ct lecture 1

7/31/2019 CT Lecture 1

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Theory of CircuitsCredits ECTS: 5,5

CONTENTS:Chapter 0: Introduction0. Presentation: Objectives and evaluation.1. Relationship between Theory of Electromagnetic Fields and Theory of Circuits.

Chapter 1: Exitation signals of regular use

0. Clasification of signals. Characteristic values.1. Signal associations.

Chapter 2: Continuos current0. Definitions and basic laws.1. Continuos current circuits.2. Initiation to circuit analysis.3. Condenser and coil.

Chapter 3: Alternating current circuits0. Study of sinusoidal functions.1. Behavior of passive basic elements in permanent sinusoidal system.2. Calculation of power in alternating current circuits.3. Nets excitation (generators).4. Dependant generators.


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Sinusoids and Phasors

Chapter Objectives:

Understand the concepts of sinusoids and phasors. Apply phasors to circuit elements.

Introduce the concepts of impedance and admittance.

Learn about impedance combinations. Apply what is learnt to phase-shifters and AC

bridges.


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Alternating (AC) Waveforms

1fT

The term alternating indicates only that the waveform alternates between two prescribed levels in a settime sequence.

Instantaneous value: The magnitude of a waveform at any instant of time; denoted by the lowercase

letters (v1, v2).

Peak amplitude: The maximum value of the waveform as measured from its average (or mean) value,

denoted by the uppercase letters Vm.

Period (T): The time interval between successive repetitions of a periodic waveform.

Cycle: The portion of a waveform contained in one period of time.

Frequency: (Hertz) the number of cycles that occur in 1 s

The sinusoidal waveform is the only alternating waveform whose shape is

unaffected by the response characteristics of R, L, and C elements.

T


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SinusoidsThe sinusoidal wave form can be derived from the length of the vertical projection of a radius vectorrotating in a uniform circular motion about a fixed point.

The velocity with which the radius vector rotates about the center, called the angular velocity, can be

determined from the following equation:

The angular velocity () is:

Since () is typically provided in radians per second, the angle obtained using = t is usually in

radians.

The time required to complete one revolution is equal to the period (T) of the sinusoidal waveform. The

radians subtended in this time interval are 2.

2or 2 f

T

t

sinmV

cosmV


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SinusoidsThe basic mathematical format for the sinusoidal waveform is:

Vmsin

Vm is the peak value of the waveform and is the unit of measure for the horizontal axis.

The equation = t states that the angle through which the rotating vector will pass is determined by

the angular velocity of the rotating vector and the length of time the vector rotates.

For a particular angular velocity (fixed ), the longer the radius vector is permitted to rotate (that is, the

greater the value of t ), the greater will be the number of degrees or radians through which the vector will

pass. The general format of a sine wave can also be as:

sin( )m

V t


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Sketch of Vmsint.

SinusoidsA SINUSOID is a signal that has the form of the sine or cosine function.

The sinusoidal current is referred to as AC. Circuits driven by AC sources are referred to as AC Circuits.

(a)As a function oft. (b)As a function oft.

Vm is the AMPLITUDE of the sinusoid.is the ANGULAR FREQUENCY in radians/s.

fis the FREQUENCY in Hertz.

Tis the period in seconds.

12 andf fT

T Period


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Phase of Sinusoids A periodic function is one that satisfies v(t) = v(t + nT),for all tand for all integers n.

1

2f Hz fT

Only two sinusoidal values with the same frequency can becompared by their amplitude and phase difference.

If phase difference is zero, they are in phase; if phase difference isnot zero, they are out of phase.


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Phase of Sinusoids The terms leadand lag are used to indicate the relationship between two

sinusoidal waveforms of the same frequency plotted on the same set of axes.

The cosine curve is said to leadthe sine curve by 90.

The sine curve is said to lag the cosine curve by 90. 90 is referred to as the phase angle between the two waveforms.

When determining the phase measurement we first note that each sinusoidal

function has the same frequency, permitting the use of either waveform to determine

the period.

Since the full period represents a cycle of 360, the following ratio can be formed:


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Phase of Sinusoids Consider the sinusoidal voltage having phase , ( ) sin( )mv t V t

v2 LEADS v1 by phase .

v1 LAGS v2 by phase .

v1 and v2 are out of phase.


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(120 V at 60 Hz) versus (220 V at 50 Hz) ACIn North and South America the most common available ac supply is 120 V at 60 Hz, whilein Europe and the Eastern countries it is 220 V at 50 Hz.

Technically there is no noticeable difference between 50 and 60 cycles per second (Hz).

The effect of frequency on the size of transformers and the role it plays in the generation and

distribution of power was also a factor.

The fundamental equation for transformer design is that the size of the transformer is

inversely proportional to frequency.

A 50 HZ transformer must be larger than a 60 Hz (17% larger) sinusoidal voltage having

phase .

Higher frequencies result in concerns about arcing, increased losses in the transformer core

due to eddy current andhysteresis losses, and skin effect phenomena.

Larger voltages (such as 220 V) raise safety issues beyond those of 120 V.

Higher voltages result in lower current for the same demand, permitting the use of smaller

conductors.

Motors and power supplies, found in common home appliances and throughout the

industrial community, can be smaller in size if supplied with a higher voltage.


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Trigonometric Identities Sine and cosine form conversions.

2 2 -1

sin( ) sin cos cos sin

cos( ) cos cos sin sin

sin( 180 ) sin

cos( 180 ) cos

sin( 90 ) cos

cos( 90 ) sin

cos sin cos( )

Where

C= A and =tan

A B A B A B

A B A B A B

t t

t t

t t

t t

A t B t C t

BB

A

cos( 90 ) sint t

sin( 180 ) sint t

Graphically relating sine

and cosine functions.


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Figure shows a pair of waveforms v1and v2 on an oscilloscope. Eachmajor vertical division represents20 V and each majordivision on the horizontal (time)scale represents 20 ms. Voltage v1

leads. Prepare a phasor diagramusing v1 as reference. Determineequations for both voltages.


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EXERCISE Voltage and current are out of phase by 40, and voltage lags. Usingcurrent as the reference, sketch the phasor diagram and thecorresponding waveforms.

ct lecture 1

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