Transformers

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Transcript Transformers 

Transformers
Single Phase Transformers
Principles of Operation – Single Phase
Flux and Voltage 90o out of phase
Relationship of induced voltage and flux:
Also Eqn 1.148
Fundamental Transformer equation
Only accurate if leakage impedance of coil is negligible
Primary winding flux divided into leakage flux and mutual flux
Ideal Transformer Relationships
Exercise 2-1
𝑒1 𝑁1
=
β‰‘π‘Ž
𝑒2 𝑁2
𝐼2 𝑁1
=
β‰‘π‘Ž
𝐼1 𝑁2
𝑍1
𝑁1
=
𝑍2
𝑁2
2
≑ π‘Ž2
Non-ideal Transformer
Equivalent Circuits of a non-ideal Transformer
Equivalent Circuits of a non-ideal Transformer
π‘Ž ≑ π‘‘π‘’π‘Ÿπ‘›π‘  π‘Ÿπ‘Žπ‘‘π‘–π‘œ > 1
𝐸1 ≑ π‘π‘Ÿπ‘–π‘šπ‘Žπ‘Ÿπ‘¦ 𝑖𝑛𝑑𝑒𝑐𝑒𝑑 π‘£π‘œπ‘™π‘‘π‘Žπ‘”π‘’
𝐸2 ≑ π‘ π‘’π‘π‘œπ‘›π‘‘π‘Žπ‘Ÿπ‘¦ 𝑖𝑛𝑑𝑒𝑐𝑒𝑑 π‘£π‘œπ‘™π‘‘π‘Žπ‘”π‘’
𝑉1 ≑ π‘π‘Ÿπ‘–π‘šπ‘Žπ‘Ÿπ‘¦ π‘‘π‘’π‘Ÿπ‘šπ‘–π‘›π‘Žπ‘™ π‘£π‘œπ‘™π‘‘π‘Žπ‘”π‘’
𝑉2 ≑ π‘ π‘’π‘π‘œπ‘›π‘‘π‘Žπ‘Ÿπ‘¦ π‘‘π‘’π‘Ÿπ‘šπ‘–π‘›π‘Žπ‘™ π‘£π‘œπ‘™π‘‘π‘Žπ‘”π‘’
𝐼1 ≑ π‘π‘Ÿπ‘–π‘šπ‘Žπ‘Ÿπ‘¦ π‘π‘’π‘Ÿπ‘Ÿπ‘’π‘›π‘‘
𝐼2 ≑ π‘ π‘’π‘π‘œπ‘›π‘’π‘Žπ‘Ÿπ‘¦ π‘π‘’π‘Ÿπ‘Ÿπ‘’π‘›π‘‘
𝐼0 ≑ π‘›π‘œ βˆ’ π‘™π‘œπ‘Žπ‘‘ π‘π‘Ÿπ‘–π‘šπ‘Žπ‘Ÿπ‘¦ π‘π‘’π‘Ÿπ‘Ÿπ‘’π‘›π‘‘
𝑅1 ≑ π‘Ÿπ‘’π‘ π‘–π‘ π‘‘π‘Žπ‘›π‘π‘’ π‘œπ‘“ π‘‘β„Žπ‘’ π‘π‘Ÿπ‘–π‘šπ‘Žπ‘Ÿπ‘¦ 𝑀𝑖𝑛𝑑𝑖𝑛𝑔
𝑅2 ≑ π‘Ÿπ‘’π‘ π‘–π‘ π‘‘π‘Žπ‘›π‘π‘’ π‘œπ‘“ π‘‘β„Žπ‘’ π‘ π‘’π‘π‘œπ‘›π‘‘π‘Žπ‘Ÿπ‘¦ 𝑀𝑖𝑛𝑑𝑖𝑛𝑔
𝑋1 ≑ π‘π‘Ÿπ‘–π‘šπ‘Žπ‘Ÿπ‘¦ π‘™π‘’π‘Žπ‘˜π‘Žπ‘”π‘’ π‘Ÿπ‘’π‘Žπ‘π‘‘π‘Žπ‘›π‘π‘’
𝑋2 ≑ π‘ π‘’π‘π‘œπ‘›π‘‘π‘Žπ‘Ÿπ‘¦ π‘™π‘’π‘Žπ‘˜π‘Žπ‘”π‘’ π‘Ÿπ‘’π‘Žπ‘π‘‘π‘Žπ‘›π‘π‘’
πΌπ‘š , π‘‹π‘š ≑ π‘šπ‘Žπ‘”π‘›π‘’π‘‘π‘–π‘§π‘–π‘›π‘” π‘π‘’π‘Ÿπ‘Ÿπ‘’π‘›π‘‘ π‘Žπ‘›π‘‘ π‘Ÿπ‘’π‘Žπ‘π‘‘π‘Žπ‘›π‘π‘’
𝐼𝑐 , 𝑅𝑐 ≑ π‘π‘’π‘Ÿπ‘Ÿπ‘’π‘›π‘‘ π‘Žπ‘›π‘‘ π‘Ÿπ‘’π‘ π‘–π‘ π‘‘π‘Žπ‘›π‘π‘’ π‘Žπ‘π‘π‘œπ‘’π‘›π‘‘π‘–π‘›π‘” π‘“π‘œπ‘Ÿ π‘‘β„Žπ‘’ π‘π‘œπ‘Ÿπ‘’ π‘™π‘œπ‘ π‘ π‘’π‘ 
Open-Circuit (or No-Load) Test
Here one winding is open-circuited and voltage---usually, rated voltage at rated
frequency--is applied to the other winding. Voltage, current, and power at the
terminals of this winding are measured. The open circuit voltage of the second winding
is also measured, and from this measurement a check on the turns ratio can be
obtained. It is usually convenient to apply the test voltage to the winding that has a
