Phasor Representaion of AC
In order to define the affects of the alternating current to a circuit,frequency, amplitude and phase need to be known. Frequency dependson the electrical grid, so the country or region. So, the voltage/currentfunctions can be defined depending on two parameters. One of thechoice in modeling alternating current / voltage is to represent themusing rotating vectors (i.e. phasors).
Alternating Current13
The projection of a rotating vector around origin in cartesian coordinatesystem is a sine function as can be seen in figure above. The length (or theradius) of the rotating vector is the amplitude of alternating voltage in thisrepresentaion. Similarly, the angle between the vector and the horizantalaxis is the phase value (θ). The angular velocity of this rotating vector isthe frequency of alternating voltage.
Phasor Representaion of AC
Alternating Current14
Phasor Representaion of AC
Alternating Current15
Phasor Representaion of AC
A complex number is a mathematical quantity representing twodimensions of magnitude and direction. Representation of alternatingvoltage as a rotating vector indicates a complex number in means ofmathematics. As known, complex numbers has two parts called real andimaginary which are represented in complex plane.The common representation of a complex number in cartesian form isequation (1) whereas phasor representation that is representation of acomplex number as a rotating vector and more used in electrical circuitanalysis is the equation (2) below.
biaz (1)
(2) rz
Alternating Current16
Phasor Representaion of AC
biaz rz
)(1
a
btg 22 bar
Alternating Current17
Phasor Representaion of AC
ibbaazz )()( 212121
ibbaazz )()( 212121
212121 rrzz
212121 rrzz
Four basic operations in complex numbers can be seen below.
Implementation of complex number arithmetics to voltage/current signalsis examination of total effects of voltage/current sources which havedifferent phases. Before calculating this effect, the state of the tworotating vectors, which have different phases, according to each othershould be understood. This relation might be described as vectors whichhave the same starting point but have different angles respect to thehorizontal line.
Alternating Current18
Phasor Representaion of AC
Alternating Current19
Phasor Representaion of AC
Created by a combination of
current/voltage sources
connected to the same circuit is
determined by the complex
numbers, addition and
subtraction operations.
Alternating Current
1 2 3
20
Resistance(R)
Coil (L)(Inductance)
Capacitance (C)(Capacitor)
Behaviors of Basic Circuit Components under AC
Alternating Current21
Behaviors of Basic Circuit Components under AC
I
VR )s in(max wtVV )s in(max wt
R
VI
Ohm’s Law can be used for resistance under the influence of alternating voltage.
According to equations above, there is no phase shift between current andvoltage on a resistor. Nevertheles, the amplitude changes due to Ohm’s Law.
Resistance (R)
Alternating Current22
Behaviors of Basic Circuit Components under AC
Coil (L) (Inductor)
In contrast with resistors, coils under alternating voltage resists against alternating current. The voltage on a coil (the voltage measured between two terminals) can be calculated using Lenz Law.
dt
tdiLtV
)()(
If this equation is studied considering the alternating current, theralationship between the current and voltage might be predicted.
Alternating Current23
Behaviors of Basic Circuit Components under AC
Coil (L) (Inductance)
dt
tdiLtV
)()(
)s in()( max wtItI
)cos()(
)( wtLdt
tdiLtV
In contrast with resistors, coils under alternating voltage resists against alternating current. The voltage on a coil (the voltage measured between two terminals) can be calculated using Lenz Law.
If this equation is studied considering the alternating current, theralationship between the current and voltage might be predicted.
Alternating Current24
Behaviors of Basic Circuit Components under AC
Alternating Current25
Behaviors of Basic Circuit Components under AC
wLXL fLXL 2
This result shows us that there is a 90 degrees of phase shift between voltage and current on a coil under AC. The voltage leads current by phase angle of 90 degree. The phase shift results with negative electrical power. Negative power denotes that the coil transfer power to the circuit. The ‘resistance’ of coils changes due to time or frequency. This is called as reactance (inductive reactance XL) for this reason.
Alternating Current26
Behaviors of Basic Circuit Components under AC
I
VX
.7699310602 2LX AX
VI .
.65262
76993
10
Ohm’s Law might be implemented easily to alternating current circuitsusing quantity, the reactance. In that case, the calculations should bemade using complex numbers instead of scalars.
Now, let us calculate the influence of total resistance of aresistor and a coil adding a 5 [Ohm] resistor to this circuit.
Alternating Current27
Behaviors of Basic Circuit Components under AC
jR 05
jXL 7699.30
016.37262.67699.35 jXRZ L
Resistor value:
Inductive reactance of the coil
The total effect is called as impedance.
Commonly, impedance in alternating voltage circuits is thecorresponding definition of resistance. Besides, it can be used asresistor in Ohm’s Law as mentioned above.
Alternating Current28
Behaviors of Basic Circuit Components under AC
I
VZ 016.37262.67699.35 jXRZ L
AI 016.37597.1016.37262.6
010
Alternating Current29
Behaviors of Basic Circuit Components under AC
Alternating Current30
Behaviors of Basic Circuit Components under AC
Parallel circuit
Alternating Current31
Behaviors of Basic Circuit Components under AC
First state:
Alternating Current32
Behaviors of Basic Circuit Components under AC
Implementing the Ohm’s Law;
Alternating Current33
Behaviors of Basic Circuit Components under AC
Implementing the Ohm’s Law;
Alternating Current34
Behaviors of Basic Circuit Components under AC
The impedance equation of parallel circuits:
Alternating Current