LECTURE NOTES

ON

ELECTRICAL CIRCUITS -II

2018 – 2019

II B. Tech III Semester

Mr. J Srinu Naick, Professor

CHADALAWADA RAMANAMMA ENGINEERING COLLEGE

(AUTONOMOUS)

Chadalawada Nagar, Renigunta Road, Tirupati – 517 506

Department of Electrical and Electronics Engineering

ELECTRICAL CIRCUITS- II

III Semester: EEE

Course Code Category Hours / Week Credits Maximum Marks

17CA02302 Foundation

L T P C CIA SEE Total

2 2 - 3 30 70 100

Contact Classes: 34 Tutorial Classes: 34 Practical Classes: Nil Total Classes: 68

OBJECTIVES:

The course should enable the students to:

I. Analyze star and delta connected three phase circuits and calculate active and reactive powers.

II. Understand the response of RL, RC and RLC circuits for DC and AC excitations and plot locus

diagrams.

III. Discuss the concept of network functions and calculate network parameters.

IV. Understand the simulation and design of various types of filters.

UNIT - I THREE PHASE CIRCUITS Classes: 07

Three phase circuits: Star and delta connections, phase sequence, relation between line and phase voltages

and currents in balanced star and delta circuits, three phase three wire and three phase four wire systems,

shifting of neutral point, analysis of balanced and unbalanced three phase circuits, measurement of active

and reactive power

UNIT - II DC AND AC TRANSIENT ANALYSIS Classes: 07

D.C Transient Analysis: Transient Response of R-L, R-C, R-L-C Series Circuits for D.C

Excitation-Initial Conditions-Solution Method Using Differential Equations and Laplace

Transforms, Response of R-L & R-C Networks to Pulse Excitation.

A.C Transient Analysis: Transient Response of R-L, R-C, R-L-C Series Circuits for Sinusoidal

Excitations-Initial Conditions-Solution Method Using Differential Equations and Laplace

Transform.

UNIT - III

TWO PORT NETWORK PARAMETERS & MAGNETIC CIRCUITS Classes: 07

Two port network parameters: Z, Y, ABCD, hybrid and inverse hybrid parameters,

conditions for symmetry and reciprocity, inter relationships of different parameters.

Magnetic Circuits: Faraday’s Laws of Electromagnetic Induction, Concept of Self and Mutual Inductance, Dot Convention, Coefficient of Coupling, Composite Magnetic Circuit-Analysis of Series and Parallel Magnetic Circuits, MMF Calculations.

UNIT - IV

FOURIER TRANSFORMS Classes: 07

Fourier Theorem- Trigonometric Form and Exponential Form of Fourier Series – Conditions of

Symmetry- Line Spectra and Phase Angle Spectra- Analysis of Electrical Circuits excited by Non

Sinusoidal sources of Periodic Waveforms. Fourier Integrals and Fourier Transforms – Properties

of Fourier Transforms and Application to Electrical Circuits.

UNIT - V FILTERS AND DIGITAL SIMULATION OF CIRCUITS

Classes: 06

Filters: Design Filter - Low pass, high pass, band pass, band elimination filters, introduction to active filter.

Digital simulation: MATLAB simulation and mathematical modeling of R, RL, RC and RLC circuits with

DC and AC excitations: steady state and transient analysis, time and frequency domain analysis, frequency

and phase spectra by Fourier analysis; basic test signals representation, filter design.

Text Books:

1. A Chakrabarthy, “Electric Circuits”, Dhanpat Rai & Sons, 6th Edition, 2010.

2. A Sudhakar, Shyammohan S Palli, “Circuits and Networks”, Tata Mc Graw Hill, 4th Edition, 2010.

3. M E Van Valkenberg, “Network Analysis”, PHI, 3rd Edition, 2014.

4. Rudrapratap, “Getting Started with MATLAB: A Quick Introduction for Scientists and Engineers”,

Oxford University Press, 1st Edition, 1999.

Reference Books:

1. John Bird, “Electrical Circuit Theory and technology”, Newnes, 2nd Edition, 2003.

2. C L Wadhwa, “Electrical Circuit Analysis including Passive Network Synthesis”, New Age

International, 2nd Edition, 2009.

3. David A Bell, “Electric Circuits”, Oxford University press, 7th Edition, 2009.

Web References:

1. https://www.igniteengineers.com

2. https://www.ishuchita.com/PDF/Matlab%20rudrapratap.pdf

3. https://www.ocw.nthu.edu.tw

4. https://www.uotechnology.edu.iq

5. https://www.crectirupati.com

E-Text Books:

1. https://www.bookboon.com/en/concepts-in-electric-circuits-ebook

2. https://www.jntubook.com

3. https://www.allaboutcircuits.com

4. https://www.archive.org

Course Home Page:

UNIT -1

3-PHASE CIRCUITS

Phase sequence

Phase sequence refers to the relation between voltages (or currents, as well) in a three-phase

system. The common representation of this relation is in terms of a phasor diagram, as below

The phasor diagram represents the phasor (or vector) relation of the three phase-ground voltages

(for simplicity, in a balanced system). The diagram is based on counter- clockwise rotation. In a

three phase system the voltage or current sinusoid attain peak values periodically one after

nother.The sinusoids are displaced 120 degrees from each other. So also phasors representing the

three sinusoids for voltage or current waves of three lines are phase displaced by 120 degrees.

The order in which the three phase voltages attain their positive peak values is known as the

phase sequence. Conventionally the three phases are designated as red-R, yellow-Y and blue-B

phases. The phase sequence is said to be RYB if R attains its peak or maximum value first with

respect to the reference as shown in the counter clockwise direction followed by Y phase 120°

later and B phase 240° later than the R phase. The phase sequence is said to be RBY if R is

followed by B phase120° later and Y phase 240° later than the R phase. By convention RYB is

considered as positive while the sequence RBY as negative. The phase sequence of the voltages

applied to a load is determined by the order in which the 3 phase lines are connected. The phase

sequence can be reversed by interchanging any one pair of lines without causing any change in

the supply sequence. Reversal of sequence results in reversal of the direction of rotation in case

of induction motor. Phase sequence is critical in measurements on power systems, and for

protective relaying, but perhaps most importantly, for rotating machines (so machines do not run

backwards). Modern microprocessor protective relays have a selectable phase-sequence setting

(often called the phase-rotation setting), so the relay adapts to the phase sequence without

ordinarily requiring changes to the wiring connections. In the historic electromechanical relays

(and meters), the wiring connections had to reflect the phase sequence to enable accurate

measurements and protection.

Inter connection of phases: The three phases can be inter connected either in star (Y)or in delta

(Δ).These connections result in a compact and a relatively economical system as the number of

conductors gets reduced by 33% for a three phase 4 - wire star system and by 50% for 3phase 3 -

wire star or delta systems when compared to independent connection of phases.

Star connection: In the Star Connection, the similar ends (either start or finish) of the three

windings are connected to a common point called star or neutral point. The three line conductors

run from the remaining three free terminals called line conductors. The wires are carried to the

external circuit, giving three phase, three wire star connected systems. However, sometimes a

fourth wire is carried from the star point to the external circuit, called neutral wire, forming

three phase, four wire star connected systems. The star connection is shown in the diagram

below.

