TRANSFORMERS
1. Types and Construction of Transformers
Types of cores for power transformer (both types are constructed from thin laminations electrically isolated from each other – minimize eddy currents)
i) Core Form : a simple rectangular laminated piece of steel with the transformer windings wrapped around two sides of the rectangle.
ii) Shell Form : a three legged laminated core with the windings wrapped around the centre leg.
The primary and secondary windings are wrapped one on top of the other with the low-voltage winding innermost, due to 2 purposes:
i) It simplifies the problem of insulating the high-voltage winding from the core.
ii) It results in much less leakage flux
Types of transformers:
i) Step up/Unit transformers – Usually located at the output of a generator. Its function is to step up the voltage level so that transmission of power is possible.
ii) Step down/Substation transformers – Located at main distribution or secondary level transmission substations. Its function is to lower the voltage levels for distribution 1st level purposes.
iii) Distribution Transformers – located at small distribution substation. It lowers the voltage levels for 2nd level distribution purposes.
iv) Special Purpose Transformers - E.g. Potential Transformer (PT) , Current Transformer (CT)
2. The Ideal Transformer
1. Definition – a lossless device with an input winding and an output winding.
2. Figures below show an ideal transformer and schematic symbols of a transformer.
3. The transformer has Np turns of wire on its primary side and Ns turns of wire on its secondary sides. The relationship between the primary and secondary voltage is as follows:
where a is the turns ratio of the transformer.
4. The relationship between primary and secondary current is:
Np ip (t) = Ns is (t)
5. Note that since both type of relations gives a constant ratio, hence the transformer only changes ONLY the magnitude value of current and voltage. Phase angles are not affected.
6. The dot convention in schematic diagram for transformers has the following relationship:
i) If the primary voltage is +ve at the dotted end of the winding wrt the undotted end, then the secondary voltage will be positive at the dotted end also. Voltage polarities are the same wrt the dots on each side of the core.
ii) If the primary current of the transformer flows into the dotted end of the primary winding, the secondary current will flow out of the dotted end of the secondary winding.
Power in an Ideal Transformer
1. The power supplied to the transformer by the primary circuit:
Pin = Vp Ip cos θp
Where θp = the angle between the primary voltage and the primary current. The power supplied by the transformer secondary circuit to its loads is given by:
Pout = Vs Is cos θs
Where θs = the angle between the secondary voltage and the secondary current.
2. The primary and secondary windings of an ideal transformer have the SAME power factor – because voltage and current angles are unaffected θp - θs = θ
3. How does power going into the primary circuit compare to the power coming out?
Pout = Vs Is cos θ
Also, Vs = Vp/a and Is = a Ip
So,
Pout = Vp Ip cos θ = Pin
The same idea can be applied for reactive power Q and apparent power S.
Output power = Input power
Impedance Transformation through a Transformer
1. The impedance of a device or an element is defined as the ratio of the phasor voltage across it to the phasor current flowing through it:
2. Definition of impedance and impedance scaling through a transformer:
3. Hence, the impedance of the load is:
4. The apparent impedance of the primary circuit of the transformer is:
5. Since primary voltage can be expressed as VP=aVS, and primary current as IP=IS/a, thus the apparent impedance of the primary is
ZL’ = a2 ZL
Analysis of Circuits containing Ideal Transformers
The easiest way for circuit analysis that has a transformer incorporated is by simplifying the transformer into an equivalent circuit.
3. Theory of Operation of Real Single-Phase Transformers
Ideal transformers may never exist due to the fact that there are losses associated to the operation of transformers. Hence there is a need to actually look into losses and calculation of real single phase transformers.
Assume that there is a transformer with its primary windings connected to a varying single phase voltage supply, and the output is open circuit.
Right after we activate the power supply, flux will be generated in the primary coils, based upon Faraday’s law,
where λ is the flux linkage in the coil across which the voltage is being induced. The flux linkage λ is the sum of the flux passing through each turn in the coil added over all the turns of the coil.
