ECE 476 - staff.fit.ac.cy

Lecture 9Transformers, Per Unit

Professor Tom OverbyeDepartment of Electrical and

Computer Engineering

ECE 476POWER SYSTEM ANALYSIS

1

Announcements

z Be reading Chapter 3z HW 3 is 4.32, 4.41, 5.1, 5.14. Due September 22 in class.

2

Transformer Equivalent Circuit

Using the previous relationships, we can derive an equivalent circuit model for the real transformer

' 2 '2 2 1 2' 2 '2 2 1 2

This model is further simplified by referring allimpedances to the primary side

r e

e

a r r r r

x a x x x x

�

�

3

Simplified Equivalent Circuit

4

Calculation of Model Parameters

z The parameters of the model are determined based upon – nameplate data: gives the rated voltages and power– open circuit test: rated voltage is applied to primary with

secondary open; measure the primary current and losses (the test may also be done applying the voltage to the secondary, calculating the values, then referring the values back to the primary side).

– short circuit test: with secondary shorted, apply voltage to primary to get rated current to flow; measure voltage and losses.

5

Transformer Example

Example: A single phase, 100 MVA, 200/80 kV transformer has the following test data:

open circuit: 20 amps, with 10 kW lossesshort circuit: 30 kV, with 500 kW losses

Determine the model parameters.

6

Transformer Example, cont’d

e

2sc e

2 2e

2

e

100 30500 , R 60200 500

P 500 kW R 2 ,

Hence X 60 2 60

200 410

200R 10,000 10,00020

sc e

e sc

c

e m m

MVA kVI A jXkV A

R I

kVR MkW

kVjX jX XA

� :

o :

� :

:

� � : :

From the short circuit test

From the open circuit test

7

Residential Distribution Transformers

Single phase transformers are commonly used in residential distribution systems. Most distributionsystems are 4 wire, with a multi-grounded, commonneutral.

8

Per Unit Calculations

z A key problem in analyzing power systems is the large number of transformers. – It would be very difficult to continually have to refer

impedances to the different sides of the transformersz This problem is avoided by a normalization of all

variables.z This normalization is known as per unit analysis.

actual quantityquantity in per unit base value of quantity

9

Per Unit Conversion Procedure, 1I

1. Pick a 1I VA base for the entire system, SB

2. Pick a voltage base for each different voltage level, VB. Voltage bases are related by transformer turns ratios. Voltages are line to neutral.

3. Calculate the impedance base, ZB= (VB)2/SB

4. Calculate the current base, IB = VB/ZB

5. Convert actual values to per unit

Note, per unit conversion on affects magnitudes, not the angles. Also, per unit quantities no longer have units (i.e., a voltage is 1.0 p.u., not 1 p.u. volts)

10

Per Unit Solution Procedure

1. Convert to per unit (p.u.) (many problems are already in per unit)

2. Solve3. Convert back to actual as necessary

11

Per Unit Example

Solve for the current, load voltage and load power in the circuit shown below using per unit analysis with an SB of 100 MVA, and voltage bases of 8 kV, 80 kV and 16 kV.

Original Circuit

12

Per Unit Example, cont’d

2

2

2

8 0.64100

80 6410016 2.56

100

LeftB

MiddleB

RightB

kVZMVAkVZMVAkVZMVA

:

:

:

Same circuit, withvalues expressedin per unit.

13


L

2*

1.0 0 0.22 30.8 p.u. (not amps)3.91 2.327

V 1.0 0 0.22 30.8p.u.

0.189 p.u.

1.0 0 0.22 30.8 30.8 p.u.

