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Energy storage elements
Inductors in Networks:
Inductors in a parallel configuration each have the same potential difference (voltage). To find
their total equivalent inductance (Leq):
The current through inductors in series stays the same, but the voltage across each
inductor can be different. The sum of the potential differences (voltage) is equal to the total
voltage. To find their total inductance:
These simple relationships hold true only when there is no mutual coupling of magnetic
fields between individual inductors
The effect of an inductor in a circuit is to oppose changes in current through it by
developing a voltage across it proportional to the rate of change of the current. An ideal inductor
would offer no resistance to a constant direct current; however, only superconducting inductors
have truly zero electrical resistance.
The relationship between the time-varying voltage v(t) across an inductor with
inductance L and the time-varying current i(t) passing through it is described by the differential
equation:
When there is a sinusoidal alternating current (AC) through an inductor, a sinusoidal voltage is
induced. The amplitude of the voltage is proportional to the product of the amplitude (IP) of the
current and the frequency (f) of the current.
In this situation, the phase of the current lags that of the voltage by π/2 (90°). For
sinusoids, as the voltage across the inductor goes to its maximum value, the current goes to zero,
and as the voltage across the inductor goes to zero, the current through it goes to its maximum
value.
If an inductor is connected to a direct current source with value I via a resistance R, and
then the current source is short-circuited, the differential relationship above shows that the
current through the inductor will discharge with an exponential decay:
Reactance
The ratio of the peak voltage to the peak current in an inductor energised from a
sinusoidal source is called the reactance and is denoted XL. The suffix is to distinguish inductive
reactance from capacitive reactance due to capacitance.
Thus,
Reactance is measured in the same units as resistance (ohms) but is not actually a
resistance. A resistance will dissipate energy as heat when a current passes. This does not happen
with an inductor; rather, energy is stored in the magnetic field as the current builds and later
returned to the circuit as the current falls. Inductive reactance is strongly frequency dependent.
At low frequency the reactance falls, and for a steady current (zero frequency) the inductor
behaves as a short-circuit. At increasing frequency, on the other hand, the reactance increases
and at a sufficiently high frequency the inductor approaches an open circuit.
Laplace circuit analysis (s-domain)
When using the Laplace transform in circuit analysis, the impedance of an ideal inductor
with no initial current is represented in the s domain by:
where
is the inductance, and
is the complex frequency.
If the inductor does have initial current, it can be represented by:
adding a voltage source in series with the inductor, having the value:
where
is the inductance, and
is the initial current in the inductor.
(Note that the source should have a polarity that is aligned with the initial current)
or by adding a current source in parallel with the inductor, having the value:
where
is the initial current in the inductor.
is the complex frequency.
Inductor networks
Main article: Series and parallel circuits
Inductors in a parallel configuration each have the same potential difference (voltage). To find
their total equivalent inductance (Leq):
The current through inductors in series stays the same, but the voltage across each inductor can
be different. The sum of the potential differences (voltage) is equal to the total voltage. To find
their total inductance:
These simple relationships hold true only when there is no mutual coupling of magnetic fields
between individual inductors.
Stored energy
Neglecting losses, the energy (measured in joules, in SI) stored by an inductor is equal to the
amount of work required to establish the current through the inductor, and therefore the magnetic
field. This is given by:
where L is inductance and I is the current through the inductor.
This relationship is only valid for linear (non-saturated) regions of the magnetic flux linkage and
current relationship. In general if one decides to find the energy stored in a LTI inductor that has
initial current in a specific time between
and can use this:
Capacitor and Capacitance
Capacitor :
A capacitor is an electrical device that can store energy
in the electric field between a pair of closely-spaced conductors (called 'plates'). A capacitor
consists of two conductive electrodes, or plates, separated by an insulator.When voltage is
applied to the capacitor, electric charges of equal magnitude, but opposite polarity, build up on
each plate.Capacitors are occasionally referred to as condensers.
Capacitors are used in electrical circuits as energy-storage devices. They can also be used to
differentiate between high-frequency and low-frequency signals and this makes them useful in
electronic filters.
Practical capacitors are often classified according to the material used as the dielectric with the
dielectrics divided into two broad categories: bulk insulators and metal-oxide films (so-called
electrolytic capacitors).
