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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.