LL L L

G

Ij

VS V IZ

S

� q �� q

� � q � �� qu ��q �� q�

� qu � q �� q�

14


To convert back to actual values just multiply theper unit values by their per unit base

LActual

ActualLActualG

MiddleB

ActualMiddle

0.859 30.8 16 kV 13.7 30.8 kV

0.189 0 100 MVA 18.9 0 MVA

0.22 30.8 100 MVA 22.0 30.8 MVA100 MVAI 1250 Amps

80 kVI 0.22 30.8 Amps 275 30.8

V

S

S

�� qu �� q

� qu � q

� qu � q

�� qu�� q�$

15

Three Phase Per Unit

1. Pick a 3I VA base for the entire system, 2. Pick a voltage base for each different voltage

level, VB. Voltages are line to line. 3. Calculate the impedance base

Procedure is very similar to 1I except we use a 3IVA base, and use line to line voltage bases

3BSI

2 2 2, , ,3 1 1

( 3 )3

B LL B LN B LNB

B B B

V V VZ

S S SI I I

Exactly the same impedance bases as with single phase!

16

Three Phase Per Unit, cont'd

4. Calculate the current base, IB

5. Convert actual values to per unit

3 1 13 1B B

, , ,

3I I3 3 3B B B

B LL B LN B LN

S S SV V V

I I II I

Exactly the same current bases as with single phase!

17

Three Phase Per Unit Example

Solve for the current, load voltage and load power in the previous circuit, assuming a 3I power base of300 MVA, and line to line voltage bases of 13.8 kV,138 kV and 27.6 kV (square root of 3 larger than the 1I example voltages). Also assume the generator is Y-connected so its line to line voltage is 13.8 kV. Convert to per unit

as before. Note thesystem is exactly thesame!

18

3I Per Unit Example, cont'd

L

2*

1.0 0 0.22 30.8 p.u. (not amps)3.91 2.327

V 1.0 0 0.22 30.8p.u.

0.189 p.u.

1.0 0 0.22 30.8 30.8 p.u.

LL L L

G

Ij

VS V IZ

S

� q �� q

� � q � �� qu ��q �� q�

� qu � q �� q�

Again, analysis is exactly the same!

19

3I Per Unit Example, cont'd

LActual

ActualLActualG

MiddleB

ActualMiddle

0.859 30.8 27.6 kV 23.8 30.8 kV

0.189 0 300 MVA 56.7 0 MVA

0.22 30.8 300 MVA 66.0 30.8 MVA300 MVAI 125 (same cur0 Amps

3138 kV

I 0.22 30.

rent!)

8

V

S

S

�� qu �� q

� qu � q

� qu � q

�� qu Amps 275 30.8�� q�$

Differences appear when we convert back to actual values

20

3I Per Unit Example 2

Assume a 3I load of 100+j50 MVA with VLL of 69 kV is connected to a source through the below network:

What is the supply current and complex power?

Answer: I=467 amps, S = 103.3 + j76.0 MVA

21

Per Unit Change of MVA Base

z Parameters for equipment are often given using power rating of equipment as the MVA base

z To analyze a system all per unit data must be on a common power base

2 2

Hence Z /

Z

base base

OriginalBase NewBasepu actual pu

OriginalBase NewBasepu puOriginalBase NewBase

BaseBaseNewBase

OriginalBase NewBaseBasepu puOriginalBase

Base

Z Z Z

V VZ

SS

S ZS

o o

u

u

22

Per Unit Change of Base Example

A 54 MVA transformer has a leakage reactance or 3.69%. What is the reactance on a 100 MVA base?

1000.0369 0.0683 p.u.54eX u

23

Transformer Reactance

z Transformer reactance is often specified as a percentage, say 10%. This is a per unit value (divide by 100) on the power base of the transformer.

z Example: A 350 MVA, 230/20 kV transformer has leakage reactance of 10%. What is p.u. value on 100 MVA base? What is value in ohms (230 kV)?

2

1000.10 0.0286 p.u.350

2300.0286 15.1100

eX u

u :

24

Three Phase Transformers

There are 4 different ways to connect 3I transformersY-Y '-'

Usually 3I transformers are constructed so all windingsshare a common core

25

3I Transformer Interconnections

'-Y Y-'

26

Y-Y Connection

Magnetic coupling with An/an, Bn/bn & Cn/cn1, ,An AB A

an ab a

V V Ia aV V I a

27

Y-Y Connection: 3I Detailed Model

28

Y-Y Connection: Per Phase Model

Per phase analysis of Y-Y connections is exactly the same as analysis of a single phase transformer.