voltage rating equal to that of the available power source. In step-up voltage
transformers, this means that the open-circuit voltage of the second winding will be
higher than the applied voltage, sometimes much higher. Care must be exercised in
guarding the terminals of this winding to ensure safety for test personnel and to
prevent these terminals from getting close to other electrical circuits, instrumentation,
grounds, and so forth.
In presenting the no-load parameters obtainable from test data, it is assumed that
voltage is applied to the primary and the secondary is open-circuited. The no-load
power loss is equal to the wattmeter reading in this test; core loss is found by
subtracting the ohmic loss in the primary, which is usually small and may be neglected
in some cases. Thus, if Po, Io and Vo are the input power, current, and voltage, then the
core loss is given by:
𝑃𝑐 = π‘ƒπ‘œ βˆ’ πΌπ‘œ 2 𝑅1
The primary induced voltage is given in phasor form by:
π‘¬πŸ = π‘‰π‘œ ∠0π‘œ βˆ’ (πΌπ‘œ βˆ πœƒ π‘œ )(𝑅1 + 𝑗𝑋1 )
where πœƒ π‘œ ≑ no-load power-factor angle = π‘π‘œπ‘  βˆ’1
Other circuit quantities are found from:
𝐸1
𝑅𝑐 =
𝑃𝑐
𝑃𝑐
𝐼𝑐 =
𝐸1
πΌπ‘š =
2
𝐼0
𝐸1
π‘‹π‘š =
πΌπ‘š
π‘‰π‘œ
aβ‰ˆ
𝐸2
2
βˆ’ 𝐼𝑐
2
π‘ƒπ‘œ
π‘‰π‘œ πΌπ‘œ
< 0.
Short-Circuit Test
In this test, one winding is short-circuited across its terminals, and a reduced voltage is
applied to the other winding. This reduced voltage is of such a magnitude as to cause a
specific value of current---usually, rated current--to flow in the short-circuited winding.
Again, the choice of the winding to be short-circuited is usually determined by the
measuring equipment available for use in the test. However, care must be taken to note
which winding is short-circuited, for this determines the reference winding for expressing
the impedance components obtained by this test. Let the secondary be short-circuited
and the reduced voltage be applied to the primary.
With a very low voltage applied to the primary winding, the cord-loss current and
magnetizing current become very small, and the equivalent circuit reduces to that of Fig.