Considering the above figure, the finish terminals a2, b2, and c2 of the three windings are

connected to form a star or neutral point. The three conductors named as R, Y and B run from

the remaining three free terminals as shown in the above figure. The current flowing through

each phase is called Phase current Iph, and the current flowing through each line conductor is

called Line Current IL. Similarly, the voltage across each phase is called Phase Voltage Eph,

and the voltage across two line conductors is known as the Line Voltage EL.

Relation Between Phase Voltage and Line Voltage in Star Connection:

The Star connection is shown in the figure below.

As the system is balanced, a balanced system means that in all the three phases, i.e., R, Y and B,

the equal amount of current flows through them. Therefore, the three voltages ENR, ENY and

ENB are equal in magnitude but displaced from one another by 120 degrees electrical. The

Phasor Diagram of Star Connection is shown below.

The arrowheads on the emfs and current indicate direction and not their actual direction at any

instant. Now,

There are two phase voltages between any two lines. Tracing the loop NRYN

To find the vector sum of ENY and –ENR, we have to reverse the vector ENR and add it with

ENY as shown in the phasor diagram above. Therefore,

Similarly,

Hence, in Star Connections Line voltage is root 3 times of phase voltage.

VRY=VNR-VNY= Vr m s∟0- Vr m s∟-120

=Vr m s[(cos 0+jsin0)-[cos(-120)+Jsin(-120)]]

Hence, in a 3 Phase system of Star Connections, the Line Current is equal to Phase Current.

Power in Star Connection

In a three phase AC circuit, the total True or Active power is the sum of the three phase power.

Or the sum of the all three phase powers is the Total Active or True Power.

Hence, total active or true power in a three phase AC system;

Total True or Active Power = 3 Phase Power

Or

P = 3 x VPH x IPH x CosФ ….. Eq … (1)

Good to Know: Where Cos Φ = Power factor = the phase angle between Phase Voltage and

Phase Current and not between Line current and line voltage. We know that the values of Phase

Current and Phase Voltage in Star Connection;

IL = IPH

VPH = VL /√3 ….. (From VL = √3 VPH)

Putting these values in power eq……. (1)

P = 3 x (VL/√3) x IL x CosФ …….…. (VPH = VL /√3)

P = √3 x√3 x (VL/√3) x IL x CosФ ….… {3 = √3x√3}

P = √3 x VL x IL x CosФ

Hence proved;

Power in Star Connection,

P = 3 x VPH x IPH x CosФ or

P = √3 x VL x IL x CosФ

Similarly,

Total Reactive Power = Q = √3 x VL x IL x SinФ

Good to know: Reactive Power of Inductive coil is taken as Positive (+) and that of a Capacitor

as Negative (-).

Also, the total apparent power of the three phases

Total Apparent Power = S = √3 x VL x IL

Or, S = √ (P2 + Q2)

Delta Connection In a 3 Phase System :

In this system of interconnection, the starting ends of the three phases or coils are connected to

the finishing ends of the coil. Or the starting end of the first coil is connected to the finishing end

of the second coil and so on (for all three coils) and it looks like a closed mesh or circuit as

shown in fig (1). In more clear words, all three coils are connected in series to form a close mesh

or circuit. Three wires are taken out from three junctions and the all outgoing currents from

junction assumed to be positive. In Delta connection, the three windings interconnection looks

like a short circuit, but this is not true, if the system is balanced, then the value of the algebraic

sum of all voltages around the mesh is zero. When a terminal is open, then there is no chance of

flowing currents with basic frequency around the closed mesh.

.

1. Line Voltages and Phase Voltages in Delta Connection

It is seen from fig 2 that there is only one phase winding between two terminals (i.e. there is one

phase winding between two wires). Therefore, in Delta Connection, the voltage between (any

pair of) two lines is equal to the phase voltage of the phase winding which is connected between

two lines. Since the phase sequence is R → Y → B, therefore, the direction of voltage from R

phase towards Y phase is positive (+), and the voltage of R phase is leading by 120°from Y

phase voltage. Likewise, the voltage of Y phase is leading by 120° from the phase voltage of B

and its direction is positive from Y towards B.

If the line voltage between; Line 1 and Line 2 = VRY ;

Line 2 and Line 3 = VYB : ; Line 3 and Line 1 = VBR

Then, we see that VRY leads VYB by 120° and VYB leads VBR by 120°.

Let‟s suppose,

VRY = VYB = VBR = VL …………… (Line Voltage)

Then VL = VPH

I.e. in Delta connection, the Line Voltage is equal to the Phase Voltage.

2. Line Currents and Phase Currents in Delta Connection

It will be noted from the below (fig-2) that the total current of each Line is equal to the vector

difference

between two phase currents flowing through that line. i.e.;

Current in Line 1= I1 = IR – IB

Current in Line 2 =I2 = IY – IR

Current in Line 3 =I3 = IB – IY

The current of Line 1 can be found by determining the vector difference between IR and IB and

we can do

that by increasing the IB Vector in reverse, so that, IR and IB makes a parallelogram. The

diagonal of that parallelogram shows the vector difference of IR and IB which is equal to Current

in Line 1= I1. Moreover, by reversing the vector of IB, it may indicate as (-IB), therefore, the

angle between IR and -IB (IB, when reversed = -IB) is 60°.

If, IR = IY = IB = IPH …. The phase currents Then;

The current flowing in Line 1 would be;

IL or IL = √(IR 2+IY 2 +2.IR*IY C0S 60 o)

i.e. In Delta Connection, The Line current is √3 times of Phase Current

Similarly, we can find the reaming two Line currents as same as above. i.e.,

I2 = IY – IR … Vector Difference = √3 IPH

I3 = IB – IY … Vector difference = √3 IPH

As, all the Line current are equal in magnitude i.e.

I1 = I2 = I3 = IL

Hence

IL = √3 IPH

It is seen from the fig above that;

The Line Currents are 120° apart from each other

Line currents are lagging by 30° from their corresponding Phase Currents

The angle Ф between line currents and respective line voltages is (30°+Ф), i.e. each line

current is

lagging by (30°+Ф) from the corresponding line voltage.

3. Power in Delta Connection

We know that the power of each phase

Power / Phase = VPH x IPH x CosФ

And the total power of three phases;

Total Power = P = 3 x VPH x IPH x CosФ ….. (1)

We know that the values of Phase Current and Phase Voltage in Delta Connection;

IPH = IL / /√3 ….. (From IL = √3 IPH)

VPH = VL

Putting these values in power eq……. (1)

P = 3 x VL x ( IL/√3) x CosФ …… (IPH = IL / /√3)

P = √3 x√3 x VL x ( IL/√3) x CosФ …{ 3 = √3x√3 }

P = √3 x VLx IL x CosФ …

Hence proved;

Power in Delta Connection,

P = 3 x VPH x IPH x CosФ …. or

P = √3 x VL x IL x CosФ

Good to Know: Where Cos Φ = Power factor = the phase angle between Phase Voltage and

Phase Current and not between Line current and line voltage.

Good to Remember:

In both Star and Delta Connections, The total power on balanced load is same.

