Revision on DC & AC Circuits Analysis

Introduction Sinusoids Phasors Phasor Relationships for Circuit Elements Conclusions

Revision on DC & AC Circuits Analysis

Future University Egypt, FUEFaculty of Engineering & Technology

Electrical Engineering Dept.

Fall 2017:Electrical Circuits 2 (EPR 263) — Lecture no. 1

Sept. 24, 2017

Dr. Waleed Al-Hanafy Fall 2017: Electrical Circuits 2 (EPR 263) — Lecture no. 1



Overview

1 Introduction

2 Sinusoids

3 Phasors

4 Phasor Relationships for Circuit Elements

5 Conclusions

Reference:C. K. Alexander and M. N. Sadiku, Fundamentals of ElectricCircuits, 5th ed. McGraw-Hill, 2013.




Introduction

We now begin the analysis of circuits in which the source voltageor current is time-varying, particularly sinusoidally excitation, orsimply, excitation by a sinusoid.

A sinusoid is a signal that has the form of the sine or cosine functionand usually is referred to as alternating current (ac).

Sinusoids is used for the following reasons:

1 It is the form of voltage generated through out the world andsupplied to homes, factories, laboratories, and so on.

2 It is the dominant form of signal in the communications andelectric power industries.

3 Through Fourier analysis, any practical periodic signal can berepresented by a sum of sinusoids.

.




Sinusoids

v(t) = Vm sin (ωt + φ) (1)

Vm is the amplitude of the sinusoidω is the angular frequency in rad/sωt + φ is the argumentφ is the phase angle in rad. or degree.The period of the sinusoid T is given by

T =2π

ω. (2)

A periodic function is one that satisfies

fn (t) = fn (t + nT ) , (3)

for all t and for all integers n.

The frequency of the sinusoid inhertz (Hz) is given by

f =1

T=

ω

2π(4)




Sinusoids — Cont’d

A sinusoid can be expressed ineither sine or cosine form.

When comparing two sinusoids, itis expedient to express both aseither sine or cosine with positiveamplitudes.

This is achieved by using thefollowing trigonometric identities:

sin (A± B) = sinA cosB ± cosA sinBcos (A± B) = cosA cosB ∓ sinA sinB,therefore:

sin (ωt ± 180◦) = − sinωtcos (ωt ± 180◦) = − cosωtsin (ωt ± 90◦) = ± cosωtcos (ωt ± 90◦) = ∓ sinωt




Phasors

Sinusoids are easily expressed in terms of phasors, which are more

convenient to work with than sine and cosine functions. Phasors provide

a simple means of analyzing linear circuits excited by sinusoidal sources.

A phasor is a complex number that represents the amplitude andphase of a sinusoid.

complex numbers revision

A complex number z can be written inrectangular form as z = x + jy and polar orexponential form as z = r∠φ = re jφ, where rand φ are the magnitude and phase of z .

r =√

x2 + y2 and φ = tan−1 yx , and x = r cosφ

and y = r sinφ.

..Dr. Waleed Al-Hanafy Fall 2017: Electrical Circuits 2 (EPR 263) — Lecture no. 1



Phasors — Cont’d

The idea of phasor representation was first introduced by CharlesSteinmetz, a German-Austrian mathematician and engineer, and is basedon the Euler’s identity e±jφ = cosφ± j sinφ, therefore we have

v (t) = Vm cos (ωt + φ)= Re

(Vme

j(ωt+φ))

= Re(Vme

jφe jωt),

or simply, we can write v(t) = Re(Ve jωt

),

where V = Vmejφ = Vm∠φ

This is equivalent to transform the sinusoid from the time domainto the phasor domain summarized as follows:

v(t) = Vm cos (ωt + φ)⇐⇒ V = Vm∠φ (5)




Phasors — Cont’d

Equation (5) demonstrated in the following Table. It is noted that to getthe phasor representation of a sinusoid, we express it in cosine form andtake the magnitude and phase.

The differences between v(t) and V should be emphasized:1. v(t) is time dependent, while V is not.2. v(t) is always real with no complex term, while V is generally complex.Finally, phasor analysis applies only when frequency is constant; it appliesfor two or more sinusoids only if they are of the same frequency.




Differentiation, Integration, and Addition of Phasors

1 Differentiating v(t) = Vm cos (ωt + φ) w.r.t. t, we get

dvdt = −ωVm sin (ωt + φ) = ωVm cos (ωt + φ+ 90◦)

= Re(ωVme

jωte jφe j90◦)

= Re(jωVe jωt

).

Thus dvdt ⇐⇒ jωV.

2 Integrating v(t) is, similarly, transformed to the phasordomain as

∫vdt ⇐⇒ V

jω .

3 Adding sinusoids of the same frequency is equivalent toadding their corresponding phasors.




The Resistor, R

We begin with the resistor. If the current through a resistor R isi = Im cos (ωt + φ), the voltage across it is given by Ohm’s law as

v = iR = RIm cos (ωt + φ)⇐⇒ V = RI

Therefore voltage and current are in phase as shown.




The Inductor, L

Assume the current through the inductor isi = Im cos (ωt + φ)⇐⇒ I = Im∠φ, the voltage across it is

v = L didt

= −ωLIm sin (ωt + φ)= ωLIm cos (ωt + φ+ 90◦)

V = ωLIm∠φ+ 90◦

= jωLI

Therefore the voltage and current are out of phase. Specifically, thecurrent lags the voltage by 90◦ as shown.




The Capacitor, C

Assume the voltage across the capacitor isv = Vm cos (ωt + φ)⇐⇒ V = Vm∠φ. The current through it is

i = C dvdt

= −ωCVm sin (ωt + φ)= ωCVm cos (ωt + φ+ 90◦)

I = ωCVm∠φ+ 90◦

= jωCV

Therefore the current and voltage are out of phase. To be specific, thecurrent leads the voltage by 90◦ as shown.




Circuit Elements Summary

The following table summarizes the time domain and phasordomain representations of the circuit elements.




Conclusions

Concluding remarks

Circuits of time-varying (in particular, sinusoid) sources arediscussed.

Phasors are used to deal with these sources and differentcircuit elements.

Relation between voltage across and current through differentcircuit elements, e.g. resistors, inductors, and capacitors areanalyized.

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Revision on DC & AC Circuits Analysis

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