Y-Y connections are common in transmission systems.

Key advantages are the ability to ground each side and there is no phase shift is introduced.

29

'-' Connection

Magnetic coupling with AB/ab, BC/bb & CA/ca1 1, ,AB AB A

ab ab a

V I IaV I a I a

30

'-' Connection: 3I Detailed Model

To use the per phase equivalent we need to usethe delta-wye load transformation

31

'-' Connection: Per Phase Model

Per phase analysis similar to Y-Y except impedances are decreased by a factor of 3.

Key disadvantage is '-' connections can not be grounded; not commonly used.

32

'-Y Connection

Magnetic coupling with AB/an, BC/bn & CA/cn

33

'-Y Connection V/I Relationships

, 3 30

30 30Hence 3 and 3

For current we get1

13 30 303

1 303

AB ABan ab an

an

AB Anab an

ABa AB

ab

A AB AB A

a A

V Va V V VV a

V VV Va a

I I a II a

I I I I

a I

o � q

� q � q

o

�� qo � q

, � q

34

'-Y Connection: Per Phase Model

Note: Connection introduces a 30 degree phase shift!

Common for transmission/distribution step-down sincethere is a neutral on the low voltage side.

Even if a = 1 there is a sqrt(3) step-up ratio

35

Y-' Connection: Per Phase Model

Exact opposite of the '-Y connection, now with a phase shift of -30 degrees.

36

Load Tap Changing Transformers

z LTC transformers have tap ratios that can be varied to regulate bus voltages

z The typical range of variation is r10% from the nominal values, usually in 33 discrete steps (0.0625% per step).

z Because tap changing is a mechanical process, LTC transformers usually have a 30 second deadband to avoid repeated changes.

z Unbalanced tap positions can cause "circulating vars"

37

LTCs and Circulating Vars

slack

1 1.00 pu

2 3

40.2 MW

40.0 MW

1.7 Mvar

-0.0 Mvar

1.000 tap 1.056 tap

24.1 MW 12.8 Mvar

24.0 MW-12.0 Mvar

A

MVA

1.05 pu 0.98 pu

24 MW 12 Mvar

64 MW 14 Mvar

40 MW 0 Mvar

0.0 Mvar

80%A

MVA

38

Phase Shifting Transformers

z Phase shifting transformers are used to control the phase angle across the transformer

z Since power flow through the transformer depends upon phase angle, this allows the transformer to regulate the power flow through the transformer

z Phase shifters can be used to prevent inadvertent "loop flow" and to prevent line overloads.

400 MVA 220/155 kV phase-shifting

transformer.

Search Wikipedia

Last edited 9 months ago by Yobot

A phase angle regulating transformer, phase

angle regulator (PAR, American usage), phase-

shifting transformer, phase shifter (West coast

American usage), or quadrature booster (quad

booster, British usage), is a specialised form of

transformer used to control the flow of real power

on three-phase electricity transmission networks.

For an alternating current transmission line, power

flow through the line is proportional to the sine of the difference in the phase angle of

the voltage between the transmitting end and the receiving end of the line.[1] Where

parallel circuits with different capacity exist between two points in a transmission grid

(for example, an overhead line and an underground cable), direct manipulation of the

phase angle allows control of the division of power flow between the paths,

preventing overload.[2] Quadrature boosters thus provide a means of relieving

overloads on heavily laden circuits and re-routing power via more favorable paths.

Alternately, where an interchange partner is intentionally causing significant

"inadvertent energy" to flow through an unwilling interchange partner's system, the

unwilling partner may threaten to install a phase shifter to prevent such "inadvertent

energy", with the unwilling partner's tactical objective being the improvement of his

system's stability at the expense of the other system's stability. As power system

reliability is really a regional or national strategic objective, the threat to install a

phase shifter is usually sufficient to cause the other system to implement the

required changes to his system to reduce or eliminate the "inadvertent energy".

