SYLLABUS
(RGPV Syllabus)
ELECTRICAL AND ELECTRONICS ENGINEERING
B.E. - 104
UNIT - I
Electrical Circuit Analysis: Voltage and current sources, dependent and independent sources, source conversion, DC circuits analysis using mesh and nodal method, Thevenin’s and superposition theorem, star-delta transformation. 1-phase AC circuits under sinusoidal steady state, active, reactive and apparent power, physical meaning of reactive power, power factor, 3phase balanced and unbalanced supply, star and delta connections.
UNIT - II
Transformers: Review of laws of electromagnetism, mmf, flux, and their relation, analysis of magnetic circuits. Single-phase transformer, basic concepts and construction features, voltage, current and impedance transformation, equivalent circuits, phasor diagram, voltage regulation, losses and efficiency, OC and SC test.
UNIT - III
Rotating Electric Machines: Constructional details of DC machine, induction machine and synchronous machine, working principle of 3-Phase induction motor, EMF equation of 3-Phase induction motor, concept of slip in 3-Phase induction motor, explanation of Torque-slip characteristics of 3-Phase induction motor, classification of self excited DC motor and generator.
UNIT - IV
Digital Electronics: Number systems used in digital electronics, decimal, binary, octal, hexadecimal, their complements, operation and conversion, floating point and signed numbers,
Demorgan’s theorem, AND, OR, NOT, NOR, NAND, EX-NOR, EX-OR gates and their representation, truth table, half and full adder circuits, R-S flip flop, J-K flip flop.
UNIT - V
Electronic Components and Circuits: Introduction to Semiconductors, Diodes, V-I characteristics, Bipolar junction transistors (BJT) and their working, introduction to CC, CB and
CE transistor configurations, different configurations and modes of operation of BJT, DC biasing of BJT.

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VOLTAGE SOURCE -is a two terminal device which can maintain a fixed voltage. An ideal voltage source can maintain the fixed voltage independent of the load resistance or the output current.
However, a real-world voltage source cannot supply unlimited current. A voltage source is the dual of a current source. Real-world sources of electrical energy, such as batteries, generators, and power systems, can be modeled for analysis purposes as a combination of an ideal voltage source.
CURRENT SOURCE -is an electronic circuit that delivers or absorbs an electric current which is independent of the voltage across it. A current source is the dual of a voltage source.The term constant-current 'sink' is sometimes used for sources fed from a negative voltage supply. Figure 1 shows the schematic symbol for an ideal current source, driving a resistor load There are two types - an independent current source (or sink) delivers a constant current. A dependent current source delivers a current which is proportional to some other voltage or current in the circuit.
Independent Sources The ideal voltage source and ideal current source discussed here come under the category of independent sources. The independent source is one which does not depend on any other quantity in the circuit.
It has a constant value i.e., the strength of voltage or current is not changed by any variation in the connected circuit. Thus, the voltage or current is fixed and is not adjustable.
Dependent Sources The source whose output voltage or current is not fixed but depends on the voltage or current in another part of the circuit is called as dependent or controlled source.

The dependent source is basically a three terminal device. The three terminals are paired with one common terminal. One pair is referred as input while the other pair as output. For example, in a transistor, the output voltage depends upon the input voltage. The dependent sources are represented by diamond shaped box as shown in Fig.
The dependent sources can be categorised as:
1. Voltage dependent voltage source
2. Current dependent voltage source
3. Voltage dependent current source
4. Current dependent current source
SOURCE CONVERSION
It is most important part of the circuit analysis. To simplify the circuit, certain rules have been framed which are given below:
(a) A voltage source having some resistance can be replaced by the current source in parallel with the resistance as shown in Fig.
(b) A current source in parallel with some resistance can be replaced by a voltage source in series with the same resistance as shown in fig.
.

Kirchoffs Current Law or KCL, states that the “ total current or charge entering a junction or node is exactly equal to the charge leaving the node as it has no other place to go except to leave, as no charge is lost within the node “. In other words the algebraic sum of ALL the currents entering and leaving a node must be equal to zero, I
(exiting)
+ I
(entering)
= 0. This idea by
Kirchoff is commonly known as the Conservation of Charge .
Here, the 3 currents entering the node, I
1
, I
2
, I
3
are all positive in value and the 2 currents leaving the node, I
4
and I
5
are negative in value. Then this means we can also rewrite the equation as;
I
1
+ I
2
+ I
3
– I
4
– I
5
= 0
The term Node in an electrical circuit generally refers to a connection or junction of two or more current carrying paths or elements such as cables and components. Also for current to flow either in or out of a node a closed circuit path must exist. We can use Kirchoff’s current law when analysing parallel circuits.
Kirchoffs Second Law – The Voltage Law, (KVL)
Kirchoffs Voltage Law or KVL, states that “ in any closed loop network, the total voltage around the loop is equal to the sum of all the voltage drops within the same loop
” which is also equal to zero. In other words the algebraic sum of all voltages within the loop must be equal to zero. This idea by Kirchoff is known as the Conservation of Energy .

