E283C2.docx

1、E283C2CHAPTER 2 TRANSFORMERSSummary:1. Types and Construction of Transformers2. The Ideal Transformer Power in an Ideal Transformer Impedance transformation through a transformer Analysis of circuits containing ideal transformer3. Theory of operation of real single-phase transformers. The voltage ra

2、tio across a transformer The magnetization current in a Real Transformer The current ratio on a transformer and the Dot Convention4. The Equivalent Circuit of a Transformer. Exact equivalent circuit Approximate equivalent circuit Determining the values pf components in the transformer model5. The Pe

3、r-Unit System of Measurement6. Transformer voltage regulation and efficiency The transformer phasor diagram Transformer efficiency7. Three phase transformers1. Types and Construction of TransformersTypes of cores for power transformer (both types are constructed from thin laminations electrically is

4、olated 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

5、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 fluxTypes of transformers:i) Step up/Unit transformers Usually located at the o

6、utput 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

7、.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 Transformer1. Definition a lossless device with an inpu

8、t 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

9、 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

10、.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 eac

11、h 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 Transformer1. The power supplied to the transformer by the primary circuit:Pin = Vp Ip c

12、os pWhere p = 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 sWhere s = the angle between the secondary voltage and the secondary current. 2. The primary and secondary windings of an ide

13、al transformer have the SAME power factor because voltage and current angles are unaffected p - s = 3. How does power going into the primary circuit compare to the power coming out?Pout = Vs Is cos Also, Vs = Vp/a and Is = a Ip So, Pout = Vp Ip cos = PinThe same idea can be applied for reactive powe

14、r Q and apparent power S.Output power = Input powerImpedance Transformation through a Transformer1. 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 tran

15、sformer: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 ZLAnalysis of Circuits containing Ideal Tra

16、nsformersThe easiest way for circuit analysis that has a transformer incorporated is by simplifying the transformer into an equivalent circuit. Example 2.1A generator rated at 480V, 60 Hz is connected a transmission line with an impedance of 0.18+j0.24. At the end of the transmission line there is a

17、 load of 4+j3. (a) If the power system is exactly as described above in Figure (a), what will the voltage at the load be? What will the transmission line losses be?(b) Suppose a 1:10 step-up transformer is placed at the generator end of the transmission line and a 10:1 step-down transformer is place

18、d at the load end of the line (Figure (b). What will the load voltage be now? What will the transmission line losses be now?3. Theory of Operation of Real Single-Phase TransformersIdeal transformers may never exist due to the fact that there are losses associated to the operation of transformers. He

19、nce 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 ge

20、nerated in the primary coils, based upon Faradays law,where is the flux linkage in the coil across which the voltage is being induced. The flux linkage is the sum of the flux passing through each turn in the coil added over all the turns of the coil.This relation is true provided on the assumption t

21、hat the flux induced at each turn is at the same magnitude and direction. But in reality, the flux value at each turn may vary due to the position of the coil it self, at certain positions, there may be a higher flux level due to combination of other flux from other turns of the primary winding.Henc

22、e the most suitable approach is to actually average the flux level asHence Faradays law may be rewritten as:The voltage ratio across a Transformerf the voltage source is vp(t), how will the transformer react to this applied voltage?Based upon Faradays Law, looking at the primary side of the transfor

23、mer, we can determine the average flux level based upon the number of turns; where, This relation means that the average flux at the primary winding is proportional to the voltage level at the primary side divided by the number of turns at the primary winding. This generated flux will travel to the

24、secondary side hence inducing potential across the secondary terminal. For an ideal transformer, we assume that 100% of flux would travel to the secondary windings. However, in reality, there are flux which does not reach the secondary coil, in this case the flux leaks out of the transformer core in

25、to the surrounding. This leak is termed as flux leakage. Taking into account the leakage flux, the flux that reaches the secondary side is termed as mutual flux.Looking at the secondary side, there are similar division of flux; hence the overall picture of flux flow may be seen as below:Primary Side

26、:= total average primary flux= flux component linking both rpimary and secondary coils= primary leakage flux For the secondary side, similar division applies. Hence, looking back at Faradays Law, Or this equation may be rewritten into: The same may be written for the secondary voltage. The primary v

27、oltage due to the mutual flux is given by And the same goes for the secondary (just replace P with S)From these two relationships (primary and secondary voltage), we haveTherefore,Magnetization Current in a Real transformerAlthough the output of the transformer is open circuit, there will still be c

28、urrent flow in the primary windings. The current components may be divided into 2 components:1) Magnetization current, iM current required to produce flux in the core.2) Core-loss current, ih+e current required to compensate hysteresis and eddy current losses.We know that the relation between curren

29、t and flux is proportional since, Therefore, in theory, if the flux produce in core is sinusoidal, therefore the current should also be a perfect sinusoidal. Unfortunately, this is not true since the transformer will reach to a state of near saturation at the top of the flux cycle. Hence at this poi

30、nt, more current is required to produce a certain amount of flux.If the values of current required to produce a given flux are compared to the flux in the core at different times, it is possible to construct a sketch of the magnetization current in the winding on the core. This is shown below:Hence

31、we can say that current in a transformer has the following characteristics:1. It is not sinusoidal but a combination of high frequency oscillation on top of the fundamental frequency due to magnetic saturation.2. The current lags the voltage at 90o3. At saturation, the high frequency components will be extreme as such that harmonic problems will occur.Looking at the core-loss current, it again is dependent upon hysteresis and eddy current flow. Since Eddy current is dependent upon the rate of change of flux, hence we can also say that the core-loss current is great