Transformers: Auto-Transformer, Ideal and Practical Transformers, Losses and Efficiency

Transformers:

Transformer is the device that transmits electrical energy from one circuit to the other. the current transmitted is the AC current. It is commonly used to increase or decrease the supply voltage without changing the frequency of AC between circuits. it works on the basic principles of electromagnetic induction and the Faradays laws.

Transformers are used in various fields like power generation grid, distribution sector, transmission and electric energy consumption. Transformers are classified in to types-

Commonly used transformer type, depending upon voltage they are classified as:

Step-up Transformer: They are used between the power generator and the power grid. The secondary output voltage is higher than the input voltage.
Step down Transformer: These transformers are used to convert high voltage primary supply to low voltage secondary output.

In a transformer, we will find different types of cores that are used.

Air core Transformer: The flux linkage between primary and secondary winding is through the air. The coil or windings wound on the non-magnetic strip.
Iron core Transformer: Windings are wound on multiple iron plates stacked together, which provides a perfect linkage path to generate flux.

Autotransformer: It will have only one winding wound over a laminated core. The primary and secondary share the same coil.

The winding AB of total turns N1 is considered as primary winding. This winding is tapped from point ′C′ and the portion BC is considered as secondary. Let’s assume the number of turns in between points ′B′ and ′C′ is N2.If V1 voltage is applied across the winding i.e. in between ′A′ and ′C′.

Hence, the voltage across the portion BC of the winding, will be,

When load is connected between secondary terminals i.e. between ′B′ and ′C′, load current I2 starts flowing. The current in the secondary winding or common winding is the difference of I2 and I1.

The advantages of an auto transformer include:

An auto transformer has higher efficiency than two winding transformer. This is because of less ohmic loss and core loss due to reduction of transformer material.
Auto transformer has better voltage regulation as voltage drop in resistance and reactance of the single winding is less.

The applications of an auto transformer include:

Compensating voltage drops by boosting supply voltage in distribution systems.
Auto transformers with a number of tapping are used for starting induction and synchronous motors.
Auto transformer is used as variance in laboratory or where continuous variable over broad ranges are required.

The electrical energy is first converted into mechanical energy. The magnetic energy flow through the magnetic core. The energy is wasted in the core during magnetization and demagnetization of the core. Iron loss and copper loss are part of losses in a transformer. The difference between iron loss and copper loss is as follows:

Iron loss: This is defined as the loss that is caused due to the alternating flux in the core of the transformer. As the loss occurs in the core, therefore the iron loss is also known as core loss. There are two types of iron losses, and they are

=>Hysteresis loss

Ph=KhfBmaxn

=>Eddy current loss

Pe=Kef2Bmaxnt2

Copper loss: This is defined as the heat that is produced by the electrical currents in the conductors of transformer windings. It is an undesirable transfer of energy resulting from the induced currents in the adjacent components.

Winding Resistances: -

As the windings of a transformer are made up of copper conductors. Therefore, both the primary and secondary windings will have winding resistances, which produce the copper loss or I2R loss in the transformer. The primary winding resistance R1 and the secondary winding resistance R2 act in series with the respective windings.

Efficiency

The efficiency of transformer is also introduced as the same ratio of output energy to the input energy (efficiency = output/input). Electrical machines like transformers are greatly efficient instruments.
efficiency of these transformers differs from 95% to 98.5%. Once a transformer is typically efficient, then the output, as well as the input, has almost the same value.

The efficiency of the transformer can be expressed as the intensity or the value of power waste through a transformer. Thus, the ratio of the secondary section’s power output to the primary section’s input can be obtained. The efficiency can be presented as the following:

Efficiency (η) = (Power Output/Power Input)

Whenever an ideal device is considered with no wastes, then the power of the transformer will be stable since the voltage V has multiplied across the stable current I. So, the power across the primary winding is identical to the power through the secondary section.

$P_{p r ima ry} = P_{seco n d a ry}$

${V}_{P}{I}_{P}Cos({\phi }_{P})={V}_{S}{I}_{S}Cos({\phi }_{S})$

Where ∅s and ∅P are secondary as well as primary phase angles.

Let us purpose x as the percentage of full or rated load ‘S’ (VA), Pcufl(watts) as the complete load copper waste, and cos(θ) as the power coefficient of the load. We can also define Pi (watts) as the core waste. As copper and iron wastes are the main losses in the transformer hence only these two forms of losses are considered while measuring efficiency. Then the efficiency of transformer can be formulated as:

The condition for maximum efficiency is:

Now we can obtain the maximum efficiency as:

Ideal and Practical Transformers

An ideal transformer is regarded as a conceptual model which possesses all the fundamental features of a practical transformer excluding the energy losses. Ideal transformer cannot be achieved in practical world.

Ideal Transformer

A ideal transformer is an imaginary transformer which has:

-> no copper losses
-> no iron loss in the core
-> no leakage flux

In ideal transformer, input power = output power.

=>Characteristics of Ideal Transformer

Zero Winding Resistance : Resistance of both primary and secondary winding is 0 i.e. both the coils are purely inductive in nature.

100% Efficiency : There are no losses in ideal transformer so the input power = output power ⇒(E1)(I1)=(E2)(I2)

No leakage flux : The whole amount of flux is linked from primary to secondary winding, so there is no leakage flux.