I.e. total power in a Three Phase System = P = √3 x VL x IL x CosФ

Good to know:

Balanced System is a system where:

All three phase voltages are equal in magnitude

All phase voltages are in phase by each other i.e. 360°/3 = 120°

All three phase Currents are equal in magnitude

All phase Currents are in phase by each other i.e. 360°/3 = 120°

A three phase balanced load is a system in which the load connected across three phases are

identical.

Analysis of Balanced Three Phase Circuits:

It is always better to solve the balanced three phase circuits on per phase basis. When the three

phase supply voltage is given without reference to the line or phase value, then it is the line

voltage which is taken into consideration.

The following steps are given below to solve the balanced three phase circuits.

Step 1 – First of all draw the circuit diagram.

Step 2 – Determine XLP = XL/phase = 2πfL.

Step 3 – Determine XCP = XC/phase = 1/2πfC.

Step 4 – Determine XP = X/ phase = XL – XC

Step 5 – Determine ZP = Z/phase = √R2 + X2 P

Step 6 – Determine cosϕ = RP/ZP; the power factor is lagging when XLP > XCP and it is

leading when XCP > XLP.

Step 7 – Determine V phase. For star connection VP = VL/√3 and for delta connection VP = VL

Step 8 – Determine IP = VP/ZP.

Step 9 – Now, determine the line current IL. For star connection IL = IP and for delta connection

IL = √3 IP

Step 10 – Determine the Active, Reactive and Apparent power.

THREE-PHASE CONNECTIONS:

The sources and loads in a three-phase system can each be connected in either a wye (Y) or delta

(Δ) configuration. Note that the wye connections are line-to-neutral while the delta connections

are line-toline with no neutral. Also note the convention on the node designations (lowercase

letters at the source connections and uppercase letters at the load connections). Both the three

phase source and the three phase load can be connected either Wye or DELTA.

We have 4 possible connection types.

1. Y-Y connection

2. Y-Δ connection

3. Δ- Y connection

4. Δ -Δ connection

Balanced Δ connected load is more common & Y connected sources are more common.

BALANCED WYE-WYE CONNECTION

The balanced three-phase wye-wye connection is shown below. Note that the line impedance for

each of the individual phases in included in the circuit. The line impedances are assumed to be

equal for all three phases. The line currents (IaA, IbB and IcC) are designated according to the

source/load node naming convention. The source current, line current, and load current are all

one in the same current for a given phase in a wye-wye connection. Wye source Wye load

Assuming a positive phase sequence, the application of Kirchoff‟s voltage law around each

phase gives where Ztotal

Assuming a positive phase sequence, the application of Kirchoff‟s voltage law around each

phase gives

Van = Vrms∟0o = Ia(Zl+ZL) = Ia Ztotal = Ia ǀZtotal Iǀ∟θZ

Vbn = Vrms∟-120o = Ib(Zl+ZL) = Ib Ztotal = Ib ǀZtotal ǀ∟θZ

Vcn = Vrms∟120o = Ic(Zl+ZL) = Ic Ztotal = Ic ǀZtotal ǀ ∟θZ

Where Ztotal is the total impedance in each phase and θZ is the phase angle associated with the

total phase impedance. The preceding equations can be solved for the line currents.

Note that the line current magnitudes are equal and each line current lags the respective line-to-

neutral voltage by the impedance phase angle 2Z. Thus, the balanced voltages yield balanced

currents. The phasor diagram for the line currents and the line-to-neutral voltages is shown

below. If we lay the line-to neutral voltage phasors end to end, they form a closed triangle (the

same property is true for the line currents). The closed triangle shows that the sum of these

phasors is zero.

The fact that the line currents sum to zero in the balanced wye-wye connection shows that the

neutral current In is zero in this balanced system. Thus, the impedance of the neutral is

immaterial to the performance of the circuit under balanced conditions. However, any imbalance

in the system (loads, line impedances, source variations, etc.) will produce a non-zero neutral

current. In any balanced three-phase system (balanced voltages, balanced line and load

impedances), the resulting currents are balanced. Thus, there is no real need to analyze all three

phases. We may analyze one phase to determine its current, and infer the currents in the other

phases based on a simple balanced phase shift (120o phase difference between any two line

currents). This technique is known as the per phase analysis.

Single Phase Equivalent of Balanced Y-Y Connection:

Balanced three phase circuits can be analyzed on “per phase “ basis.We look at one phase, say

phase a and analyze the single phase equivalent circuit. Because the circuit is balanced, we can

easily obtain other phase values using their phase relationships.

In addition to the wye-wye three-phase connection, there are three other possible configurations

of wye and delta sources and loads. The most efficient way to handle three-phase circuits

containing delta sources and/or loads is to transform all delta connections into wye connections.

Delta Source - Source Voltage and Source Current Calculations: Source Voltage:

Given that a delta source is defined in terms of line-to-line voltages while the wye source is

defined in terms of line-to-neutral voltages, we can use the previously determined relationship

between line-to-line voltages and line-to neutral voltages to perform the transformation.

Two Wattmeter Method of Power Measurement:

Two Wattmeter Method can be employed to measure the power in a 3 phase, 3 wire star or

delta connected balanced or unbalanced load. In Two wattmeter method the current coils of the

wattmeter are connected with any two lines, say R and Y and the potential coil of each wattmeter

is joined across the same line, the third line i.e. B as shown below in the figure A.

Measurement of Power by Two Wattmeter Method in Star Connection:

Considering the above figure A in which Two Wattmeter W1 and W2 are connected, the

instantaneous Ncurrent through the current coil of Wattmeter, W1 is given by the equation

shown below.

Instantaneous potential difference across the potential coil of Wattmeter, W1 is given as

Instantaneous power measured by the Wattmeter, W1 is

The instantaneous current through the current coil of Wattmeter, W2 is given by the equation

Instantaneous potential difference across the potential coil of Wattmeter, W2 is given as

Instantaneous power measured by the Wattmeter, W2 is

Therefore, the Total Power Measured by the Two Wattmeters W1 and W2 will be obtained by

adding the equation (1) and (2).

Where, P is the total power absorbed in the three loads at any instant.

Hence, the total instantaneous power absorbed by the three loads Z1, Z2 and Z3, are equal to the

sum of the powers measured by the Two watt meters, W1 and W2.

Measurement of Power by Two Wattmeter Method in Delta Connection

Considering the delta connected circuit shown in the figure below,

The instantaneous current through the coil of the Wattmeter, W1 is given by the equation

Instantaneous voltage measured by the Wattmeter, W1 will be

Therefore, the instantaneous power measured by the Wattmeter, W1 will be given as

The instantaneous current through the current coil of the Wattmeter, W2 is given as

The instantaneous potential difference across the potential coil of Wattmeter, W2 is

Therefore, the instantaneous power measured by Wattmeter, W2 will be

Hence, to obtain the total power measured by the Two Wattmeter the two equations, i.e. equation

(3) and (4) has to be added.

Where, P is the total power absorbed in the three loads at any instant. The power measured by

the Two Wattmeter at any instant is the instantaneous power absorbed by the three loads

connected in three phases. In fact, this power is the average power drawn by the load since the

Wattmeter reads the average power because of the inertia of their moving system.

