FUNDAMENTALS OF ELECTRICAL ENGINEERING [ ENT 163 ] LECTURE #7 INTRODUCTION TO AC CIRCUITS HASIMAH ALI Programme of Mechatronics, School of Mechatronics Engineering, UniMAP.

CONTENTS Introduction Sinusoids Phasors Impedance and Admittance AC Power Analysis

INTRODUCTION  Dc circuit- excited by constant or time-invariant sources.  Historically, dc sources were the main means of providing electric power till the late 1800s.  End of century – started to introduce an ac.  Due to the advantages (more efficient and economical to transmit over long distance) ac system is accepted.

SINUSOIDS  A sinusoid current is referred to as alternating current (ac). Such a current reverse at regular time intervals and has alternately positive and negative values.  Circuits driven by sinusoidal current or voltage sources are called ac circuits.  Consider the sinusoidal voltage: A sinusoid is a signal that has the form of the sine or cosine function. V m = the amplitude of the sinusoid, ω = the angular frequency in rads/s ωt = the argument of the sinusoid

 The sinusoid repeats itself every T seconds; thus T is called the period of the sinusoid.  Period; T is the required for a given sine wave to complete one full cycle.  Frequency f is the number of cycles that a sine wave complete in one second SINUSOIDS

 Peak to peak Voltage V p-p : is the value of twice the peak voltage (amplitude).  The root-mean-square (RMS) value of a sinusoidal voltage is equal to the dc voltage that produces the same amount of heat in a resistance as does the sinusoidal voltage.  The average value of a sine wave – defined over a half cycle of the wave. or

SINUSOIDS Let consider a more general expression for the sinusoid: Where is the phase VmVm -V m

SINUSOIDS  Therefore we say that v 2 leads v 1 by or v 1 lag v 2 by.  For we could also say that v 1 and v 2 are out of phase.  If, then v 1 and v 2 are said to be in phase. Example: Find the amplitude, phase, period and frequency of the sinusoid

SINUSOIDS Example: Calculate the phase angle between and. State which sinusoid is leading.

PHASORS A phasor is a complex number that represents the amplitude and phase of a sinusoid. Three representation of complex number:  Rectangular form:  Polar form:  Exponential form: j=(√-1), x= real part of z, y=imaginary part of z, r=magnitude of z, Ø=phase of z. or

 Important operations involving complex numbers:  Addition  Subtraction  Multiplication  Division  Reciprocal PHASORS

 Square root  Complex conjugate PHASORS Phasor representation

PHASORS Representation of

PHASORS Phasor diagram:  The phasors are rotating anticlockwise as indicated by the arrowed circle. A is leading B by 90 degrees.  Since the two voltages are 90 degrees apart, then the resultant can be found by using Pythagoras, as shown..

Sinusoid-phasor transformation PHASORS Transformation of sinusoid from the time domain to phasor domain Time-domainPhase-domain

Phasor-domain= frequency domain Differences between v(t) and V: v(t) is the instantaneous or time-domain representation, while V is the frequency or phasor-domain representation. v(t) is time dependent, while V is not. v(t) is always real with no complex term, while V is generally complex PHASORS

PHASORS RELATIONSHIPS FOR CIRCUIT ELEMENTS 1.Resistor If the current through a resistor R is then, the voltage, Phasor form

PHASORS Phasor Relationships for Circuit Elements. ElementTime-domain representation Phasor-domain representation R L C

IMPEDANCE AND ADMITTANCE Previously, we know that the voltage-current relations for the three passive elements as: V=IR,, The above equation can be written in terms of the ratio of phasor voltage to phasor current as: V/I=R V/I= JωL V/I= 1/JωC From here, we obtain Ohm’s Law in phasor as Z=V/I V=ZI Impedance, Z is the ratio of the phasor voltage V to the phasor current I, measured in ohms (Ω)

IMPEDANCE AND ADMITTANCE ElementImpedanceAdmittance RZ=RY=1/R CZ=jωLY=1/jωL LZ=1/jωCY=jωC When ω=0 (dc source), Z L =0 and Z c =∞ (confirm- inductor acts like short circuit, capacitors acts like open circuit). When ω=∞ (high frequencies ) Z L = ∞ and Zc=0 (indicate- inductor acts like open circuit, capacitors acts like short circuit).

Further Reading Fundamentals of electric circuit. (2th Edition), Alexander, Sadiku, MagrawHill. (chapter 9).