Kirchoffs Voltage Law
Thevenins Theorem
In the previous three tutorials we have looked at solving complex electrical circuits using
Kirchhoff’s Circuit Laws
, Mesh Analysis and finally Nodal Analysis but there are many more
“Circuit Analysis Theorems” available to choose from which can calculate the currents and voltages at any point in a DC circuit. In this tutorial we will look at one of the more common circuit analysis theorems (next to Kirchhoff’s) that has been developed, Thevenins Theorem .
Thevenins Theorem states that “ Any linear circuit containing several voltages and resistances can be replaced by just a Single Voltage in series with a Single Resistor
“. In other words, it is possible to simplify any “Linear” circuit, no matter how complex, to an equivalent circuit with just a single voltage source in series with a resistance connected to a load as shown below.
Thevenins Theorem is especially useful in the Circuit Analysis of power or battery systems and other interconnected circuits were it will have an effect on the adjoining part of the circuit.
Thevenins equivalent circuit.
As far as the load resistor R
L
is concerned, any “one-port” network consisting of resistive circuit elements and energy sources can be replaced by one single equivalent resistance Rs and

equivalent voltage Vs, where Rs is the source resistance value looking back into the circuit and
Vs is the open circuit voltage at the terminals.
For example, consider the circuit from the previous section
Superposition Theorem.
If a number of voltage or current source are acting simultanously in a linear network, the resultant current in any branch is the algebraic sum of the currents that would be produced in it, when each source acts alone replacing all other independent sources by their internal resistances.
Circuit Diagram:
In a given figure apply superposition theorem , let us first take the sources V
1 alone at first replacing V
2
by short circuit.

Here,
Next, removing V
1
by short circuit, let the circuit be energized by V
2 only

Here,
As per superposition theorem,
Firstly, we have to remove the centre 40Ω resistor and short out (not physically as this would be dangerous) all the emf´s connected to the circuit, or open circuit any current sources. The value of resistor Rs is found by calculating the total resistance at the terminals A and B with all the emf´s removed, and the value of the voltage required Vs is the total voltage across terminals A and B with an open circuit and no load resistor Rs connected. Then, we get the following circuit.

Find the Equivalent Resistance (Rs)
Find the Equivalent Voltage (Vs)
We now need to reconnect the two voltages back into the circuit, and as V
S
= V
AB
the current flowing around the loop is calculated as: so the voltage drop across the 20Ω resistor can be calculated as:

V
AB
= 20 – (20Ω x 0.33amps) = 13.33 volts. or
V
AB
= 10 + (10Ω x 0.33amps) = 13.33 volts, the same.
Then the Thevenins Equivalent circuit is shown below with the 40Ω resistor connected. and from this the current flowing in the circuit is given as: which again, is the same value of 0.286 amps, we found using
Kirchhoff’s
circuit law in the previous tutorial.
Star Delta Transformation
Star Delta Transformations allow us to convert impedances connected together from one type of connection to another. We can now solve simple series, parallel or bridge type resistive networks using Kirchhoff’s Circuit Laws , mesh current analysis or nodal voltage analysis techniques but in a balanced 3-phase circuit we can use different mathematical techniques to simplify the analysis of the circuit and thereby reduce the amount of math’s involved which in itself is a good thing.
Standard 3-phase circuits or networks take on two major forms with names that represent the way in which the resistances are connected, a Star connected network which has the symbol of the letter, Υ and a
Delta connected network which has the symbol of a triangle, Δ (delta).

If a 3-phase, 3-wire supply or even a 3-phase load is connected in one type of configuration, it can be easily transformed or changed it into an equivalent configuration of the other type by using either the Star Delta Transformation or Delta Star Transformation process.
A resistive network consisting of three impedances can be connected together to form a T or
“Tee” configuration but the network can also be redrawn to form a
Star or Υ type network as shown below.
T-connected and Equivalent Star Network
As we have already seen, we can redraw the T resistor network to produce an equivalent Star or
Υ type network. But we can also convert a Pi or π type resistor network into an equivalent Delta or Δ type network as shown below.
Pi-connected and Equivalent Delta Network.