Alternate method:

THREE PHASE REAL POWER MEASUREMENTS

The three phase real power is given by,

UNIT-2

Transients

Introduction

There are many reasons for studying initial and final conditions. The most important reason is

that the initial and final conditions evaluate the arbitrary constants that appear in the general

solution of a differential equation. In this chapter, we concentrate on finding the change in

selected variables in a circuit when a switch is thrown open from closed position or vice versa.

The time of throwing the switch is considered to be = 0, and we want to determine the value of

the variable at = 0 and at = 0+, immediately before and after throwing the switch. Thus a

switched circuit is an electrical circuit with one or more switches that open or close at time = 0.

We are very much interested in the change in currents and voltages of energy storing elements

after the switch is thrown since these variables along with the sources will dictate the circuit

behaviour for 0. Initial conditions in a network depend on the past history of the circuit (before =

0 ) and structure of the network at = 0+, (after switching). Past history will show up in the form

of capacitor voltages and inductor currents. The computation of all voltages and currents and

their derivatives at = 0+ is the main aim of this chapter.

TRANSIENT RESPONSE OF RC CIRCUITS Ideal and real capacitors: An ideal capacitor has an infinite dielectric resistance and plates (made

of metals) that have zero resistance. However, an ideal capacitor does not exist as all dielectrics

have some leakage current and all capacitor plates have some resistance. A capacitor’s of how

much charge (current) it will allow to leak through the dielectric medium. Ideally, a

charged capacitor is not supposed to allow leaking any current through the dielectric medium and

also assumed not to dissipate any power loss in capacitor plates resistance. Under this situation,

the model as shown in fig. 10.16(a) represents the ideal capacitor. However, all real or practical

capacitor leaks current to some extend due to leakage resistance of dielectric medium. This

leakage resistance can be visualized as a resistance connected in parallel with the capacitor and

power loss in capacitor plates can be realized with a resistance connected in series with

capacitor. The model of a real capacitor is shown in fig. Let us consider a simple series

RC−circuit shown in fig. 10.17(a) is connected through a switch ‘S’ to a constant voltage source

.

The switch ‘S’ is closed at time‘t=0’ It is assumed that the capacitor is initially charged with a

voltage and the current flowing through the circuit at any instant of time ‘’ after closing the

switch is Current decay in source free series RL circuit: -

At t = 0- , , switch k is kept at position ‘a’ for very long time. Thus, the network is in steady

state. Initial current through inductor is given as,

Because current through inductor can not change instantaneously Assume that at t = 0 switch k is

moved to position 'b'. Applying KVL,

Rearranging the terms in above equation by separating variables

Integrating both sides with respect to corresponding variables

Where k’ is constant of integration. To find- k’: Form equation 1, at t=0, i=I0 Substituting the values in equation 3

Substituting value of k’ from equation 4 in

fig. shows variation of current i with respect to time

From the graph, H is clear that current is exponentially decaying. At point P on graph. The

current value is (0.363) times its maximum value. The characteristics of decay are determined by

values R and L which are two parameters of network. The voltage across inductor is given by

TRANSIENT RESPONSE OF RL CIRCUITS:

So far we have considered dc resistive network in which currents and voltages were independent

of time. More specifically, Voltage (cause input) and current (effect output) responses displayed

simultaneously except for a constant multiplicative factor (VR). Two basic passive elements

namely, inductor and capacitor are introduced in the dc network. Automatically, the question will

arise whether or not the methods developed in lesson-3 to lesson-8 for resistive circuit analysis

are still valid. The voltage/current relationship for these two passive elements are defined by the

derivative (voltage across the inductor

Our problem is to study the growth of current in the circuit through two stages, namely; (i)

dc transient response (ii) steady state response of the system

D.C Transients: The behavior of the current and the voltage in the circuit switch is closed

until it reaches its final value is called dc transient response of the concerned circuit. The

response of a circuit (containing resistances, inductances, capacitors and switches) due to sudden

application of voltage or current is called transient response. The most common instance of a

transient response in a circuit occurs when a switch is turned on or off –a rather common event

in an electric circuit

Growth or Rise of current in R-L circuit

To find the current expression (response) for the circuit shown in fig. 10.6(a), we can write the

KVL equation around the circuit

The table shows how the current i(t) builds up in a R-L circuit.

Consider network shown in fig. the switch k is moved from position 1 to 2 at

reference time t = 0.

Now before switching take place, the capacitor C is fully charged to V volts and it

discharges through resistance R. As time passes, charge and hence voltage across capacitor i.e.

Vc decreases gradually and hence discharge current also decreases gradually from maximum to

zero exponentially.

After switching has taken place, applying kirchoff’s voltage law,

Where VR is voltage across resistor and VC is voltage across capacitor.

Above equation is linear, homogenous first order differential equation. Hence rearranging we

have,

Integrating both sides of above equation we have

Now at t = 0, VC =V which is initial condition, substituting in equation we have,

Where Q is total charge on capacitor

Similarly at any instant, VC = q/c where q is instantaneous charge.

Thus charge behaves similarly to voltage across capacitor.

Now discharging current i is given by

But VR = VC when there is no source in circuit.

The above expression is nothing but discharge current of capacitor. The variation of this current

with respect to time is shown in fig.

This shows that the current is exponentially decaying. At point P on the graph. The current value

is (0.368) times its maximum value. The characteristics of decay are determined by values R and

C, which are 2 parameters of network. For this network, after the instant t = 0, there is no driving

voltage source in circuit, hence it is called un driven RC circuit.

TRANSIENT RESPONSE OF RLC CIRCUITS

In the preceding lesson, our discussion focused extensively on dc circuits having resistances

with either inductor L or capacitor C (i.e., single storage element) but not both. Dynamic

response of such first order system has been studied and discussed in detail. The presence of

resistance, inductance, and capacitance in the dc circuit introduces at least a second order

differential equation or by two simultaneous coupled linear first order differential equations. We

shall see in next section that the complexity of analysis of second order circuits increases

significantly when compared with that encountered with first order circuits. Initial conditions for

the circuit variables and their derivatives play an important role and this is very crucial to

analyze a second order dynamic system.

Response of a series R-L-C circuit

Consider a series RLC circuit as shown in fig.11.1, and it is excited with a dc voltage

source Vs. Applying around the closed path for,

The current through the capacitor can be written as Substituting the current ‘’expression in

eq.(11.1) and rearranging the terms,

The above equation is a 2nd-order linear differential equation and the parameters associated with

the differential equation are constant with time. The complete solution of the above differential

equation has two components; the transient response and the steady state response.

Mathematically, one can write the complete solution as

Since the system is linear, the nature of steady state response is same as that of forcing function

(input voltage) and it is given by a constant value. Now, the first part of the total response is

completely dies out with time while and it is defined as a transient or natural response of the

system. The natural or transient response (see Appendix in Lesson-10) of second order

differential equation can be obtained from the homogeneous equation (i.e., from force free

system) that is expressed by

and solving the roots of this equation (11.5) on that associated with transient part of the complete

solution (eq.11.3) and they are given below.