The capital cost of a quadrature booster can be high: as much as four to six million

GBP (6–9 million USD) for a unit rated over 2 GVA. However, the utility to

transmission system operators in flexibility and speed of operation, and particularly

Quadrature booster

39

Phase Shifter Example 3.13

slack

Phase Shifting Transformer

345.00 kV 341.87 kV

0.0 deg 216.3 MW 216.3 MW

283.9 MW 283.9 MW

1.05000 tap

39.0 Mvar 6.2 Mvar

93.8 Mvar 125.0 Mvar

500 MW

164 Mvar 500 MW 100 Mvar

40

ComED Control Center

41

ComED Phase Shifter Display

42

Autotransformers

z Autotransformers are transformers in which the primary and secondary windings are coupled magnetically and electrically.

z This results in lower cost, and smaller size and weight.

z The key disadvantage is loss of electrical isolation between the voltage levels. Hence auto-transformers are not used when a is large. For example in stepping down 7160/240 V we do not ever want 7160 on the low side!

Home (http://www.electronics-tutorials.ws) » Transformers (http://www.electronics-tutorials.ws/category/transformer) » The AutotransformerSearch

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The Autotransformer

Autotransformer Basics

Unlike the previous voltage transformer which has two electrically isolated windings called: the primary and the secondary, an Autotransformer

has only one single voltage winding which is common to both sides. This single winding is “tapped” at various points along its length to provide a

percentage of the primary voltage supply across its secondary load. Then the autotransformer has the usual magnetic core but only has one

winding, which is common to both the primary and secondary circuits.

Therefore in an autotransformer the primary and secondary windings are linked together both electrically and magnetically. The main advantage of this type

of transformer design is that it can be made a lot cheaper for the same VA rating, but the biggest disadvantage of an autotransformer is that it does not have

the primary/secondary winding isolation of a conventional double wound transformer.

The section of winding designated as the primary part of the winding is connected to the AC power source with the secondary being part of this primary

winding. An autotransformer can also be used to step the supply voltage up or down by reversing the connections. If the primary is the total winding and is

connected to a supply, and the secondary circuit is connected across only a portion of the winding, then the secondary voltage is “stepped-down” as shown.

Autotransformer Design

When the primary current I is !owing through the single winding in the direction of the arrow as shown, the secondary current, I , !ows in the opposite

direction. Therefore, in the portion of the winding that generates the secondary voltage, V the current !owing out of the winding is the di"erence of I and I .P S

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direction. Therefore, in the portion of the winding that generates the secondary voltage, V the current !owing out of the winding is the di"erence of I and I .

The Autotransformer can also be constructed with more than one single tapping point. Auto-transformers can be used to provide di"erent voltage points

along its winding or increase its supply voltage with respect to its supply voltage V as shown.

Autotransformer with Multiple Tapping Points

The standard method for marking an auto-transformer windings is to label it with capital (upper case) letters. So for example, A, B, Z etc to identify the supply

end. Generally the common neutral connection is marked as N or n. For the secondary tapping’s, su#x numbers are used for all tapping points along the

auto-transformers primary winding. These numbers generally start at number 1 and continue in ascending order for all tapping points as shown.

Autotransformer Terminal Markings

An autotransformer is used mainly for the adjustments of line voltages to either change its value or to keep it constant. If the

voltage adjustment is by a small amount, either up or down, then the transformer ratio is small as V and V are nearly equal.

Currents I and I are also nearly equal.

Therefore, the portion of the winding which carries the di"erence between the two currents can be made from a much smaller

conductor size, since the currents are much smaller saving on the cost of an equivalent double wound transformer.

However, the regulation, leakage inductance and physical size (since there is no second winding) of an autotransformer for a

given VA or KVA rating are less than for a double wound transformer.