Many types of capacitor are available commercially, with capacitances ranging from the
picofarad range to more than a farad, and voltage ratings up to hundreds of kilovolts. In general,
the higher the capacitance and voltage rating, the larger the physical size of the capacitor and the
higher the cost. Tolerances in capacitance value for discrete capacitors are usually specified as a
percentage of the nominal value. Tolerances ranging from 50% (electrolytic types) to less than
1% are commonly available.
Insulator :
An insulator is a material or object which contains no free electrons to permit the flow of
electricity. When a voltage is placed across an insulator, no charge/current flows.
Capacitance :
The capacitor's capacitance (C) is a measure of the amount of charge (Q) stored on each plate for
a given potential difference or voltage (V) which appears between the plates :
In SI units, a capacitor has a capacitance of one farad when one coulomb of charge
causes a potential difference of one volt across the plates. Since the farad is a very large unit,
values of capacitors are usually expressed in microfarads (µF), nanofarads (nF), or picofarads
(pF).
The capacitance is proportional to the surface area of the conducting plate and inversely
proportional to the distance between the plates. It is also proportional to the permittivity of the
dielectric (that is, non-conducting) substance that separates the plates.
The capacitance of a parallel-plate capacitor is given by:
where ε is the permittivity of the dielectric, A is the area of the plates and d is the spacing
between them.
Energy Stored in a capacitor: As opposite charges accumulate on the plates of a capacitor due
to the separation of charge, a voltage develops across the capacitor owing to the electric field of
these charges. Ever-increasing work must be done against this ever-increasing electric field as
more charge is separated. The energy (measured in joules, in SI) stored in a capacitor is equal to
the amount of work required to establish the voltage across the capacitor, and therefore the
electric field. The energy stored is given by:
where V is the voltage across the capacitor.
The maximum energy that can be (safely) stored in a particular capacitor is limited by the
maximum electric field that the dielectric can withstand before it breaks down. Therefore, all
capacitors made with the same dielectric have about the same maximum energy density (joules
of energy per cubic meter).
AC Voltage, Average, RMS values, Form factor, Crest factor
AC Voltage:
A voltage in which the polarity alternates.
Measurement of AC magnitude :
AC voltage alternates in polarity and AC current alternates in direction. We know that
AC can alternate in a variety of different ways, and by tracing the alternation over time we can
plot it as a "waveform." We can measure the rate of alternation by measuring the time it takes for
a wave to evolve before it repeats itself (the "period"), and express this as cycles per unit time, or
"frequency." In music, frequency is the same as pitch, which is the essential property
distinguishing one note from another.
One way to express the intensity, or magnitude (also called the amplitude), of an AC
quantity is to measure its peak height on a waveform graph. This is known as the peak or crest
value of an AC waveform:
Another way is to measure the total height between opposite peaks. This is known as the peakto-peak (P-P) value of an AC waveform:
This is the measurement in case of a sinusoidal wave either it may be voltage or current.
The measurement definition is same fo the ramining wave shapes also.
Average Value :
One way of expressing the amplitude of different waveshapes in a more equivalent
fashion is to mathematically average the values of all the points on a waveform's graph to a
single, aggregate number. This amplitude measure is known simply as the average value of the
waveform. If we average all the points on the waveform algebraically (that is, to consider their
sign, either positive or negative), the average value for most waveforms is technically zero,
because all the positive points cancel out all the negative points over a full cycle:
This, of course, will be true for any waveform having equal-area portions above and below the
"zero" line of a plot. However, as a practical measure of a waveform's aggregate value, "average"
is usually defined as the mathematical mean of all the points' absolute values over a cycle.
Root Mean Square (RMS) Value :
RMS value means , first square all the values, then find the average (mean) of these
square values over a complete cycle, and find the square root of this average. That is the RMS
value.
The value of an AC voltage is continually changing from zero up to the positive peak,
through zero to the negative peak and back to zero again. Clearly for most of the time it is less
than the peak voltage, so this is not a good measure of its real effect.
Instead we use the root mean square voltage (VRMS) which is 0.7 of the peak voltage
(Vpeak):
VRMS
=
0.7
×
Vpeak
and
Vpeak
=
1.4
×
VRMS
These equations also apply to current. They are only true for sine waves (the most common type
of AC) because the 0.7 and 1.4 are different values for other shapes.