The roots of the characteristic equation are classified in three groups depending upon the

values of the parameters, RL and of the circuit

Case-A (over damped response): That the roots are distinct with negative real parts. Under this

situation, the natural or transient part of the complete solution is written as

and each term of the above expression decays exponentially and ultimately reduces to zero as

and it is termed as over damped response of input free system. A system that is over damped

responds slowly to any change in excitation. It may be noted that the exponential term

t→∞11tAeαtakes longer time to decay its value to zero than the term21tAeα. One can introduce

a factor ξ that provides information about the speed of system response and it is defined by

damping ratio

Discharging of capacitor through resistor in source free series RC circuit. Or Analysis

of un driven or source free series RC circuits.

Consider network shown in fig. the switch k is moved from position 1 to 2 at

reference time t = 0. Now before switching take place, the capacitor C is fully charged to V volts

and it discharges through resistance R. As time passes, charge and hence voltage across capacitor

i.e. Vc decreases gradually and hence discharge current also decreases gradually from maximum

to zero exponentially. After switching has taken place, applying kirchoff’s voltage law,

Where VR is voltage across resistor and VC is voltage across capacitor.

Above equation is linear, homogenous first order differential equation. Hence rearranging we

have,

The above expression is nothing but discharge current of capacitor. The variation of this current

with respect to time is shown in fig. This shows that the current is exponentially decaying. At

point P on the graph. The current value is (0.368) times its maximum value. The characteristics

of decay are determined by values R and C, which are 2 parameters of network. For this

network, after the instant t = 0, there is no driving voltage source in circuit, hence it is called un

driven RC circuit.

Analysis of undriven series RL circuits (or) Analysis of source free series RL circuits

Current decay in source free series RL circuit: -

At t = 0- , switch k is kept at position ‘a’ for very long time. Thus, the network is in steady state.

Initial current through inductor is given as,

Because current through inductor can not change instantaneously

Assume that at t = 0 switch k is moved to position ‘b’,

From the graph, H is clear that current is exponentially decaying. At point P on graph. The

current value is (0.363) times its maximum value. The characteristics of decay are determined by

values R and L which are two parameters of network.

The voltage across inductor is given by

UNIT-3

TWO PORT NETWORK & MAGNETIC CIRCUITS

INTRODUCTION

A pair of terminals through which a current may enter or leave a network is known as a port.

Two-terminal devices or elements (such as resistors, capacitors, and inductors) result in one-port

networks. Most of the circuits we have dealt with so far are two-terminal or one-port circuits,

represented in Figure 2(a). We have considered the voltage across or current through a single

pair of terminals—such as the two terminals of a resistor, a capacitor, or an inductor. We have

also studied four-terminal or two-port circuits involving op amps, transistors, and transformers,

as shown in Figure 2(b). In general, a network may have n ports. A port is an access to the

network and consists of a pair of terminals; the current entering one terminal leaves through the

other terminal so that the net current entering the port equals zero.

A pair of terminals through which a current may enter or leave a network is known as a port. A

port is an access to the network and consists of a pair of terminals; the current entering one

terminal leaves through the other terminal so that the net current entering the port equals zero.

There are several reasons why we should study two-ports and the parameters that describe them.

For example, most circuits have two ports. We may apply an input signal in one port and obtain

an output signal from the other port. The parameters of a two-port network completely describes

its behavior in terms of the voltage and current at each port. Thus, knowing the parameters of a

two port network permits us to describe its operation when it is connected into a larger network.

Two-port networks are also important in modeling electronic devices and system components.

For example, in electronics, two-port networks are employed to model transistors and Op-amps.

Other examples of electrical components modeled by two-ports are transformers and

transmission lines. Four popular types of two-ports parameters are examined here: impedance,

admittance, hybrid, and transmission. We show the usefulness of each set of parameters,

demonstrate how they are related to each other

IMPEDANCE PARAMETERS:

Impedance and admittance parameters are commonly used in the synthesis of filters. They are

also useful in the design and analysis of impedance-matching networks and power distribution

networks. We discuss impedance parameters in this section and admittance parameters in the

next section. two-port network may be voltage-driven as in Figure 3 (a) or current-driven as in

Figure 3(b). From either Figure 3(a) or (b), the terminal voltages can be related to the terminal

currents as

V1=Z11I1+Z12I2

V2=Z21I1+Z22I2

Or in matrix form as

Where the z terms are called the impedance parameters, or simply z parameters, and have units

of ohms.

The values of the parameters can be evaluated by setting I1 = 0 (input port open-circuited) or I2

= 0 (output port open-circuited).

Since the z parameters are obtained by open-circuiting the input or output port, they are also

called the open-circuit impedance parameters. Specifically,

z11 = Open-circuit input impedance

z12 = Open-circuit transfer impedance from port 1 to port 2 z21 = Open-circuit transfer

impedance from port 2 to port 1 z22 = Open-circuit output impedance

We obtain z11 and z21 by connecting a voltage V1 (or a current source I1) to port 1 with port 2

open-circuited as in Figure 4 and finding I1 and V2; we then get

The above procedure provides us with a means of calculating or measuring the z parameters.

Sometimes z11 and z22 are called driving-point impedances, while z21 and z12 are called

transfer impedances. A driving-point impedance is the input impedance of a two-terminal (one-

port) device. Thus, z11 is the input driving-point impedance with the output port open-circuited,

while z22 is the output driving-point impedance with the input port open circuited. When z11 =

z22, the two-port network is said to be symmetrical. This implies that the network has mirror like

symmetry about some center line; that is, a line can be found that divides the network into two

similar halves. When the two-port network is linear and has no dependent sources, the transfer

impedances are equal (z12 = z21), and the two-port is said to be reciprocal. This means that if

the points of excitation and response are interchanged, the transfer impedances remain the same.

A two-port is reciprocal if interchanging an ideal voltage source at one port with an ideal

ammeter at the other port gives the same ammeter reading.

ADMITTANCE PARAMETERS:

In the previous section we saw that impedance parameters may not exist for a two-port network.

So there is a need for an alternative means of describing such a network. This need is met by the

second set of parameters, which we obtain by expressing the terminal currents in terms of the

terminal voltages. In either Figure 5(a) or (b), the terminal currents can be expressed in terms of

the terminal voltages as

Determination of the y parameters: (a) finding y11 and y21, (b) finding y12 and y22.

I1=Y11V1+Y12V2

I2=Y21V1+Y22V2

The y terms are known as the admittance parameters (or, simply, y parameters) and have units

of iemens The values of the parameters can be determined by setting V1 = 0 (input port short-

circuited) or V2 = 0 (output port short-circuited). Thus,

Since the y parameters are obtained by short-circuiting the input or output port, they are also

called the short-circuit admittance parameters. Specifically,

y11 = Short-circuit input admittance

y12 = Short-circuit transfer admittance from port 2 to port 1 y21 = Short-circuit transfer

admittance from port 1 to port 2 y22 = Short-circuit output admittance

We obtain y11 and y21 by connecting a current I1 to port 1 and short-circuiting port 2 and

finding V1And I2.

