1.6 Real Single-Phase Transformer.

The ideal transformer in Section 1.5 can never been made. The real transformer has many imperfections.

The real transformers consists of two or more coils of wire physically wrapped around the ferromagnetic core. The real transformer approximate the characteristic of the ideal transformer.

Operation if the real transformer; (i) It consists of two coils of wire wrapped around a transformer core.(ii) The primary of the transformer is connected to an ac power source, and the secondary winding is an open-circuited.(iii) Figure 1.5 is the hysteresis of the transformer.(iv) Basic operation from the faraday law,

dt

deind

l is the flux linkage in the coil across which the voltage is being induced.

The sum of the flux passing through each turn in the coil added over all the turns of the coil is;

The average flux per turns is given by ;

And Faraday’s law can be written as ,

N

N

ii

1

dt

dNeind

1.7 The Equivalent Circuit of a Real Transformer.

Copper losses are resistive loses in the windings of the transformer core.

Copper losses are modeled by placing a resistor Rp in the

primary circuit of the transformer and a resistor Rs in the

secondary circuit.

m

p_leakage

s_leakage

Leakage flux in the real transformer

1.7 The Equivalent Circuit of a Real Transformer.

The leakage flux in the windings is,

Core excitation effect can be model as

i) magnetization reactance, XM

ii) core resistance, Rc

Figure 1.7 is an exact model of a transformer. To analyze the transformer it is necessary to convert the entire circuit to an equivalent circuit at a single voltage level as in Figure 1.8.

dt

diLte

dt

diLte

ssLS

ppLP

)(

)(

Figure 1.7: Model of a Real Transformer.

Figure 1.8: (a) The Transformer Model Referred to its Primary Windings (top).

(b) The Transformer Model Referred to its Secondary Voltage Level (bottom).

Symbols used for the Exact Equivalent Circuit above;

1.7 The Equivalent Circuit of a Real Transformer.

Figure below is an exact model of a transformer.

Transferring impedances through a transformer

Vac Zload

T

VP VS

IP IS

S

S

S

S

P

PP I

VIV

I

VZ 2a

a

a

loadP ZZ 2aThévenin equivalents of transformer circuit

S

S

I

VZZ sload

Vac a2ZloadVP

IP

Vac/a ZloadVS

IS

Equivalent circuit when secondary impedance is transferred to primary side and ideal transformer eliminated

Equivalent circuit when primary source is transferred to secondary side and ideal transformer eliminated

Transferring impedances through a transformer

To analyze practical circuits containing transformer, it is necessary to convert the entire circuit to an equivalent circuit at a single voltage level.

Therefore, the equivalent circuit must be referred either to its primary side or to its secondary side in problem solving.

Cont’d…

The Transformer Model Referred to its Primary Windings

The Transformer Model Referred to its Secondary Voltage Level.

Cont’d…

Cont’d…

The Transformer Model Referred to its Primary Windings

Based on the above equations and assuming a zero degree reference angle for V2, the phasor diagram is shown as

Cont’d…

1.8 The Approximate Equivalent Circuit of a Transformer. In the Approximate model the voltage drop in Rp and Xp is negligible because the current is very small.

Approximate Transformer Model Referred to the Primary Side.

The voltage in the primary series impedance (r1 + jx1) is

small, even at full load. Also, the no load current (I0) is so

small that its effect on the voltage drop in the primary series impedance is negligible.

Therefore, it matters little if the shunt branch of Rc in

parallel with Xm is connected before the primary series

impedance or after it. The core loss and magnetizing currents are not greatly affected by the move.

Connecting the shunt components right at the input terminals has the great advantage of permitting the two series impedance to be combined into one complex impedance.

The equivalent impedance referred to the primary side is;

Cont’d…

Figure 1.11 shows the approximate equivalent circuit of transformer referred to the secondary side.

Approximate Circuit Model of a Transformer Referred to the Secondary.

The equivalent impedance is;

Cont’d…

1.9 Transformer Voltage Regulation and Efficiency.

Voltage regulation is a measure of the change in the terminal voltage of the transformer with respect to loading. Therefore the voltage regulation is defined as:

For ideal transformer, VR = 0. It is a good practice to have as small voltage regulator as possible.