The RMS value is the effective value of a varying voltage or current. It is the equivalent
steady DC (constant) value which gives the same effect. AC voltmeters and ammeters show the
RMS value of the voltage or current.
For pure Sinusoidal : RMS = 0.707 (Peak) , AVG = 0.637 (Peak) , P-P = 2 (Peak).
For pure Square : RMS = Peak , AVG = Peak , P-P = 2 (Peak).
For pure Triangular : RMS = 0.577 (Peak) , AVG = 0.5 (Peak) , P-P = 2 (Peak).
The Crest factor of an
Crest factor = Peak Value / RMS Value
AC
wave
form
for
instance
is
given
by
The form factor of an AC waveform is the ratio of its peak value divided by its average
value
Form factor = Peak Value / Average Value
Inductors Series and Parallel
Inductors in Networks :
Inductors in a parallel configuration each have the same potential difference (voltage). To
find their total equivalent inductance (Leq):
The current through inductors in series stays the same, but the voltage across each inductor can
be different. The sum of the potential differences (voltage) is equal to the total voltage. To find
their total inductance:
These simple relationships hold true only when there is no mutual coupling of magnetic fields
between individual inductors
Capacitor and Capacitance
Capacitor :
A capacitor is an electrical device that can store energy in the electric field between a pair
of closely-spaced conductors (called 'plates'). A capacitor consists of two conductive electrodes,
or plates, separated by an insulator.When voltage is applied to the capacitor, electric charges of
equal magnitude, but opposite polarity, build up on each plate.Capacitors are occasionally
referred to as condensers.
Capacitors are used in electrical circuits as energy-storage devices. They can also be used
to differentiate between high-frequency and low-frequency signals and this makes them useful in
electronic filters.
Practical capacitors are often classified according to the material used as the dielectric
with the dielectrics divided into two broad categories: bulk insulators and metal-oxide films (socalled electrolytic capacitors).
Many types of capacitor are available commercially, with capacitances ranging from the
picofarad range to more than a farad, and voltage ratings up to hundreds of kilovolts. In general,
the higher the capacitance and voltage rating, the larger the physical size of the capacitor and the
higher the cost. Tolerances in capacitance value for discrete capacitors are usually specified as a
percentage of the nominal value. Tolerances ranging from 50% (electrolytic types) to less than
1% are commonly available.
Insulator :
An insulator is a material or object which contains no free electrons to permit the flow of
electricity. When a voltage is placed across an insulator, no charge/current flows.
Capacitance :
The capacitor's capacitance (C) is a measure of the amount of charge (Q) stored on each
plate for a given potential difference or voltage (V) which appears between the plates :
In SI units, a capacitor has a capacitance of one farad when one coulomb of charge
causes a potential difference of one volt across the plates. Since the farad is a very large unit,
values of capacitors are usually expressed in microfarads (µF), nanofarads (nF), or picofarads
(pF).
The capacitance is proportional to the surface area of the conducting plate and inversely
proportional to the distance between the plates. It is also proportional to the permittivity of the
dielectric
(that
is,
non-conducting)
substance
that
separates
the
plates.
The capacitance of a parallel-plate capacitor is given by:
where ε is the permittivity of the dielectric, A is the area of the plates and d is the spacing
between them.
Energy Stored in a capacitor :As opposite charges accumulate on the plates of a capacitor due
to the separation of charge, a voltage develops across the capacitor owing to the electric field of
these charges. Ever-increasing work must be done against this ever-increasing electric field as
more charge is separated. The energy (measured in joules, in SI) stored in a capacitor is equal to
the amount of work required to establish the voltage across the capacitor, and therefore the
electric field. The energy stored is given by:
where V is the voltage across the capacitor
The maximum energy that can be (safely) stored in a particular capacitor is limited by the
maximum electric field that the dielectric can withstand before it breaks down. Therefore, all
capacitors made with the same dielectric have about the same maximum energy density (joules
of energy per cubic meter).
AC Phase :
Things start to get complicated when we need to relate two or more AC voltages or
currents that are out of step with each other. By "out of step," I mean that the two waveforms are
not synchronized: that their peaks and zero points do not match up at the same points in time.