Similarly, we obtain y12 and y22 by connecting a current source I2 to port 2 and short-circuiting

port 1 and finding I1 and V2, and then getting

This procedure provides us with a means of calculating or measuring the y parameters. The

impedance and admittance parameters are collectively referred to as immittance parameters

HYBRID PARAMETERS:

The z and y parameters of a two-port network do not always exist. So there is a need for

developing another set of parameters. This third set of parameters is based on making V1 and I2

the dependent variables. Thus, we obtain

Or in matrix form,

The h terms are known as the hybrid parameters (or, simply, h parameters) because they are a

hybrid combination of ratios. They are very useful for describing electronic devices such as

transistors; it is much easier to measure experimentally the h parameters of such devices than to

measure their z or y parameters. The hybrid parameters are as follows.

It is evident that the parameters h11, h12, h21, and h22 represent an impedance, a voltage gain, a

current gain, and an admittance, respectively. This is why they are called the hybrid parameters.

To be specific,

h11 = Short-circuit input impedance h12 = Open-circuit reverse voltage gain h21 = Short-circuit

forward current gain h22 = Open-circuit output admittance

The procedure for calculating the h parameters is similar to that used for the z or y parameters.

We apply a voltage or current source to the appropriate port, short-circuit or open-circuit the

other port, depending on the parameter of interest, and perform regular circuit analysis

TRANSMISSION PARAMETERS:

Since there are no restrictions on which terminal voltages and currents should be considered

independent and which should be dependent variables, we expect to be able to generate many

sets of parameters. Another set of parameters relates the variables at the input port to those at the

output port.

Thus,

The above Equations are relating the input variables (V1 and I1) to the output variables (V2 and

−I2). Notice that in computing the transmission parameters, −I2 is used rather than I2, because

the current is considered to be leaving the network, as shown in Figure 6. This is done merely for

conventional reasons; when you cascade two-ports (output to input), it is most logical to think of

I2 as leaving the two-port. It is also customary in the power − industry to consider I2 as leaving

the two-port.

Terminal variables used to define the ABCD parameters.

The two-port parameters in above Eqs. provide a measure of how a circuit transmits voltage and

current from a source to a load. They are useful in the analysis of transmission lines (such as

cable and fiber) because they express sending-end variables (V1 and I1) in terms of the

receiving-end variables (V2 and −I2). For this reason, they are called transmission parameters.

They are also known as ABCD parameters. They are used in the design of telephone systems,

microwave networks, and radars.

The transmission parameters are determined as

Thus, the transmission parameters are called, specifically,

A = Open-circuit voltage ratio

B= Negative short-circuit transfer impedance

C = Open-circuit transfer admittance

D = Negative short-circuit current ratio

A and D are dimensionless, B is in ohms, and C is in siemens. Since the transmission parameters

provide a direct relationship between input and output variables, they are very useful in cascaded

networks.

Condition of symmetry:-

A two port network is said to be symmetrical if the ports can be interchanged without port

voltages and currents

Condition of reciprocity:-

A two port network is said to be reciprocal, if the rate of excitation to response is invariant to an

interchange of the position of the excitation and response in the network. Network containing

resistors, capacitors and inductors are generally reciprocal

Condition for reciprocity and symmetry in two port parameters

In Z parameters

A network is termed to be reciprocal if the ratio of the response to the excitation remains

unchanged even if the positions of the response as well as the excitation are interchanged. A two

port network is said to be symmetrical it the input and the output port can be interchanged

without altering the port voltages or currents.

Two-port networks may be interconnected in various configurations, such as series, parallel, or

cascade connection, resulting in new two-port networks. For each configuration, certain set of

parameters may be more useful than others to describe the network. A series connection of two

two-port networks a and b with open-circuit impedance parameters Za and Zb, respectively. In

this configuration, we use the Z-parameters since they are combined as a series connection of

two impedances.

The Z-parameters of the series connection are Z 11= Z11A + Z11B

Or in the matrix form [Z]=[ZA]+[ZB]

Parallel Connection

[Y] = [YA] + [YB]

Cascade Connection

RELATIONSHIPS BETWEEN PARAMETERS:

Since the six sets of parameters relate the same input and output terminal variables of the same

two-port network, they should be interrelated. If two sets of parameters exist, we can relate one

set to the other set. Let us demonstrate the process with two examples.

Given the z parameters, let us obtain the y parameters

Magnetic Circuits

Introduction

Although the lines of magnetic flux have no physical existence, they do form a very convenient

and useful basis for explaining various magnetic effects and to calculate the magnitudes of

various magnetic quantities. The complete closed path followed by any group of magnetic flux

lines is referred as magnetic circuit. The lines of magnetic flux never intersect, and each line

forms a closed path. Whenever a current is flowing through the coil there will be magnetic flux

produced and the path followed by the magnetic flux is known as magnetic circuit. The operation

of all the electrical devices like generators, motors, transformers etc. depend upon the magnetism

produced by this magnetic circuit. Therefore, to obtain the required characteristics of these

devices, their magnetic circuits have to be designed carefully.

Magneto Motive Force (MMF)

The magnetic pressure which sets up or tends to set up magnetic flux in a magnetic circuit is

known as MMF.

1. Magneto motive force is the measure of the ability of a coil to produce flux.

2. The magnetic flux is due to the existence of the MMF caused by a current flowing through a

coil having no. of turns.

3. A coil with ‘N’ turns carrying a current of ‘I’ amperes represents a magnetic circuit producing

an MMF of NI MMF=NI

4. Units of MMF = Ampere turns (AT)

Magnetic Flux:

1. The amount of magnetic lines of force set-up in a magnetic circuit is called magnetic flux.

2. The magnetic flux that is established in a magnetic circuit is proportional to the MMF and the

proportional constant is the reluctance of the magnetic circuit.

Magnetic flux α MMF

3. The unit of magnetic flux is Weber.

Reluctance:

1. The opposition offered to the flow of magnetic flux in a magnetic circuit is called reluctance

2. Reluctance of a magnetic circuit is defined as the ratio of magneto motive force to the flux

established.

3. Reluctance depends upon length(l), area of cross-section(a) and permeability of the material

that makes up the magnetic circuit.(S l, S a, S l/a )

4. The unit of reluctance is AT/ Wb

Magnetic field strength(H)

1. If the magnetic circuit is homogeneous, and of uniform cross-sectional area, the magnetic field

strength is defined as the magneto motive force per unit length of magnetic circuit.

2. The unit of magnetic field strength is AT/m

Magnetic flux density(B)

1. The magnetic flux density in any material is defined as the magnetic flux established per unit

area of cross-section.

2.The unit of magnetic flux density is wb/m2 or TESLA

Relative permeability

1. It is defined as the ratio of flux density established in magnetic material to the flux density

established in air or vacuum for the same magnetic field strength.

INTRODUCTION TO ELECTROMAGNETIC INDUCTION::

When a conductor moves in a magnetic field, an EMF is generated; when it carries current in a

magnetic field, a force is produced. Both of these effects may be deduced from one of the most

fundamental principles of electromagnetism, and they provide the basis for a number of devices

in which conductors move freely in a magnetic field. It has already been mentioned that most

electrical machines employ a different form of construction.

FARADAY'S LAW OF ELECTROMAGNETIC INDUCTION:

In 1831, Michael Faraday, an English physicist gave one of the most basic laws of

electromagnetism called Faraday's law of electromagnetic induction. This law explains the

working principle of most of the electrical motors, generators, electrical transformers and

inductors. This law shows the relationship between electric circuit and magnetic field.