Having now defined exactly what is a Star and Delta connected network it is possible to transform the Υ into an equivalent Δ circuit and also to convert a Δ into an equivalent Υ circuit using a the transformation process. This process allows us to produce a mathematical relationship between the various resistors giving us a Star Delta Transformation as well as a
Delta Star Transformation .
These Circuit Transformations allow us to change the three connected resistances (or impedances) by their equivalents measured between the terminals 1-2, 1-3 or 2-3 for either a star or delta connected circuit. However, the resulting networks are only equivalent for voltages and currents external to the star or delta networks, as internally the voltages and currents are different but each network will consume the same amount of power and have the same power factor to each other.
Delta Star Transformation
To convert a delta network to an equivalent star network we need to derive a transformation formula for equating the various resistors to each other between the various terminals. Consider the circuit below.
Delta to Star Network.
Compare the resistances between terminals 1 and 2.
Resistance between the terminals 2 and 3.

Resistance between the terminals 1 and 3.
This now gives us three equations and taking equation 3 from equation 2 gives:
Then, re-writing Equation 1 will give us:
Adding together equation 1 and the result above of equation 3 minus equation 2 gives:

From which gives us the final equation for resistor P as:
Then to summarize a little about the above maths, we can now say that resistor P in a Star network can be found as Equation 1 plus (Equation 3 minus Equation 2) or Eq1 + (Eq3 – Eq2).
Similarly, to find resistor Q in a star network, is equation 2 plus the result of equation 1 minus equation 3 or Eq2 + (Eq1 – Eq3) and this gives us the transformation of Q as: and again, to find resistor R in a Star network, is equation 3 plus the result of equation 2 minus equation 1 or Eq3 + (Eq2 – Eq1) and this gives us the transformation of R as:
When converting a delta network into a star network the denominators of all of the transformation formulas are the same: A + B + C, and which is the sum of ALL the delta resistances. Then to convert any delta connected network to an equivalent star network we can summarized the above transformation equations as:

Delta to Star Transformations Equations
If the three resistors in the delta network are all equal in value then the resultant resistors in the equivalent star network will be equal to one third the value of the delta resistors, giving each branch in the star network as: R
STAR
= 1/3R
DELTA
Delta – Star Example No1
Convert the following Delta Resistive Network into an equivalent Star Network.
Star Delta Transformation
Star Delta transformation is simply the reverse of above. We have seen that when converting from a delta network to an equivalent star network that the resistor connected to one terminal is the product of the two delta resistances connected to the same terminal, for example resistor P is the product of resistors A and B connected to terminal 1.
By rewriting the previous formulas a little we can also find the transformation formulas for converting a resistive star network to an equivalent delta network giving us a way of producing a star delta transformation as shown below.

Star to Delta Transformation
The value of the resistor on any one side of the delta, Δ network is the sum of all the two-product combinations of resistors in the star network divide by the star resistor located “directly opposite” the delta resistor being found. For example, resistor A is given as: with respect to terminal 3 and resistor B is given as: with respect to terminal 2 with resistor C given as: with respect to terminal 1.
By dividing out each equation by the value of the denominator we end up with three separate transformation formulas that can be used to convert any Delta resistive network into an equivalent star network as given below.
Star Delta Transformation Equations

Star Delta Transformation allows us to convert one type of circuit connection into another type in order for us to easily analyse the circuit and star delta transformation techniques can be used for either resistances or impedances.
One final point about converting a star resistive network to an equivalent delta network. If all the resistors in the star network are all equal in value then the resultant resistors in the equivalent delta network will be three times the value of the star resistors and equal, giving:
R
DELTA
= 3R
STAR
A.C.CIRCUIT
Frequency
When we first started looking at SHM we defined period as the amount of time it takes for one cycle to complete... seconds per cycle
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Frequency is the same sort of idea, except we’re just going to flip things around.
Frequency is a measurement of how many cycles can happen in a certain amount of time… cycles per second.
 If a motor is running so that it completes 50 revolutions in one second, I would say that it has a frequency of 50 Hertz.
 Hertz is the unit of frequency, and just means how many cycles per second. o It is abbreviated as Hz . o It is named after Heinrich Hertz , one member of the Hertz family that made many important contributions to physics.
 In formulas frequency appears as an "f".
Since frequency and period are exact inverses of each other, there is a very basic pair of formulas you can use to calculate one if you know the other…
It is very easy to do these calculations on calculators using the x
-1
button.
Example 1: The period of a pendulum is 4.5s. Determine the frequency of this pendulum.
The period means that it will take 4.5 seconds for the pendulum to swing back and forth once.
So, I expect that my frequency will be a decimal, since it will complete a fraction of a swing per second.