%100,

,,

fls

flsnls

V

VVVR

Transformer Efficiency, efficiency of a transformer is defined as follows;

For Non-Ideal transformer, the output power is less than the input power because of losses.

These losses are the winding or I2R loss (copper losses) and the core loss (hysteresis and eddy-current losses).

Thus, in terms of the total losses, Plosses, the above

equation may be expressed as;

1

2

P

P

PowerOutput

PowerOutput

Cont’d…

corecopperlosses

losses

PPP

P

PP

P

P

PP

2

2

2

2

1

\11

Ex 3: Transformer Voltage Regulation.

Cont’d…Ex.3

1.10 Open Circuit and Short Circuit. - Determination of transformer parameter by measurement

Open Circuit Test. Provides magnetizing reactance and core loss resistance Obtain components are connected in parallel The open circuit test is conducted by applying rated

voltage at rated frequency to one of the windings, with the other windings open circuited.

The input power and current are measured. For reasons of safety and convenience, the measurements

are made on the low-voltage (LV) side of the transformer.

The secondary / high voltage (HV) side is open, the input current is equal to the no load current or exciting current (I0), and is quite

small. The input power is almost equal to the core loss at rated voltage

and frequency.

Cont’d…

Equivalent Circuit of the Open-Circuit Test.

Cont’d…

Open circuit test evaluation

Short Circuit Test.

The short-circuit test is used to determine the equivalent series resistance and reactance.

Provides combined leakage reactance and winding resistance

One winding is shorted at its terminals, and the other winding is connected through proper meters to a variable, low-voltage, high-current source of rated frequency.

The source voltage is increased until the current into the transformer reaches rated value. To avoid unnecessary high currents, the short-circuit measurements are made on the high-voltage side of the transformer.

Cont’d…

Equivalent Circuit of the Short-Circuit Test.

Cont’d…

2_

2__ HVeqHVeqHVeq RZX

sc

scHVeq I

VZ _HVeqscsc RIP _

2

2_sc

scHVeq I

PR

Equivalent circuit obtained by measurement

Equivalent circuit for a real transformer resulting from the open and short circuit tests.

Xm_p Rc_p

Xe_s Re_s

1.11 Three Phase Transformer.

Almost all the major power generation and distribution systems in the world today are three-phase ac system.

Two ways of constructing transformer of three-phase circuit;

(i) Three single phase transformers are connected in three-phase bank.

(ii) Make a three-phased transformer consisting of three sets of windings wrapped on a common core.

The three-phased transformer on a common core (ii) is preferred because it is lighter, smaller, cheaper and slightly more efficient.

Three-Phase Transformer Connections.

A three-phase transformer consists of three transformers either separate or combined on one core.

3 separate core

combined on one core.

Three-Phase Transformer Connections.

There are four possible connections between the secondary and primary of a three-phase transformer.

(1) Wye-Wye (Y-Y). - don't use causes harmonics problems

(2) Wye-Delta (Y-). - use : high voltage transmissions

(3) Delta-Wye (Y) - use : most common; commercial and industrial.

(4) Delta-Delta (). - use: industrial applications

Figure 1.15: Three-Phase Transformer Connections and Wiring Diagram

(1) Wye-Wye Connection.

(2) Wye-Delta Connection.

(3) Delta-Wye Connection.

(4) Delta-Delta Connection.

YaV

V

LS

LP 3

YaV

V

LS

LP 3

aV

V

V

V

S

P

LS

LP

YYaV

V

LS

LP

Ex. 2.6: Three-Phase Transformer.What should be the ratings (voltages and currents) and turns ratio of a three-phase transformer to transform 10 MVA from 230 kV to 4160 V, if the transformer is to be connected:a) wye-delta, b) delta-wye, and c) delta-delta?Solution

For both delta and wye connections, the line currents can be obtained as:

.

Cont’d…

Ex. 7: Voltage Regulation at Full Load.A 7200V/208V, 50kVA, three-phase distribution transformer is connected delta-wye. The transformer has 1.2% resistance and 5% reactance. Find the voltage regulation at full load, 0.8 power factor lagging.

Voltage Regulation.

Ex.8: Transformer Efficiency.

If the core loss of the transformer in Example 7 is 1kW, find the efficiency of this transformer at full load and 0.8 power factor.

Solution