FARADAY'S FIRST LAW

Any change in the magnetic field of a coil of wire will cause an emf to be induced in the coil.

This emf induced is called induced emf and if the conductor circuit is closed, the current will

also circulate through the circuit and this current is called induced current. Method to change

magnetic field:

By moving a magnet towards or away from the coil

By moving the coil into or out of the magnetic field.

By changing the area of a coil placed in the magnetic field

By rotating the coil relative to the magnet.

Faraday's Second Law

It states that the magnitude of emf induced in the coil is equal to the rate of change of flux that

linkages with the coil. The flux linkage of the coil is the product of number of turns in the coil

and flux associated with the coil.

Faraday Law Formula

Where, flux Φ in Wb = B*A

B = magnetic field strength

A = area of the coil

Lenz's law obeys Newton's third law of motion (i.e. to every action there is always an equal and

opposite reaction) and the conservation of energy (i.e. energy may neither be created nor

destroyed and therefore the sum of all the energies in the system is a constant).

Lenz's law : It states that when an emf is generated by a change in magnetic flux according to

Faraday's Law, the polarity of the induced emf is such, that it produces a current that's

magnetic field opposes the change which produces it.

The negative sign used in Faraday's law of electromagnetic induction, indicates that the induced

emf and the change in magnetic flux have opposite signs.

There exists a definite relation between the direction of the induced current, the direction of

the flux and the direction of motion of the conductor. The direction of the induced current may

be found easily by applying either Fleming’s Right-hand Rule

There exists a definite relation between the direction of the induced current, the

direction of the flux and the direction of motion of the conductor. The direction of the

induced current may be found easily by applying either Fleming's Left-hand Rule

HOW TO INCREASE EMF INDUCED IN A COIL:

By increasing the number of turns in the coil i.e. N, from the formulae derived above it is

easily seen that if number of turns in a coil is increased, the induced emf also gets

increased.

By increasing magnetic field strength i.e. B surrounding the coil- Mathematically,

if magnetic field increases, flux increases and if flux increases emf induced will also get

increased. Theoretically, if the coil is passed through a stronger magnetic field, there will

be more lines of force for coil to cut and hence there will be more emf induced.

By increasing the speed of the relative motion between the coil and the magnet - If the

relative speed between the coil and magnet is increased from its previous value, the coil

will cut the lines of flux at a faster rate, so more induced emf would be produced.

APPLICATIONS OF FARADAY'S LAW:

Faraday law is one of the most basic and important laws of electromagnetism. This law finds its

application in most of the electrical machines, industries and medical field etc.

Electrical Transformers work on Faraday's law of mutual induction.

The basic working principle of electrical generator is Faraday's law of electromagnetic

induction.

The Induction cooker is a fastest way of cooking. It also works on principle of mutual

induction. When current flows through the coil of copper wire placed below a cooking container,

it produces a changing magnetic field. This alternating or changing magnetic field induces an

emf and hence the current in the conductive container, and we know that flow of current always

produces heat in it.

Electromagnetic Flow Meter is used to measure velocity of certain fluids. When a magnetic

field is applied to electrically insulated pipe in which conducting fluids are flowing, then

according to Faraday's law, an electromotive force is induced in it. This induced emf is

proportional to velocity of fluid flowing.

Form the bases of Electromagnetic theory; Faraday's idea of lines of force is used in well

known Maxwell's equations. According to Faraday's law, change in magnetic field gives rise to

change in electric field and the converse of this is used in Maxwell's equations. It is also used

in musical instruments like electric guitar, electric violin etc.

SELF INDUCTANCE:

Inductance is the property of electrical circuits containing coils in which a change in the

electrical current induces an electromotive force (emf). This value of induced emf opposes the

change in current in electrical circuits and electric current 'I' produces a magnetic field which

generates magnetic flux acting on the circuit containing coils. The ratio of the magnetic flux to

the current is called the self-inductance.

𝐿 =𝜓/ 𝐼 = 𝑁∅

The phenomenon of inducing an emf in a coil whenever a current linked with coil changes is

called induction. Here units of L are Weber per ampere which is equivalent to Henry.

‘Ø’ denotes the magnetic flux through the area spanned by one loop, ‘I’ is the current flowing

through the coil and N is the number of loops (turns) in the coil.

MUTUAL INDUCTANCE: Mutual Inductance is the ratio between induced Electro Motive

Force across a coil to the rate of change of current of another adjacent coil in such a way that two

coils are in possibility of flux linkage. Mutual induction is a phenomenon when a coil gets

induced in EMF across it due to rate of change current in adjacent coil in such a way that the flux

of one coil current gets linkage of another coil. Mutual inductance is denoted as (M), it is called

co-efficient of Mutual Induction between two coils

Mutual inductance for two coils gives the same value when they are in mutual induction with

each other. Induction in one coil due to its own rate of change of current is called self inductance

(L), but due to rate of change of current of adjacent coil it gives mutual inductance (M)

From the above figure, first coil carries current i1 and its self inductance is L1. Along with its

self

inductance it has to face mutual induction due to rate of change of current i2 in the second coil.

Same case happens in the second coil also. Dot convention is used to mark the polarity of the

mutual induction. Suppose two coils are placed nearby

Coil 1 carries I1 current having N1 number of turn. Now the flux density created by the coil 1 is

B1. Coil 2 with N2 number of turn gets linked with this flux from coil 1. So flux linkage in coil 2

is

N2. φ21 [φ21 is called leakage flux in coil 2 due to coil 1].

Now it can be written from these equations,

DOT CONVENTION:

Dot convention is used to determine the polarity of a magnetic coil in respect of other

Magnetic coil.

Dot convention is normally used to determine the total or equivalent inductance (Leq).

Suppose two coils are in series with same place dot.

When 2 dots are at the same place of both inductors (while at entering place or leaving

Place) as shown in below figure i.e. the total mutual inductance gets aided (added)

Mutual inductance between them is positive.

SERIES OPPOSING:

Suppose two coils are in series with opposite place dot.

When 2 dots are at the opposite place of both inductors(while one at entering place and other

at leaving place)as shown in below figure i.e. the total mutual inductance gets differed

Mutual inductance between them is negative.

PARALLEL AIDING:

Suppose two coils are in parallel with same place dot.

When 2 dots are at the same place of both inductors(while at entering place or leaving

place)as shown in below figure i.e. the total mutual inductance gets aided(added)

PARALLEL OPPOSING:

Suppose two coils are in parallel with opposite place dot.

When 2 dots are at the opposite place of both inductors(while one at entering place and

other at leaving place)as shown in below figure i.e. the total mutual inductance gets

differed

COEFFICIENT OF COUPLING:

The fraction of magnetic flux produced by the current in one coil that links with the other coil is

called coefficient of coupling between the two coils. It is denoted by (k).

Two coils are taken coil A and coil B, when current flows through one coil it produces flux; the

whole flux may not link with the other coil coupled, and this is because of leakage flux by a

fraction (k) known as Coefficient of Coupling.

k=1 when the flux produced by one coil completely links with the other coil and is called

magnetically tightly coupled. k=0 when the flux produced by one coil does not link at all with

the other coil and thus the coils are said to be magnetically isolated.