The following graph illustrates an example of this:
The two waves shown above (A versus B) are of the same amplitude and frequency, but they are
out of step with each other. In technical terms, this is called a phase shift.
A sampling of different phase shifts is given in the following graphs to better illustrate this
concept:




Phase shift is where two or more waveforms are out of step with each other.
The amount of phase shift between two waves can be expressed in terms of degrees, as defined
by the degree units on the horizontal axis of the waveform graph used in plotting the
trigonometric sine function.
A leading waveform is defined as one waveform that is ahead of another in its evolution. A
lagging waveform is one that is behind another.
Calculations for AC circuit analysis must take into consideration both amplitude and phase
shift of voltage and current waveforms to be completely accurate. This requires the use of a
mathematical system called complex numbers.
Properties of AC electrical signal :
An electrical signal is a voltage or current which conveys information, usually it means a
voltage. The term can be used for any voltage or current in a circuit.
The voltage-time graph on the right shows various properties of an electrical signal. In addition
to the properties labelled on the graph, there is frequency which is the number of cycles per
second.
The diagram shows a sine wave but these properties apply to any signal with a constant shape.





Amplitude is the maximum voltage reached by the signal. It is measured in volts, V.
Peak voltage is another name for amplitude.
Peak-peak voltage is twice the peak voltage (amplitude). When reading an oscilloscope trace
it is usual to measure peak-peak voltage.
Time period is the time taken for the signal to complete one cycle. It is measured in seconds
(s), but time periods tend to be short so milliseconds (ms) and microseconds (µs) are often
used. 1ms = 0.001s and 1µs = 0.000001s.
Frequency is the number of cycles per second. It is measured in hertz (Hz), but frequencies
tend to be high so kilohertz (kHz) and megahertz (MHz) are often used. 1kHz = 1000Hz and
1MHz = 1000000Hz.
frequency = 1 / time period and time period = 1 / frequency
3-phase Power
Since the phase impedances of a balanced star- or delta-connected load contain equal
currents, the phase power is one-third of the total power. As a definition, the voltage across the
load impedance and the current in the impedance can be used to compute the power per phase.
Let's assume that the angle between the phase voltage and the phase current is θ, which is
equal to the angle of the impedance. Considering the load configurations given in below fig, the
phase power and the total power can be estimated easily.
In case of delta conneted load in above fig(a) the total active power is equal to three times
the power of one phase.
Since
the
line
current
in
the
balanced
delta
connected
loads
is
substituting this eq the total active power becomes
In star connected load in fig(b) the impedances contain the line currents Iline (= phase current,
Iphase) and the phase voltages ). Therefore, the phase active power and the total active power are
Since the phase voltage in the balanced star connected loads is
substituting this in Ptotal we wil get the same equation as that for the delta connected load.
Similarly, the total reactive and the total apparent power in the three-phase balanced ac circuits
can be given by
AC Power,Real,Reactive,Apparent
Power is defined as the rate of flow of energy past a given point. In alternating current circuits,
energy storage elements such as inductance and capacitance may result in periodic reversals of
the direction of energy flow. The portion of power flow that, averaged over a complete cycle of
the AC waveform, results in net transfer of energy in one direction is known as real power. On
the other hand, the portion of power flow due to stored energy, which returns to the source in
each cycle, is known as reactive power. AC power flow has the three components:



Real power (P), measured in watts (W)
Reactive power (Q), measured in reactive volt-amperes (VAr).
Complex power (S), measured in volt-amperes (VA).S, the modulus of complex power, is
referred to as apparent power
In the diagram, P is the real power, Q is the reactive power (in this case negative), and the
length of S is the apparent power.
The unit for all forms of power is the watt (symbol: W). In practice, however, this is
generally reserved for the real power component. Apparent power is conventionally expressed in
volt-amperes (VA) since it is the simple product of rms voltage and current. The unit for reactive
power is given the special name "VAR", which stands for volt-amperes-reactive.