2-6. Thus, if Ps, Is, and Vs are the input power, current, and voltage under short circuit,
then, referred to the primary,
𝑍𝑠 =
𝑉𝑠
𝐼𝑠
𝑅1 + π‘Ž2 𝑅2 ≑ 𝑅𝑠 =
𝑃𝑠
𝐼𝑠 2
𝑋1 + π‘Ž2 𝑋2 ≑ 𝑋𝑠 =
𝑍𝑠 2 βˆ’ 𝑅𝑠 2
Given R1and a, R2 can be found. It is usually assumed that the leakage
reactance is divided equally between the primary and the secondary; that is:
𝑋1 = π‘Ž2 𝑋2 =
1
𝑋
2 𝑠
Short Circuit Test:
(Impedance or
copper-loss test)
Open Circuit Test:
Core-loss test
Usually done on the side of
the transformer with the
lower rated voltage
Transformer
Characteristics:
Inrush
8 to 12 times rated
current,
duration about
100 ms
Efficiency
Regulation
kVA Rating
Apparent power or kVA is always inscribed on the nameplate
As long as the transformer delivers rated or reduced kVA, it will
operate with nominal heat
If it is cooled, it can safely deliver higher-than-rated kVA
A transformer of 1000 kVA supplied with fans can deliver 1333kVA
Protective equipment such as circuit breakers and fuses, must
be capable of safely withstanding the forces produced by the
short-circuit currents.
It is customary in industry to rate the short-circuit capacity of
equipment in terms of short-circuit MVA.
Three-Phase, Two-Winding Transformers
Review of Three-Phase Systems
P = 3VLβˆ’L IL cos ΞΈ
Q = 3VLβˆ’L IL sin ΞΈ
S = 3VLβˆ’L IL *
Delta-Connected Load
Star-Connected Load
Electric Circuit Analysis
Star-Delta
Star-Star
Auto Transformers
Understanding and analyzing autotransformers is simplified by
noting the following:
1. The ampere-turns of each coil are the same whether the
transformer is connected as an autotransformer or as a
two-winding transformer. That is,
(NI)= constant
In other words, the magnitude of the current through each coil
remains the same, regardless of whether the transformer
operates as an autotransformer or as a conventional twowinding transformer.
2. Since the current through each autotransformer winding
is the same as in the conventional two-winding
transformer, the winding loss remains the same.
However, the efficiency of the autotransformer is
increased if the output power is increased.
3. Neglecting losses, the complex power (kVA) at the input
is equal to the complex power to the output. The kVA
transformation capability of the autotransformer is the
same as that of the two-winding transformer. An
autotransformer, however, delivers higher kVA than the
conventional transformer because of the direct electrical
connections between the primary and secondary
windings. In other words, part of the output kVA is
conducted from the primary to the secondary winding.
The conducted kVA is referred to as the untransformed
kVA.
4. The two-winding conventional transformer has its
primary and secondary circuits electrically isolated, while in
the autotransformer electrical disturbances in the primary
can be easily passed to the secondary through their direct
electrical connections.
Disadvantages of Autotransformers
The disadvantages of autotransformers are as follows:
They are not economical for voltage ratios larger than 2.
They require primary protective devices of higher capacity.
1. Because of the electrical conductivity of the primary and
secondary windings, the lower voltage circuit is liable to be
impressed upon by higher voltage. To avoid breakdown in
the lower voltage circuit, it becomes necessary to design the
low-voltage circuit to withstand higher voltage.
2. The autotransformer has a common terminal between the
primary and the secondary windings and when the
secondary is shorted, the voltage applied to the primary is
much higher than its rated voltage. This results in higher
short-circuit currents and thus requires more expensive
protective devices.
3. The connections on the primary and secondary sides must
necessarily be the same, except when using interconnected
starring connections. This introduces complications due to
changing primary and secondary phase angles, particularly in
the case-by-case of the delta-delta connection.
4. Because of a common neutral in a star-star connected
autotransformer, it is not possible to ground the neutral of one
side only. Both if its sides must have its neutrals either
grounded or isolated.
5. It is more difficult to preserve the electromagnetic balance of
the winding when voltage adjustment tappings are provided. It
should be known that the provision of adjusting tapping on an
autotransformer increases the frame size of the transformer
considerably.
Parallel Operation of Transformers:
Equal impedance
Equal turns ratio
Equal phase shift between
primary and secondary
open-circuited voltages
Same phase rotation