Wavelength
Wavelength is a property of a wave that most people (once they know what to look for) can spot quickly and easily, and use it as a way of telling waves apart. Look at the following diagram...
Figure 1
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Any of the parts of the wave that are pointing up like mountains are called crests. Any part that is sloping down like a valley is a trough.
Wavelength is defined as the distance from a particular height on the wave to the next spot on the wave where it is at the same height and going in the same direction. o Usually it is measured in metres, just like any length.
 There isn’t a special spot you have to start on a wave to measure wavelength, just make sure you are back to the same height going in the same direction. Most people do like to measure from one crest to the next crest (or trough to trough), just because they are easy to spot.
Figure 2
On a longitudinal wave, the wavelength is measured as the distance between the middles of two compressions, or the middles of two expansions.

Figure 3
This leads us to one of the most important formulas you will use when studying waves.
 Frequency tells us how many waves are passing a point per second, the inverse of time .
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Wavelength tells us the length of those waves in metres, almost like a displacement .
If we multiply these two together, we are really multiplying 1/s and m… which gives us m/s, the velocity of the wave! v = velocity of the wave (m/s) f = frequency (Hz)
λ = wavelength (m)
Example 2: A wave is measured to have a frequency of 60Hz. If its wavelength is 24cm, determine how fast it is moving.
Example 3: The speed of light is always 3.00e8 m/s. Determine the frequency of red light which has a wavelength of 700nm.
Be careful when changing the 700nm into metres. Some people get really caught up with changing it into regular scientific notation with only one digit before the decimal. Why bother?
It's only being used in a calculation. You’ll probably just make a mistake changing the power of
10, so just substitute in the power for the prefix and leave everything else alone…700 nm = 700 x 10
-9
m since “nano” is 10
-9
.

Amplitude
Amplitude is a measure of how big the wave is.
 Imagine a wave in the ocean. It could be a little ripple or a giant tsunami. o o
What you are actually seeing are waves with different amplitudes.
They might have the exact same frequency and wavelength, but the amplitudes of the waves can be very different.
The amplitude of a wave is measured as:
1.
the height from the equilibrium point to the highest point of a crest or
2.
the depth from the equilibrium point to the lowest point of a trough
Figure 4
When you measure the amplitude of a wave, you are really looking at the energy of the wave.
 It takes more energy to make a bigger amplitude wave.
 Anytime you need to remember this, just think of a home stereo’s amplifier… it makes the amplitude of the waves bigger by using more electrical energy.
We know that reactive loads such as inductors and capacitors dissipate zero power, yet the fact that they drop voltage and draw current gives the deceptive impression that they actually do dissipate power. This “phantom power” is called reactive power , and it is measured in a unit called Volt-Amps-Reactive (VAR), rather than watts. The mathematical symbol for reactive power is (unfortunately) the capital letter Q. The actual amount of power being used, or dissipated, in a circuit is called true power , and it is measured in watts (symbolized by the capital letter P, as always). The combination of reactive power and true power is called apparent power , and it is the product of a circuit's voltage and current, without reference to phase angle. Apparent power is measured in the unit of Volt-Amps (VA) and is symbolized by the capital letter S.
As a rule, true power is a function of a circuit's dissipative elements, usually resistances (R).
Reactive power is a function of a circuit's reactance (X). Apparent power is a function of a

circuit's total impedance (Z). Since we're dealing with scalar quantities for power calculation, any complex starting quantities such as voltage, current, and impedance must be represented by their polar magnitudes , not by real or imaginary rectangular components. For instance, if I'm calculating true power from current and resistance, I must use the polar magnitude for current, and not merely the “real” or “imaginary” portion of the current. If I'm calculating apparent power from voltage and impedance, both of these formerly complex quantities must be reduced to their polar magnitudes for the scalar arithmetic.
There are several power equations relating the three types of power to resistance, reactance, and impedance (all using scalar quantities):
Please note that there are two equations each for the calculation of true and reactive power.
There are three equations available for the calculation of apparent power, P=IE being useful only for that purpose. Examine the following circuits and see how these three types of power interrelate for: a purely resistive load in Figure below , a purely reactive load in Figure below , and a resistive/reactive load in Figure below .

True power, reactive power, and apparent power for a purely resistive load.
Reactive load only:
True power, reactive power, and apparent power for a purely reactive load.
Resistive/reactive load:

True power, reactive power, and apparent power for a resistive/reactive load.
These three types of power -- true, reactive, and apparent -- relate to one another in trigonometric form. We call this the power triangle : (Figure below ).