Autotransformer’s are clearly much cheaper than conventional double wound transformers of the same VA rating. When

deciding upon using an autotransformer it is usual to compare its cost with that of an equivalent double wound type.

This is done by comparing the amount of copper saved in the winding. If the ratio “n” is de$ned as the ratio of the lower voltage

to the higher voltage, then it can be shown that the saving in copper is: n.100%. For example, the saving in copper for the two

autotransformer’s would be:

Autotransformer Example No1

An autotransformer is required to step-up a voltage from 220 volts to 250 volts. The total number of coil turns on the transformer main winding is 2000.

Determine the position of the primary tapping point, the primary and secondary currents when the output is rated at 10KVA and the economy of copper

saved.

S P S

P

P S

P S

Then the primary current is 45.4 amperes, the secondary current drawn by the load is 40 amperes and 5.4 amperes !ows through the common winding. The

economy of copper is 88%.

Disadvantages of an Autotransformer

The main disadvantage of an autotransformer is that it does not have the primary to secondary winding isolation of a conventional double wound

transformer. Then autotransformer’s can not safely be used for stepping down higher voltages to much lower voltages suitable for smaller loads.

If the secondary side winding becomes open-circuited, current stops !owing through the primary winding stopping the transformer action resulting

in the full primary voltage being applied to the secondary circuit.

If the secondary circuit su"ers a short-circuit condition, the resulting primary current would be much larger than an equivalent double wound

transformer due to the increased !ux linkage damaging the autotransformer.

Since the neutral connection is common to both the primary and secondary windings, earthing of the secondary winding automatically Earths the

primary as there is no isolation between the two windings. Double wound transformers are sometimes used to isolate equipment from earth.

The autotransformer has many uses and applications including the starting of induction motors, used to regulate the voltage of transmission lines, and can be

used to transform voltages when the primary to secondary ratio is close to unity.

An autotransformer can also be made from conventional two-winding transformers by connecting the primary and secondary windings together in series and

depending upon how the connection is made, the secondary voltage may add to, or subtract from, the primary voltage.

The Variable AutotransformerAs well as having a $xed or tapped secondary that produces a voltage output at a speci$c level, there is another useful application of the auto transformer

type of arrangement which can be used to produce a variable AC voltage from a $xed voltage AC supply. This type of Variable Autotransformer is generally

used in laboratories and science labs in schools and colleges and is known more commonly as the Variac.

The construction of a variable autotransformer, or variac, is the same as for the $xed type. A single primary winding

wrapped around a laminated magnetic core is used as in the auto transformer but instead of being $xed at some

predetermined tapping point, the secondary voltage is tapped through a carbon brush.

This carbon brush is rotated or allowed to slide along an exposed section of the primary winding, making contact with it as

it moves supplying the required voltage level.

Then a variable autotransformer contains a variable tap in the form of a carbon brush that slides up and down the

primary winding which controls the secondary winding length and hence the secondary output voltage is fully variable

from the primary supply voltage value to zero volts.

The variable autotransformer is usually designed with a signi$cant number of primary windings to produce a secondary

voltage which can be adjusted from a few volts to fractions of a volt per turn. This is achieved because the carbon brush or

slider is always in contact with one or more turns of the primary winding. As the primary coil turns are evenly spaced

along its length. Then the output voltage becomes proportional to the angular rotation.

Variable Autotransformer

We can see that the variac can adjust the voltage to the load smoothly from zero to the rated supply voltage. If the supply voltage was tapped at some point

along the primary winding, then potentially the output secondary voltage could be higher than the actual supply voltage. Variable autotransformer’s can also

be used for the dimming of lights and when used in this type of application, they are sometimes called “dimmerstats”.

Variac’s are also very useful in Electrical and Electronics (http://www.amazon.com/gp/aws/cart/add.html?