DERIVATION:

Consider two magnetic coils A and B. When current I1 flows through coil A.

The above equation (A) shows the relationship between mutual inductance and self inductance

between two the coils

SERIES MAGNETIC CIRCUIT:

A series magnetic circuit is analogous to a series electric circuit. A magnetic circuit is said

to be series, if the same flux is flowing through all the elements connected in a magnetic circuit.

Consider a circular ring having a magnetic path of ‘l’ meters, area of cross section ‘a’ m2 with a

mean radius of ‘R’ meters having a coil of ‘N’ turns carrying a current of ‘I’ amperes wound

uniformly as shown in below fig

The flux produced by the circuit is given by

Magnetic flux= In the above equation NI is the MMF of the magnetic circuit, which is analogous

to EMF in the electrical circuit.

PARALLEL MAGNETIC CIRCUIT

A magnetic circuit which has more than one path for magnetic flux is called a parallel

magnetic circuit. It can be compared with a parallel electric circuit which has more than one path

for electric current. The concept of parallel magnetic circuit is illustrated in fig. 2. Here a coil of

‘N’ turns wounded on limb ‘AF’ carries a current of ‘I’ amperes. The magnetic flux ‘φ1’ set up

by the coil divides at ‘B’ into two paths namely

Magnetic flux passes ‘φ2’ along the path ‘BE’

Magnetic flux passes ‘φ3’ along the path ‘BCDE’ i.e φ1= φ2 + φ3

The magnetic paths ‘BE’ and ‘BCDE’ are in parallel and form a parallel magnetic circuit. The

AT

required for this parallel circuit is equal to AT required for any one of the paths. Let S1=

reluctance of path EFAB

Let, S1= reluctance of path EFAB

S2= reluctance of path BE

S3= reluctance of path BCDE

Total MMF= MMF for path EFAB + MMF for path BE or path BCD

NI=Ф1S1+ Ф2 S2= Ф1 S1+ Ф3 S3

COMPOSITE MAGNETIC CIRCUIT:

Consider a magnetic circuit which consists of two specimens of iron arranged as shown in

figure. Let ℓ1 and ℓ2 be the mean lengths of specimen 1 and specimen2 in meters, A1 and A2 be

their respective cross sectional areas in square meters, and 1 and 2 be their respective relative

permeability’s.

The reluctance of specimen 1 is given as

and that for specimen 2 is

If a coil of N turns carrying a current I is wound on the specimen 1 and if the magnetic flux is

assumed to be confined to iron core then the total reluctance is given by the sum of the

individual reluctances S1 and S2. This is equivalent to adding the resistances of a series

circuit. Thus the total reluctance is given by

And the total flux is given by

Problem :Sketch the dotted equivalent circuit for the coupled coil shown in the fig. and find

the equivalent inductive?

Solution: The dotted equivalent circuit is

The equivalent inductive reactance is

j Xeq = j3 + j5 + j6 - 2 x j2 -2x j3+ 2x j4) = j14 - j2 = j12

Problem: Sketch the dotted equivale1lt circuit for the coupled coils shown in figure and find the

equivalent inductance at the terminals AB. All coupling coefficie1lt are 0.5.

UNIT-4

FOURIERTRANSFORMATION

This means that a function generator that generates square waves through the addition of

sinusoidal waveforms needs to have a bandwidth (max. freq. it can generate) that is large

compared to the frequency of the square-wave that is generated. 2. If f(t) is continuous (although

possibly with discontinuous derivatives) the nth coincident decreases as 1=n2. There is another

consequence of a discontinuity in f(t) that can cause trouble in practical applications, where one

necessarily only adds a _nite number of sinusoidal terms. The nth partial sum of the Fourier

series of a piecewise continuously di_erentiable periodic function f behaves at a jump

discontinuity in a peculiar manner. It has large oscillations near the jump, which might increase

the maximum of the partial sum above that of the function itself. It turns out that the Fourier

series exceeds the height of a square wave by about 9 percent. This is the so-called Gibbs

phenomenon, shown in Fig. 2. Increasing the number of terms in the partial sum does not

decrease the magnitude of the overshoot but moves the overshoot extremism point closer and

closer to the jump discontinuity.

Fourier series representation in Trigonometric form

Fourier series in trigonometric form can be easily derived from its exponential form. The

complex exponential Fourier series representation of a periodic signal x(t) with fundamental

Period to be given by

Since sine and cosine can be expressed in exponential form. Thus by manipulating the

exponential Fourier series, we can obtain its Trigonometric form.

The trigonometric Fourier series representation of a periodic signal x (t) with fundamental

period T, is given by

Where ak and bk are Fourier coefficients given by

a0 is the dc component of the signal and is given by

Properties of Fourier series

Analysis of Exponential Fourier series

1. If x(t) is an even function i.e. x(- t) = x(t), then bk = 0 and

2. If x(t) is an even function i.e. x(- t) = - x(t), then a0 = 0, ak = 0 and

3. If x(t) is half symmetric function i.e. x (t) = -x(t ± T0/2), then a0 = 0, ak = bk = 0 for k even,

4. Linearity

5. Time shifting

6. Time reversal

7. Multiplication

8. Conjugation

9. Differentiation

10. Integration

11. Periodic convolution

Relationship between coefficients of exponential form and coefficients of trigonometric

form

When x (t) is real, then a, and b, are real, we have

Effect of Shifting Axis of the Signal

• On shifting the waveform to the left right with respect to the reference time axis t = 0 only the

phase values of the spectrum changes but the magnitude spectrum remains same.

• On shifting the waveform upward or downward w.r.t time axis changes only the DC value of

the function.

UNIT-5

FILTERS and SIMULATION

A filter is a circuit that is designed to pass signals with desired frequencies and reject or attenuate

others. A filter is a passive filter if it consists of only passive elements R, L, and C.

Low pass filter passes low frequencies and stops high frequencies.

High pass filter passes high frequencies and rejects low.

Band pass filter passes frequencies within a frequency band and blocks or attenuates

frequencies outside the band.

Band stop (band reject) filter passes frequencies outside a frequency band and blocks or

attenuates frequencies within the band.

RC high pass filter circuit

Since the impedance of the RC series circuit depends on

frequency, as indicated above, the circuit can be used to

filter out unwanted low frequencies.

The complex e.m.f. supplied is:

The complex potential across the output resistor is:

The physical potential across the output resistor is:

A graph of output versus frequency gives:

The output potential is zero for a D.C. potential, and Em for very high frequency. Low

frequencies are suppressed and high frequencies are not really affected. The cut-off frequency is

arbitrarily chosen as the frequency where only half the input power is output.

The half power angular frequency is the reciprocal of the time constant RC. The phase will be

π/4 at the half power frequency.

RC low pass filter circuit

As above, the complex e.m.f. supplied is:

The complex potential across the output capacitor is:

The physical potential across the output capacitor is: A graph of output potential versus frequency gives:

The output potential is Em for a D.C. potential, and zero for very high frequency. High

frequencies are suppressed and low frequencies are not really affected. The cut-off frequency is

also chosen as the frequency where only half the input power is output.

The half power angular frequency is again the reciprocal of the time constant RC. The phase will

also be π/4 at the half power frequency.