Power triangle relating appearant power to true power and reactive power.
Using the laws of trigonometry, we can solve for the length of any side (amount of any type of power), given the lengths of the other two sides, or the length of one side and an angle.
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REVIEW:
Power dissipated by a load is referred to as true power . True power is symbolized by the letter P and is measured in the unit of Watts (W).
 Power merely absorbed and returned in load due to its reactive properties is referred to as reactive power . Reactive power is symbolized by the letter Q and is measured in the unit of Volt-Amps-Reactive (VAR).
 Total power in an AC circuit, both dissipated and absorbed/returned is referred to as
 apparent power . Apparent power is symbolized by the letter S and is measured in the unit of Volt-Amps (VA).
These three types of power are trigonometrically related to one another. In a right triangle, P = adjacent length, Q = opposite length, and S = hypotenuse length. The opposite angle is equal to the circuit's impedance (Z) phase angle.
We know from the tutorials about Inductors , that inductors are basically coils or loops of wire that are either wound around a hollow tube former (air cored) or wound around some ferromagnetic material (iron cored) to increase their inductive value called inductance . Inductors store their energy in the form of a magnetic field that is created when a DC voltage is applied across the terminals of an inductor.
The growth of the current flowing through the inductor is not instant but is determined by the inductors own self-induced or back emf value. Then for an inductor coil, this back emf voltage
V
L
is proportional to the rate of change of the current flowing through it.
This current will continue to rise until it reaches its maximum steady state condition which is around five time constants when this self-induced back emf has decayed to zero. At this point a steady state current is flowing through the coil, no more back emf is induced to oppose the current flow and therefore, the coil acts more like a short circuit allowing maximum current to flow through it.
However, in an alternating current circuit which contains an AC Inductance , the flow of current through an inductor behaves very differently to that of a steady state DC voltage. Now in an AC circuit, the opposition to the current flowing through the coils windings not only depends upon the inductance of the coil but also the frequency of the applied voltage waveform as it varies from its positive to negative values.
The actual opposition to the current flowing through a coil in an AC circuit is determined by the
AC Resistance of the coil with this AC resistance being represented by a complex number. But to distinguish a DC resistance value from an AC resistance value, which is also known as
Impedance, the term Reactance is used.

Like resistance, reactance is measured in Ohm’s but is given the symbol “X” to distinguish it from a purely resistive “R” value and as the component in question is an inductor, the reactance of an inductor is called Inductive Reactance , ( X
L
) and is measured in Ohms. Its value can be found from the formula.
Where: X
L
is the Inductive Reactance in Ohms, ƒ is the frequency in Hertz and L is the inductance of the coil in Henries.
We can also define inductive reactance in radians, where Omega, ω equals 2πƒ.
So whenever a sinusoidal voltage is applied to an inductive coil, the back emf opposes the rise and fall of the current flowing through the coil and in a purely inductive coil which has zero resistance or losses, this impedance (which can be a complex number) is equal to its inductive reactance. Also reactance is represented by a vector as it has both a magnitude and a direction
(angle). Consider the circuit below.
This simple circuit above consists of a pure inductance of L Henries ( H ), connected across a sinusoidal voltage given by the expression: V(t) = V max
sin ωt. When the switch is closed this sinusoidal voltage will cause a current to flow and rise from zero to its maximum value. This rise or change in the current will induce a magnetic field within the coil which in turn will oppose or restrict this change in the current.

But before the current has had time to reach its maximum value as it would in a DC circuit, the voltage changes polarity causing the current to change direction. This change in the other direction once again being delayed by the self-induced back emf in the coil, and in a circuit containing a pure inductance only, the current is delayed by 90 o
.
The applied voltage reaches its maximum positive value a quarter ( 1/4ƒ ) of a cycle earlier than the current reaches its maximum positive value, in other words, a voltage applied to a purely inductive circuit “LEADS” the current by a quarter of a cycle or 90 o
as shown below.
The inductive reactance of an inductor increases as the frequency across it increases therefore inductive reactance is proportional to frequency ( X
L
α ƒ ) as the back emf generated in the inductor is equal to its inductance multiplied by the rate of change of current in the inductor.
Also as the frequency increases the current flowing through the inductor also reduces in value.
We can present the effect of very low and very high frequencies on a the reactance of a pure AC
Inductance as follows:

In the RL series circuit above, we can see that the current is common to both the resistance and the inductance while the voltage is made up of the two component voltages, V
R
and V
L
. The resulting voltage of these two components can be found either mathematically or by drawing a vector diagram. To be able to produce the vector diagram a reference or common component must be found and in a series AC circuit the current is the reference source as the same current flows through the resistance and the inductance. The individual vector diagrams for a pure resistance and a pure inductance are given as:
We can see from above and from our previous tutorial about AC Resistance that the voltage and current in a resistive circuit are both in phase and therefore vector V
R
is drawn superimposed to scale onto the current vector. Also from above it is known that the current lags the voltage in an
AC inductance (pure) circuit therefore vector V
L
is drawn 90 o
in front of the current and to the same scale as V
R
as shown.

