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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 Current

Alternating Voltage

6

videos\AC-Generator.mp4

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


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.


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.


Phase


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).


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.




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



biaz rz

)(1

a

btg 22 bar



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.



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)




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)



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.



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.



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.



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.



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.



I

VZ 016.37262.67699.35 jXRZ L

AI 016.37597.1016.37262.6

010



Parallel circuit



First state:



Implementing the Ohm’s Law;



The impedance equation of parallel circuits:

Alternating Current

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