ASIN.1=0750634707&Quantity.1=1&AWSAccessKeyId=AKIAIOB4VMPIMBIMN7NA&AssociateTag=basicelecttut-20) workshops and labs as they can be used to

provide a variable AC supply. But caution needs to be taken with suitable fuse protection to ensure that the higher supply voltage is not present at the

secondary terminals under fault conditions.

The Autotransformer have many advantages over conventional double wound transformers. They are generally more e#cient for the same VA rating, are

smaller in size, and as they require less copper in their construction, their cost is less compared to double wound transformers of the same VA rating. Also,

their core and copper losses, I R are lower due to less resistance and leakage reactance giving a superior voltage regulation than the equivalent two winding

transformer.

In the next tutorial about Transformers we will look at another design of transformer which does not have a conventional primary winding wound around its

core. This type of transformer is commonly called a Current Transformer (http://www.electronics-tutorials.ws/transformer/current-transformer.html)

and is used to supply ammeters and other such electrical power indicators.

Matching TransformerFor induction heating, 20-350kHz frequency range, 8% loss, 500kVA

« Multiple Winding Transformers (http://www.electronics-tutorials.ws/transformer/multiple-winding-transformers.html) | The Current Transformer

(http://www.electronics-tutorials.ws/transformer/current-transformer.html) »

Other Good Tutorials in this CategoryMultiple Winding Transformers (http://www.electronics-tutorials.ws/transformer/multiple-winding-transformers.html)

The Autotransformer (http://www.electronics-tutorials.ws/transformer/auto-transformer.html)

The Current Transformer (http://www.electronics-tutorials.ws/transformer/current-transformer.html)

Three Phase Transformers (http://www.electronics-tutorials.ws/transformer/three-phase-transformer.html)

Transformer Basics (http://www.electronics-tutorials.ws/transformer/transformer-basics.html)

Transformer Construction (http://www.electronics-tutorials.ws/transformer/transformer-construction.html)

Transformer Loading (http://www.electronics-tutorials.ws/transformer/transformer-loading.html)

5 Responses to “The Autotransformer”

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Taonga

Good but on the disadvantage which says when secondary windings becomes open circulted current stops

!owing resulting in full primary voltage being applied to secondary circult,how come for the voltage to be

applied to the secondary side while the secondary side has been open circulted meaning that current amd

voltage do not pass to the other side?

September 19th, 2014 (http://www.electronics-tutorials.ws/transformer/auto-transformer.html/comment-page-1#comment-2087)Reply (/transformer/auto-transformer.html?replytocom=2087#respond)

Wayne Storr (http://www.electronics-tutorials.ws)

As stated above, autotransformers consist of just one winding so the secondary output is not isolated from the

primary input. If the secondary becomes open-circuited with the primary connected to the supply, the voltage

on the secondary tapping can increase to the same level as the primary voltage as there is nothing to control

it. No current, no voltage drop.

September 19th, 2014 (http://www.electronics-tutorials.ws/transformer/auto-transformer.html/comment-page-1#comment-2088)Reply (/transformer/auto-transformer.html?replytocom=2088#respond)

Rajat

@Taonga. Voltage and current are two di"erent entities. Voltage is simple the di"erence in the power levels of

the electrons, i.e, potential di"erence. When these two sides are joined together by a conductor the

current(which is the rate of !ow of electrons) starts to !ow. In an open circuit, the voltage is present, but since

no conductor is there the current doesn’t !ow.

September 22nd, 2014 (http://www.electronics-tutorials.ws/transformer/auto-transformer.html/comment-page-1#comment-2120)Reply (/transformer/auto-transformer.html?replytocom=2120#respond)

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uwiincht

Electrical and electronic education course.

May 28th, 2014 (http://www.electronics-tutorials.ws/transformer/auto-transformer.html/comment-page-1#comment-905)Reply (/transformer/auto-transformer.html?replytocom=905#respond)

subbi

good

April 20th, 2014 (http://www.electronics-tutorials.ws/transformer/auto-transformer.html/comment-page-1#comment-457)Reply (/transformer/auto-transformer.html?replytocom=457#respond)

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