From the vector diagram above, we can see that line OB is the horizontal current reference and line OA is the voltage across the resistive component which is in-phase with the current. Line
OC shows the inductive voltage which is 90 o
in front of the current therefore it can still be seen that the current lags the purely inductive voltage by 90 o
. Line OD gives us the resulting supply voltage. Then:
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V equals the r.m.s value of the applied voltage.
 I equals the r.m.s. value of the series current.
V
R
equals the I.R voltage drop across the resistance which is in-phase with the current.
 V
L
equals the I.X
L
voltage drop across the inductance which leads the current by 90 o .
As the current lags the voltage in a pure inductance by exactly 90 o
the resultant phasor diagram drawn from the individual voltage drops V
R
and V
L
represents a right angled voltage triangle shown above as OAD. Then we can also use Pythagoras’s theorem to mathematically find the value of this resultant voltage across the resistor/inductor ( RL ) circuit.
As V
R
= I.R and V
L
= I.X
L
the applied voltage will be the vector sum of the two as follows:

The quantity represents the impedance , Z of the circuit.
Impedance, Z is the “TOTAL” opposition to current flowing in an AC circuit that contains both
Resistance, ( the real part ) and Reactance ( the imaginary part ). Impedance also has the units of
Ohms, Ω‘s. Impedance depends upon the frequency, ω of the circuit as this affects the circuits reactive components and in a series circuit all the resistive and reactive impedance’s add together.
Impedance can also be represented by a complex number, Z = R + jX
L
but it is not a phasor, it is the result of two or more phasors combined together. If we divide the sides of the voltage triangle above by I, another triangle is obtained whose sides represent the resistance, reactance and impedance of the circuit as shown below.

Then: ( Impedance ) 2 = ( Resistance ) 2 + ( j Reactance ) 2 where j represents the 90 o phase shift.
This means that the positive phase angle, θ between the voltage and current is given as.
When capacitors are connected across a direct current DC supply voltage they become charged to the value of the applied voltage, acting like temporary storage devices and maintain or hold this charge indefinitely as long as the supply voltage is present. During this charging process, a

charging current, ( i ) will flow into the capacitor opposing any changes to the voltage at a rate that is equal to the rate of change of the electrical charge on the plates.
This charging current can be defined as: i = CdV/dt. Once the capacitor is “fully-charged” the capacitor blocks the flow of any more electrons onto its plates as they have become saturated.
However, if we apply an alternating current or AC supply, the capacitor will alternately charge and discharge at a rate determined by the frequency of the supply. Then the Capacitance in AC circuits varies with frequency as the capacitor is being constantly charged and discharged.
We know that the flow of electrons onto the plates of a Capacitor is directly proportional to the rate of change of the voltage across those plates. Then, we can see that capacitors in AC circuits like to pass current when the voltage across its plates is constantly changing with respect to time such as in AC signals, but it does not like to pass current when the applied voltage is of a constant value such as in DC signals. Consider the circuit below.
In the purely capacitive circuit above, the capacitor is connected directly across the AC supply voltage. As the supply voltage increases and decreases, the capacitor charges and discharges with respect to this change. We know that the charging current is directly proportional to the rate of change of the voltage across the plates with this rate of change at its greatest as the supply voltage crosses over from its positive half cycle to its negative half cycle or vice versa at points,
0 o
and 180 o
along the sine wave.
Consequently, the least voltage change occurs when the AC sine wave crosses over at its maximum or minimum peak voltage level, ( Vm ). At these positions in the cycle the maximum or minimum currents are flowing through the capacitor circuit and this is shown below.