The mathematical relationship among them can be represented by vectors and is typically
expressed using complex numbers:
S = P + jQ (where j is the imaginary unit)
In the case of a perfectly sinusoidal waveform, P, Q and S can be expressed as vectors that form
a vector triangle with phase angle φ such that:
where
S = VI
P = VI cosφ = S cosφ
Q =VI sinφ = S sinφ
Consider an ideal alternating current (AC) circuit consisting of a source and a generalized
load, where both the current and voltage are sinusoidal,using trigonometric identities, the
instantaneous power may be expressed as the sum of two sinusoids of twice the frequency.
If the load is purely resistive, the two quantities reverse their polarity at the same time; the
direction of energy flow does not reverse; and only real power flows.
If the load is purely inductive or capacitive, then the voltage and current are 90 degrees out of
phase (for a capacitor, current leads voltage; for an inductor, current lags voltage) and there is no
net power flow. This energy flowing backwards and forwards is known as reactive power.
If a capacitor and an inductor are placed in parallel, then the currents caused by the inductor and
the capacitor are in antiphase with each other and therefore partially cancel out rather than
adding to each other. Conventionally, capacitors are considered to generate reactive power and
inductors to consume it. In reality, the load is likely to have resistive, inductive, and capacitive
parts; and so both real and reactive power will flow to the load. The apparent power is the result
of a naive calculation of power from the voltage and current in which the RMS voltage is simply
multiplied by the rms current. Apparent power is handy for rough sizing of generators or wiring,
especially when the power factor is close to 1. However, adding the apparent power for two
loads will not give the total apparent power unless the two loads have the same phase difference
between voltage and current.
The above graph shows the instantaneous and average power calculated from AC voltage
and current with a lagging power factor (φ=45, cosφ=0.71).Average power is the real power and
instantaneous power is the apparent power
Power Factor :
It is defined in several ways
(i)
In alternating-current power transmission and distribution, the cosine of the phase angle
between the voltage and current "cosφ".
(ii)
In AC networks power factor is the ratio of resistane to impedance of the circuit ( R/Z ).
(iii) The power factor of an AC electric power system is defined as the ratio of the real
power to the apparent power ( P/S ), and is a number between 0 to 1 inclusive .
Power factors other than unity have deleterious effects on power transmission systems,
including excessive transmission losses and reduced system capacity.
When the load is inductive, e.g., an induction motor, the current lags the applied voltage,
and the power factor is said to be a lagging power factor. When the load is capacitive,
e.g., a synchronous motor or a capacitive network, the current leads the applied voltage,
and the power factor is said to be a leading power factor.Power factor equals unity (1)
when the voltage and current are in phase, and is zero when the current leads or lags the
voltage by 90 degrees.
Capacitive circuits cause reactive power with the current waveform leading the voltage
wave by 90 degrees, while inductive circuits cause reactive power with the current
waveform lagging the voltage waveform by 90 degrees. The result of this is that
capacitive and inductive circuit elements tend to cancel each other out. By convention,
capacitors are said to generate reactive power while inductors are said to consume it (this
probably comes from the fact that most real-life loads are inductive and so reactive power
has to be supplied to them from power factor correction capacitors).
In power transmission and distribution, significant effort is made to control the reactive
power flow. This is typically done automatically by switching inductors or capacitor
banks in and out, by adjusting generator excitation, and by other means. Electricity
retailers may use electricity meters which measure reactive power to financially penalise
customers with low power factor loads. This is particularly relevant to customers
operating highly inductive loads such as motors at water pumping stations.
B-H Curves
If an alternating magnetic field is applied to a soft magnetic material, the magnetic
induction(B) changes with the magnetic field(H) as shown Fig.1
The hysteresis loop, describing the relation between H and B, is called the magnetization curve.
Initial permeability
The initial permrability μi defines rhe relative permeability at low excitation level.
according to IEC 60401 test condition : F < 1-KHz, B < 0.25mT, 25°.
Saturation magnetization, Bs
At high field strength, the induction flattens out at a value, called the saturation flux density, Bs.
Residual flux density, Br
Residual flux density Br, is a magnetic flux density remaining in material when the
magnetic field strength is decreased to
zero after being magnetized to its saturation point.
Coercive field strength, Hc
The residual flux redueses to zero at a certain reverse field strength which is referred to a
corecivity Hc.
