EE202 Circuit Theory II - akademik.adu.edu.tr · ... RC and RL Circuits, Natural and Step Responses...

EE202 – Circuit Theory II2017-2018, Spring

Dr. Yılmaz KALKAN

I. Introduction & Review of First Order Circuits (Chapter 7 of Nilsson - 3 Hrs.)

Introduction, RC and RL Circuits,

Natural and Step Responses of Series and Parallel RL/RC Circuits

II. Second Order Circuits (Chapter 8 of Nilsson - 9 Hrs.)

The Natural Response of a Parallel RLC Circuit,

The Forms of Natural Response of a Parallel RLC Circuit, The Step Response of a Parallel RLCCircuit, Natural and Step Responses of a Series RLC Circuit

III. Sinusoidal Steady-State Analysis (Chapter 9 of Nilsson - 9 Hrs.)

The Sinusoidal Source, the Sinusoidal Response, the Phasor. The Passive Circuit Elements in theFreq. Domain, Kirchhoff’s Laws in the Freq. Domain. Series, Parallel, and Delta-to-WyeSimplifications, Source Transformations Thevenin and Norton Equivalent Circuits, the Node-Voltage Method, the Mesh-Current Method. The Transformer, Mutual Coupling

IV. Sinusoidal Steady-State Power Calculations (Chapter 10 of Nilsson - 6 Hrs.)

Instantaneous Power, Average and Reactive Power, The rms Value and Power Calculations,Maximum Power Transfer

V. Balanced Three-Phase Circuits (Chapter 11 of Nilsson - 6 Hrs.)

Balanced Three-Phase Voltage, Three-Phase Voltage Sources, Analysis of the Wye-Wye and Wye-Delta Circuits, Power Calculations in Balanced Three-Phase Circuits, Measuring Average Power inThree-Phase Circuits

VI. The Laplace Transformation in Circuit Analysis (Chapter 12-13 of Nilsson - 6 Hrs.)

Circuit Elements in the s Domain, Circuit Analysis in the s Domain, The Transfer Function, TheTransfer Function in Partial Fraction Expansions, The Transfer Function and the ConvolutionIntegral, The Transfer Function and the Steady-State Sinusoidal Response, The Impulse Functionin Circuit Analysis

12.02.2018EE202 - Circuit Theory II

EE202 - Circuit Theory II

Thus far our analysis has been limited for the most part to dccircuits: those circuits excited by constant or time-invariantsources.

We have restricted the forcing function to dc sources for the sakeof simplicity, for pedagogic reasons, and also for historic reasons.

Historically, dc sources were the main means of providing electricpower up until the late 1800s. At the end of that century, thebattle of direct current versus alternating current began.

Both had their advocates among the electrical engineers of thetime. Because ac is more efficient and economical to transmit overlong distances, ac systems ended up the winner.

We now begin the analysis of circuits in which the source voltageor current is time-varying.


In this chapter, we are particularly interested in sinusoidally time-varying excitation, or simply, excitation by a sinusoid.

A sinusoid is a signal that has the form of the sine or cosine function.

A sinusoidal current is usually referred to as alternating current(ac). Circuits driven by sinusoidal current or voltage sources are calledac circuits. We are interested in sinusoids for a number of reasons.

Nature itself is characteristically sinusoidal. A sinusoidal signal is easy to generate and transmit. Through Fourier analysis, any practical periodic signal canbe represented by a sum of sinusoids. A sinusoid is easy to handle mathematically.


Consider the sinusoidal voltage;Vm = the amplitude of the sinusoidω = the angular frequency in radians/sωt = the argument of the sinusoid


Sinusoids

wtVtv m sin)(

It is evident that the sinusoid repeats itself every T seconds; thus, T is called the period of the sinusoid. We observe that ωT = 2π,


Sinusoids

)sin()( wtVtv m

The period “T” of the periodic function is the time of one completecycle or the number of seconds per cycle. The reciprocal of this quantity is the number of cycles per second,known as the cyclic frequency “f “ of the sinusoid. Thus,

Let us now consider a more general expression for the sinusoid,


Sinusoids

or

byvleadsv 12

byvlagsv 21

.,0

.,0

21

21

phaseinarevandvif

phaseofoutarevandvif

We can transform a sinusoid from sine form to cosine form usingtrigonometric identities.

A graphical approach may be used to relate or compare sinusoidsas an alternative to using the trigonometric identities


Sinusoids

The graphical technique can also be used to add two sinusoids ofthe same frequency when one is in sine form and the other is incosine form.


Sinusoids


Sinusoids

Example:


Sinusoids

Solution:

Example:


Sinusoids

Exercise:


Phasors

Sinusoids are easily expressed in terms of phasors, which aremore convenient to work with than sine and cosine functions.

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

Phasors provide a simple means of analyzing linear circuitsexcited by sinusoidal sources; solutions of such circuits would beintractable otherwise. The notion of solving ac circuits usingphasors was first introduced by Charles Steinmetz in 1893.


Phasors

A complex number z can be written in different forms as


Phasors


Phasors

The idea of phasor representation is based on Euler’s identity. Ingeneral,

where

Given a sinusoid

V is the phasor representationof the sinusoid v(t). In otherwords, a phasor is a complexrepresentation of themagnitude and phase of asinusoid.


Phasors


Phasors

As a complex quantity, aphasor may be expressedin rectangular form, polarform, or exponentialform. Since a phasor hasmagnitude and phase(“direction”), it behavesas a vector and is printedin boldface.


Phasors


Example:

Phasors


Exercises:

Phasors


Phasor Relationships For Circuit Elements

Resistor



Inductor



Capacitor



R, L, C Summary


Example:

Phasors

Exercise:


Impedance & Admittance

We obtained the voltage-current relations for the three passiveelements as

These equations may be written in terms of the ratio of thephasor voltage to the phasor current as

From these three expressions, we obtain Ohm’s law in phasorform for any type of element as

jwCjwLR

IVIVIV ,,

jwCjwLR

1,,

I

V

I

V

I

V

ZIVI

VZ or



“Z” is a frequency-dependent quantity known as impedance,measured in ohms.

ZIVI

VZ or

The impedance Z of a circuit is the ratio of the phasor voltage V to the phasor current I, measured in ohms ().

The admittance Y of a circuit is the reciprocal of Z, measured in siemens (S).

V

IY

ZY or

1


Impedance & Admittance The impedance represents theopposition which the circuit exhibits tothe flow of sinusoidal current.

Although the impedance is the ratio oftwo phasors, it is not a phasor, because itdoes not correspond to a sinusoidallyvarying quantity.

From the table;

wC

jjwL CL ZZ ,



As a complex quantity, the impedance may be expressed in rectangularform as

jXR Zwhere R = Re {Z} is the resistance and X = Im {Z} is the reactance.

The reactance X may be positive or negative.

We say that the impedance is inductive when X is positive or capacitivewhen X is negative.

Thus, impedance Z = R + jX is said to be inductive or lagging sincecurrent lags voltage, while impedance Z = R − jX is capacitive or leadingbecause current leads voltage.

The impedance, resistance, and reactance are all measured in ohms.The impedance may also be expressed in polar form as



As a complex quantity, the admittance may be expressed in rectangularform as

jBG Ywhere G= Re {Y} is the conductanceand B = Im {Y} is the susceptance.


Example:

Phasors


Exercise:

Phasors

END OF CHAPTER 3 (Part I)

Dr. Yılmaz KALKAN


EE202 Circuit Theory II - akademik.adu.edu.tr · ... RC and RL Circuits, Natural and Step Responses...

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