Capacitive Reactance in a purely capacitive circuit is the opposition to current flow in AC circuits only. Like resistance, reactance is also measured in Ohm’s but is given the symbol X to distinguish it from a purely resistive value. As reactance is a quantity that can also be applied to
Inductors as well as Capacitors, when used with capacitors it is more commonly known as
Capacitive Reactance .
For capacitors in AC circuits, capacitive reactance is given the symbol Xc. Then we can actually say that Capacitive Reactance is a capacitors resistive value that varies with frequency. Also, capacitive reactance depends on the capacitance of the capacitor in Farads as well as the frequency of the AC waveform and the formula used to define capacitive reactance is given as:
Where:
F is in Hertz and C is in Farads.
2πF can also be expressed collectively as the Greek letter Omega , ω to denote an angular frequency.
From the capacitive reactance formula above, it can be seen that if either of the Frequency or
Capacitance where to be increased the overall capacitive reactance would decrease. As the frequency approaches infinity the capacitors reactance would reduce to zero acting like a perfect conductor.
However, as the frequency approaches zero or DC, the capacitors reactance would increase up to infinity, acting like a very large resistance. This means then that capacitive reactance is

“
Inversely proportional
” to frequency for any given value of Capacitance and this shown below:
The capacitive reactance of the capacitor decreases as the frequency across it increases therefore capacitive reactance is inversely proportional to frequency.
The opposition to current flow, the electrostatic charge on the plates (its AC capacitance value) remains constant as it becomes easier for the capacitor to fully absorb the change in charge on its plates during each half cycle.
Also as the frequency increases the current flowing through the capacitor increases in value because the rate of voltage change across its plates increases.
In the previous tutorials we have looked at resistors, their connections and used
Ohm’s Law
to calculate the voltage, current and power associated with them. In all cases both the voltage and current has been assumed to be of a constant polarity, flow and direction, in other words Direct
Current or DC .
But there is another type of supply known as Alternating Current or AC whose voltage switches polarity from positive to negative and back again over time and also whose current with respect to the voltage oscillates back and forth. The oscillating shape of an AC supply follows that of the mathematical form of a “sine wave” which is commonly called a
Sinusoidal Waveform .
Therefore, a sinusoidal voltage can be defined as V(t) = V max
sin ωt.
When using pure resistors in AC circuits that have negligible values of inductance or capacitance, the same principals of
Ohm’s Law
, circuit rules for voltage, current and power (and even Kirchoff’s Laws) apply as they do for DC resistive circuits the only difference this time is in the use of the instantaneous “peak-to-peak” or “rms” quantities.
When working with AC alternating voltages and currents it is usual to use only “rms” values to avoid confusion. Also the schematic symbol used for defining an AC voltage source is that of a
“wavy” line as opposed to a battery symbol for DC and this is shown below.

Resistors are “passive” devices, that is they do not produce or consume any electrical energy, but convert electrical energy into heat. In DC circuits the linear ratio of voltage to current in a resistor is called its resistance. However, in AC circuits this ratio of voltage to current depends upon the frequency and phase difference or phase angle ( φ ) of the supply. So when using resistors in AC circuits the term Impedance , symbol Z is the generally used and we can say that
DC resistance = AC impedance, R = Z.
It is important to note, that when used in AC circuits, a resistor will always have the same resistive value no matter what the supply frequency from DC to very high frequencies, unlike capacitor and inductors.
For resistors in AC circuits the direction of the current flowing through them has no effect on the behaviour of the resistor so will rise and fall as the voltage rises and falls. The current and voltage reach maximum, fall through zero and reach minimum at exactly the same time. i.e, they rise and fall simultaneously and are said to be “in-phase” as shown below.
We can see that at any point along the horizontal axis that the instantaneous voltage and current are in-phase because the current and the voltage reach their maximum values at the same time, that is their phase angle θ is 0 o
. Then these instantaneous values of voltage and current can be

compared to give the ohmic value of the resistance simply by using ohms law. Consider below the circuit consisting of an AC source and a resistor.
The instantaneous voltage across the resistor, V
R
is equal to the supply voltage, V t
and is given as:
The instantaneous current flowing in the resistor will therefore be:
As the voltage across a resistor is given as V
R
= I.R, the instantaneous voltage across the resistor above can also be given as:
In purely resistive series AC circuits, all the voltage drops across the resistors can be added together to find the total circuit voltage as all the voltages are in-phase with each other. Likewise, in a purely resistive parallel AC circuit, all the individual branch currents can be added together to find the total circuit current because all the branch currents are in-phase with each other.