Hard magnetic materials
Permanent magnets are used to produce strong fields without applying a current to coil.
Therefore, they should exhibit a strong net magnetization, and is stable in the presence of
external fields, which requires high coercivity. In hard magnetic materials uniaxial magnetic
anisotropy is necessary and the following magnetic properties are required:
1) High coercivity
2) Large magnetization
3) Rectangular hysteresis loop
Two hysteresis loops, I-H and B-H loops.
I-H loop is useful for physical understanding of the magnetic properties. While, B-H
curve is effective for practical use. It is noticed that
Basic construction, working and types
Electrical transformer is a static electrical machine which transforms electrical power from one
circuit to another circuit, without changing the frequency. Transformer can increase or decrease
the voltage with corresponding decrease or increase in current.
Working principle of transformer
The basic principle behind working of a transformer is the phenomenon of mutual induction
between two windings linked by common magnetic flux. Basically a transformer consists of two
inductive coils; primary winding and secondary winding. The coils are electrically separated but
magnetically linked to each other. When, primary winding is connected to a source of alternating
voltage, alternating magnetic flux is produced around the winding. The core provides magnetic
path for the flux, to get linked with the secondary winding. Most of the flux gets linked with the
secondary winding which is called as 'useful flux' or main 'flux', and the flux which does not get
linked with secondary winding is called as 'leakage flux'. As the flux produced is alternating (the
direction of it is continuously changing), EMF gets induced in the secondary winding according
to Faraday's law of electromagnetic induction. This emf is called 'mutually induced emf', and the
frequency of mutually induced emf is same
as that of supplied emf. If the secondary winding is closed circuit, then mutually induced current
flows through it, and hence the electrical energy is transferred from one circuit (primary) to
another circuit (secondary).
Basic construction of transformer
Basically a transformer consists of two inductive windings and a laminated steel core. The coils
are insulated from each other as well as from the steel core. A transformer may also consist of a
container for winding and core assembly (called as tank), suitable bushings to take our the
terminals, oil conservator to provide oil in the transformer tank for cooling purposes etc.
In all types of transformers, core is constructed by assembling (stacking) laminated sheets of
steel, with minimum air-gap between them (to achieve continuous magnetic path). The steel used
is having high silicon content and sometimes heat treated, to provide high permeability and low
hysteresis loss. Laminated sheets of steel are used to reduce eddy current loss. The sheets are cut
in the shape as E,I and L. To avoid high reluctance at joints, laminations are stacked by
alternating the sides of joint. That is, if joints of first sheet assembly are at front face, the joints
of following assemble are kept at back face.
Types of transformers
Transformers can be classified on different basis, like types of construction, types of cooling etc.
(A) On the basis of construction, transformers can be classified into two types as; (i) Core type
transformer and (ii) Shell type transformer.
(i) Core type transformer
In core type transformer, windings are cylindrical former wound, mounted on the core limbs as
shown in the figure above. The cylindrical coils have different layers and each layer is
insulated from each other. Materials like paper, cloth or mica can be used for insulation. Low
voltage windings are placed nearer to the core, as they are easier to insulate.
(ii) Shell type transformer
The coils are former wound and mounted in layers stacked with insulation between them. A shell
type transformer may have simple rectangular form as shown in above fig, or it may have a
distributed form.
(B) On the basis of their purpose
1. Step up transformer: Voltage increases (with subsequent decrease in current) at
secondary.
2. Step down transformer: Voltage decreases (with subsequent increase in current) at
secondary.
(C) On the basis of type of supply
1. Single phase transformer
2. Three phase transformer
(D) On the basis of their use
1. Power transformer: Used in transmission network, high rating
2. Distribution transformer: Used in distribution network, comparatively lower rating than
that of power transformers.
3. Instrument transformer: Used in relay and protection purpose in different instruments in
industries
o
Current transformer (CT)
o
Potential transformer (PT)
(E) On the basis of cooling employed
1. Oil-filled self cooled type
2. Oil-filled water cooled type
3. Air blast type (air cooled)
LECTURER-17
Voltage Transformation Ratio
In a transformer, source of alternating current is applied to the primary winding. Due to this, the
current in the primary winding (called as magnetizing current) produces alternating flux in the
core of transformer. This alternating flux gets linked with the secondary winding, and because of
the phenomenon of mutual induction an emf gets induced in the secondary winding. Magnitude
of this induced emf can be found by using the following EMF equation of the transformer.