1. Types and Construction of Transformers
Types of cores for power transformer (both types are constructed from thin laminations electrically isolated from each other – minimize eddy currents)
i) Core Form : a simple rectangular laminated piece of steel with the transformer windings wrapped around two sides of the rectangle.
ii) Shell Form : a three legged laminated core with the windings wrapped around the centre leg.
The primary and secondary windings are wrapped one on top of the other with the low-voltage winding innermost, due to 2 purposes:
i) It simplifies the problem of insulating the high-voltage winding from the core.
ii) It results in much less leakage flux
Types of transformers:
i) Step up/Unit transformers – Usually located at the output of a generator. Its function is to step up the voltage level so that transmission of power is possible.
ii) Step down/Substation transformers – Located at main distribution or secondary level transmission substations. Its function is to lower the voltage levels for distribution 1st level purposes.
iii) Distribution Transformers – located at small distribution substation. It lowers the voltage levels for 2nd level distribution purposes.
iv) Special Purpose Transformers - E.g. Potential Transformer (PT) , Current Transformer (CT)
2. The Ideal Transformer
1. Definition – a lossless device with an input winding and an output winding.
2. Figures below show an ideal transformer and schematic symbols of a transformer.
3. The transformer has Np turns of wire on its primary side and Ns turns of wire on its secondary sides. The relationship between the primary and secondary voltage is as follows:
where a is the turns ratio of the transformer.
4. The relationship between primary and secondary current is:
Np ip (t) = Ns is (t)
5. Note that since both type of relations gives a constant ratio, hence the transformer only changes ONLY the magnitude value of current and voltage. Phase angles are not affected.
6. The dot convention in schematic diagram for transformers has the following relationship:
i) If the primary voltage is +ve at the dotted end of the winding wrt the undotted end, then the secondary voltage will be positive at the dotted end also. Voltage polarities are the same wrt the dots on each side of the core.
ii) If the primary current of the transformer flows into the dotted end of the primary winding, the secondary current will flow out of the dotted end of the secondary winding.
Power in an Ideal Transformer
1. The power supplied to the transformer by the primary circuit:
Pin = Vp Ip cos θp
Where θp = the angle between the primary voltage and the primary current. The power supplied by the transformer secondary circuit to its loads is given by:
Pout = Vs Is cos θs
Where θs = the angle between the secondary voltage and the secondary current.
2. The primary and secondary windings of an ideal transformer have the SAME power factor – because voltage and current angles are unaffected θp - θs = θ
3. How does power going into the primary circuit compare to the power coming out?
Pout = Vs Is cos θ
Also, Vs = Vp/a and Is = a Ip
So,
Pout = Vp Ip cos θ = Pin
The same idea can be applied for reactive power Q and apparent power S.
Output power = Input power
Impedance Transformation through a Transformer
1. The impedance of a device or an element is defined as the ratio of the phasor voltage across it to the phasor current flowing through it:
2. Definition of impedance and impedance scaling through a transformer:
3. Hence, the impedance of the load is:
4. The apparent impedance of the primary circuit of the transformer is:
5. Since primary voltage can be expressed as VP=aVS, and primary current as IP=IS/a, thus the apparent impedance of the primary is
ZL’ = a2 ZL
Analysis of Circuits containing Ideal Transformers
The easiest way for circuit analysis that has a transformer incorporated is by simplifying the transformer into an equivalent circuit.
3. Theory of Operation of Real Single-Phase Transformers
Ideal transformers may never exist due to the fact that there are losses associated to the operation of transformers. Hence there is a need to actually look into losses and calculation of real single phase transformers.
Assume that there is a transformer with its primary windings connected to a varying single phase voltage supply, and the output is open circuit.
Right after we activate the power supply, flux will be generated in the primary coils, based upon Faraday’s law,
where λ is the flux linkage in the coil across which the voltage is being induced. The flux linkage λ is the sum of the flux passing through each turn in the coil added over all the turns of the coil.