Since for resistors in AC circuits the phase angle φ between the voltage and the current is zero, then the power factor of the circuit is given as cos 0 o
= 1.0. The power in the circuit at any instant in time can be found by multiplying the voltage and current at that instant.
Then the power (P), consumed by the circuit is given as P = Vrms Ι cos Φ in watt’s. But since cos Φ = 1 in a purely resistive circuit, the power consumed is simply given as, P = Vrms Ι the same as for Ohm’s Law .
This then gives us the “Power” waveform and which is shown below as a series of positive pulses because when the voltage and current are both in their positive half of the cycle the resultant power is positive. When the voltage and current are both negative, the product of the two negative values gives a positive power pulse.
Then the power dissipated in a purely resistive load fed from an AC rms supply is the same as that for a resistor connected to a DC supply and is given as:


Where:


P is the average power in Watts
 V rms
is the rms supply voltage in Volts
I rms
is the rms supply current in Amps
R is the resistance of the resistor in Ohm’s (Ω) – should really be Z to indicate impedance

The heating effect produced by an alternating current with a maximum value of Imax is not the same as that of a DC current of the same value. To compare the AC heating effect to an equivalent DC the rms values must be used. Any resistive heating element such as Electric Fires,
Toasters, Kettles, Irons, Water Heaters etc can be classed as a resistive AC circuit and we use resistors in AC circuits to heat our homes and water
Following are the requirements that must be satisified in order for a set of 3 sinusoidal variables
(usually voltages or currents) to be a "balanced 3-phase set"
1.
All 3 variables have the same amplitude
2.
All 3 variables have the same frequency
3.
All 3 variables are 120 o
in phase
In terms of the time domain, a set of balance 3-phase voltages has the following general form. v a
= V m
cos ( t + ) v b
= V m
cos ( t + - 120 o
) v c
= V m
cos ( t + - 240 o ) = V m
cos ( t + +120 o )
Notice that we have assumed (and will continue to assume) positive (abc) phase sequence, i.e., phase "b" follows 120 o
behind "a" & phase "c" follows 120 o
behind phase "b"
Figure 1 below illustrates the balanced 3-phase voltages in time domain.
Figure 1: Balanced 3-Phase Variables in Time Domain
In terms of phasors, we write the same balanced set as follows. Note that the phasors are in rms, as will be assumed throughout this course.
V a
= V m m
V b
= V m
- 120 o

Figure 2 below illustrates the balanced 3-phase phasors graphically.
Requirements of a Balanced 3-Phase Circuit
Following are the requirements that must be satisfied in order for a 3-phase system or circuit to be balanced
1.
All 3 sources are represented by a set of balanced 3-phase variables
2.
All loads are 3-phase with equal impedances
3.
Line impedances are equal in all 3 phases
Having a balanced circuit allows for simplified analysis of the 3-phase circuit. In fact, if the circuit is balanced, we can solve for the voltages, currents, and powers, etc. in one phase using circuit analysis. The values of the corresponding variables in the other two phases can be found using some basic equations. This type of solution is accomplished using a "one-line diagram" , which will be discussed later. If the circuit is not balanced, all three phases should be analyzed in detail.
Figure 3 illustrates a balanced 3-phase circuit and some of the naming conventions to be used in this course
Figure 3: A Balanced 3-Phase Circuit

Phase describes or pertains to one element or device in a load, line, or source. It is simply a
"branch" of the circuit and could look something like this .
Line refers to the "transmission line" or wires that connect the source (supply) to the load. It may be modeled as a small impedance (actually 3 of them), or even by just a connecting line.
Neutral the 4th wire in the 3-phase system. It's where the phases of a Y connection come together.
Phase Voltages & Phase Currents the voltages and currents across and through a single branch (phase) of the circuit. Note this definition depends on whether the connection is Wye or Delta!
Line Currents the currents flowing in each of the lines (I a
, I b
, and I c
). This definition does not change with connection type.
Line Voltages the voltages between any two of the lines (V ab
, V bc
, and V ca
). These may also be referred to as the line-to-line voltages. This definition does not change with connection type.
Line to Neutral Voltages the voltages between any lines and the neutral point (V a
, V b
, and V c
). This definition does not change with connection type, but they may not be physically measureable in a Delta circuit.
Line to Neutral Currents same as the line currents (I a
, I b
, and I c
). f the Neutral points in Figure 12 are actually the same point, Figure 12 can be redrawn as shown in Figure 13.
Figure 13: ReDrawn All-Y Circuit

When we have an unbalanced circuit, we CANNOT use the one-line diagram to solve for "a" phase values and then get the answers for the other phases by adding or subtracting 120 o
.
In general, a unbalanced three-phase circuit requires that you draw the complete circuit including all 3-phase and single-phase loads and perform a circuit analysis of the whole thing. Normal methods such as "meshes" or "node voltages" may be used. If you have the simple case in which a balanced 3-phase load is connected directly to a source and a single phase load is connected in parallel to the same source, you may calculate the currents in the balanced load using a one-line method. The single phase current is calculated separately and then individual line currents can be found by summing the currents at certain nodes in the system.
Remember any circuit that does not have all loads with the same impedance in all three branches is an unbalanced circuit.

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