EMF equation of the Transformer
Let,
N1 = Number of turns in primary winding
N2 = Number of turns in secondary winding
Φm = Maximum flux in the core (in Wb) = (Bm x A)
f = frequency of the AC supply (in Hz)
As, shown in the fig., the flux rises sinusoidally to its maximum value Φm from 0. It reaches to
the maximum value in one quarter of the cycle i.e in T/4 sec (where, T is time period of the sin
wave of the supply = 1/f).
Therefore,
average rate of change of flux = Φm /(T/4) = Φm /(1/4f)
Therefore,
average rate of change of flux = 4f Φm
....... (Wb/s).
Now,
Induced emf per turn = rate of change of flux per turn
Therefore, average emf per turn = 4f Φm ..........(Volts).
Now, we know, Form factor = RMS value / average value
Therefore, RMS value of emf per turn = Form factor X average emf per turn.
As, the flux Φ varies sinusoidally, form factor of a sine wave is 1.11
Therefore, RMS value of emf per turn = 1.11 x 4f Φm = 4.44f Φm.
RMS value of induced emf in whole primary winding (E1) = RMS value of emf per turn X
Number of turns in
primary winding
E1 = 4.44f N1 Φm
............................. eq 1
Similarly, RMS induced emf in secondary winding (E2) can be given as
E2 = 4.44f N2 Φm.
............................ eq 2
from the above equations 1 and 2,
𝐸1
𝑁1
=
𝐸2
𝑁2
=4.44f Φm
This is called the emf equation of transformer, which shows, emf / number of turns is same for
both primary and secondary winding.
For an ideal transformer on no load, E1 = V1 and E2 = V2
.
where, V1 = supply voltage of primary winding
V2 = terminal voltage of secondary winding
Voltage Transformation Ratio (K)
As derived above,
𝐸2
𝐸1
=
𝑁2
𝑁1
=K
Where, K = constant
This constant K is known as voltage transformation ratio.

If N2 > N1, i.e. K > 1, then the transformer is called step-up transformer.
If N2 < N1, i.e. K < 1, then the transformer is called step-up transformer
Q-1) A transformer has a primary voltage of 480 volts and a secondary voltage of 120 volts.
If the primary windings have 700 turns, how many turns are in the secondary windings?
Cross multiply:
Q-2) A transformer has 20 primary windings and 100 secondary windings. If the secondary
voltage is 25 v, find the primary voltage.
ANS.
100/20=25/X
CROSS MULTIPLY
100X=20*25
100X=500
X=500/100
X=5 VOLTS IN THE PRIMARY.
Q-3) If the primary to secondary voltage ratio is 4:1 for a transformer, what is the secondary
voltage if the primary voltage is 460 volts?
ANS.
primary/secondary=4/1=460/x
> 4x=460
So
--> x=115
So the secondary voltage is 115 Volts
Q-4) An ideal transformer with a 300 turn primary connected to a 480 V, 60 Hz supply line
needs to output 120 V from the secondary. If a 100 Ω resistor is connected across the
secondary, determine: A) How many turns the secondary must have to output the desired
voltage. B) The current drawn through the primary. C) The maximum flux in the core of
the transformer
Solution:
A):The ratio of primary voltage to secondary voltage is directly proportional to the ratio of
number
of
Where
turns
on
the
primary
Voltage across primary,
turns in primary,
to
number
of
turns
on
Voltage across secondary,
the
secondary:
Number of
Number of turns in secondary
To solve for the number of turns required for the secondary, the equation is rearranged solving
for
:
B):The ratio of primary current to secondary current is inversely proportional to the ratio of
number
of
Where
turns
on
the
primary
Current in primary,
primary,
to
number
of
turns
Current in secondary,
on
the
secondary:
Number of turns in
Number of turns in secondary
Rearranging to solve for
:
C)The induced emf of the secondary can be calculated by:
Solving for
, we can calculate the maximum flux in the core:
Where
max flux in core,
Number of turns in secondary
Voltage across secondary,
Frequency of line,