No Iron loss : As the iron core is subjected to alternating flux there occurs eddy current and hysteresis loss in it. These two losses together are called Iron loss. It is 0 in ideal transformer.

When an alternating voltage V1 is supplied to the primary winding of an ideal transformer, counter emf E1 is induced in the primary winding. Since there is no resistance, this induced emf E1 will be exactly equal to the applied voltage but in 180 degree opposite in phase.
The current drawn from the source produces required magnetic flux. As the primary winding resistance is 0, the current lags emf E1 by 90 degree. This is current is called Magnetizing current Iμ. This magnetizing current produces alternating magnetic flux φ. This flux gets linked with the secondary winding and emf E2 is induced by mutual induction. This E2 is in phase with E1. If the circuit is closed at secondary winding, then secondary current I2 is produced.

=>Phasor diagram

Practical Transformer

In practical transformer, we have winding resistance, leakage flux, and iron losses, etc.
Here we have two cases:
(a) No load
(b) On load

=>Practical Transformer on No Load

The circuit on the secondary side is open.

V1 is the primary voltage and I1 is the primary current. Now I1 has two components:

a) One component is responsible for generation of magnetic flux. This is called Magnetizing component of I1 and is denoted by Iμ.

b) Second component which is responsible for magnetic losses (Hysteresis and Eddy current losses) and primary winding losses. This is called Core loss component of Ic . So its equivalent circuit diagram is:

where on primary side,

V1 is Primary Voltage
R1 is Primary Winding Resistance
X1 is Primary Leakage Reactance
I0 is No Load Primary Current
Ic is Core Loss component of I1
Iμ is Magnetizing Component of I1
Rc is Core loss resistance
Xm is Magnetizing Reactance
N1 is Number of turns in Primary Winding
E1 is Primary induced Emf

On secondary side:

R2 is Secondary winding resistance
X2 is Secondary leakage reactance
N2 is Secondary winding turns
E2 is Secondary induced emf
V2 is Secondary terminal voltage
=>Phasor Diagram
First we will start with reference line which is common to both primary and secondary curcuit, here it is flux φ. Now using KVL in primary circuit V1=−E1+I0R1+jI0X1

There is no direct relation between φ and V1 so that we could directly draw phasor. But we have relation with E1 and φ, as φ is responsible for both E1 and E2. E1 and E2 both lag φ by 90 degrees. Here we will consider E1 < E2.
φ is produced due to magnetizing current Iμ therefore Iμ is in phase with φ. As we have -E1 in our equation (because it is in opposite direction of magnetizing current) so we will draw a phasor opposite to E1. Ic is 90 degrees leading from Iμ so it is in phase with
-E1 As I0 = Ic + Iμ , hence the phasor I0.
Now I0R1 is voltage drop in R1 and it is in phase with I0 , and as it is added to -E1 so the phasor of I0R1 will be drawn at the head of -E1. I0X1 is voltage drop across X1 and is 90 degrees leading I0R1, so it is drawn perpendicular to I0R1.
Now following the equation, V1 phasor is drawn adding all the quantities given in the equation.
=>Practical Transformer on Load
The secondary side is closed-circuited with a load.
Its equivalent circuit diagram:
where on primary side,
- V1 is Primary Voltage
- I1 is Primary Current
- I'2 is Primary Current to neutralize the demagnetizing effects of I2, I'2 = K I2
- I0 is No Load Primary Current
- R1 is Primary Winding Resistance
- X1 is Primary Leakage Reactance
- Ic is Core Loss component of I1
- Iμ is Magnetizing Component of I1
- I'1 component of I1 (doubtful)
- Rc is Core loss resistance
- Xm is Magnetizing Reactance
- N1 is Number of turns in Primary Winding
- E1 is Primary induced Emf
- R2 is Secondary winding resistance
- X2 is Secondary leakage reactance
- N2 is Secondary winding turns
- E2 is Secondary induced emf
- V2 is Secondary voltage
- I2 is Secondary current
=>Phasor Diagram
Again we will start with reference line, here it is φ.
Using KVL in secondary circuit,
$V_{2} = E_{2} - I_{2} R_{2} - j I_{2} X_{2}$
As there is no direct relation between V1 and φ so we will use the equation to draw all three quantities and then add them to get V1.
E2 and E1 lag φ by 90 degrees. Here we will consider E2 < E1.
I2 lags E2 by phase difference of φ2
Now voltage drop I2R2 across R2 will be in phase with I2 but we need -I2R2 so we will it in opposite direction of I2. As it is to be added to E2 so it will be drawn at the head of E2.
I2X2 is voltage drop across X2 and is leading by 90 degrees from I2R2. Again it is -ve in magnitude. Adding all the quantities we have V2.
Now in primary circuit,
V1 = - E1 + I1R1 + jI1X1
I1 = I'2 + I0 where I'2 = -KI2
I0 = Iμ + Ic
Again we will draw Iμ first as it is directly connected to φ, then Ic and then I0 .
I'2 is opposite to I2 and is added to I0. The resultant phasor is I1.
I1R1 is in phase with I1 and is added to -E1. I1X1 is 90 degrees leading I1R1.
Phase difference between V1 and I1 is φ1
Power Factor = cosφ1
Input Power = V1I1cosφ1

Phase difference between V2 and I2 is φ2
Power Factor = cosφ2
Input Power = V2I2cosφ2

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