ELECTRICAL TECHNOLOGY ERT 105/3

ELECTRICAL TECHNOLOGY ERT 105/3 Define and analyze the principle of transformer, its parameters and structure. Describe and analyze Ideal transformer, equivalent circuit, and phasor diagram Calculate and justify efficiency, losses, performance

TRANSFORMERS (CHAPTER 5)

Introduction • This chapter covers: • Mutual inductance that exists between coils of the same or different dimensions. • Basic to the operation of a transformer • Three of the basic operations of a transformer. • step up/down • impedance matching • isolation • The dot convention.

FUNCTION OF A TRANSFORMER • To transfer electrical energy from one side (primary) to the other side (secondary). • The secondary current and voltage may or may not be at the same level as that of the primary current and voltage. The energy is transferred by means of magnetic coupling. • The magnetic flux produced by the current in primary winding links to the secondary winding. Since the flux varies with time, this flux linkage results in an induced voltage in the secondary winding. If the secondary winding is terminated with a load, the induced voltage will drive a secondary current through the load.

Mutual Inductance • Transformers are constructed of two coils placed so that the charging flux developed by one will link to the other. • The coil to which the source is applied is called the primary coil. • The coil to which the load is applied is called the secondary coil.

Mutual Inductance Defining the component of a transformer

Mutual Inductance • Primary transformer formula for voltage using Faraday’s Law: • Voltage induced across the primary and self-inductance is directly related to the number of turns in the coil and the rate of change of magnetic flux linking the primary coil:

Mutual Inductance • The magnitude of eS , the voltage inducted across the secondary is determined by; where NS is the number of turns in the secondary winding and m is the portion of the primary flux, Pthat links the secondary winding

Mutual Inductance • If all of the flux linking the primary links the secondary, then: • The coefficient of coupling (k) between two coils is determined by;

Mutual Inductance • Since the maximum level of m is P, the coefficient of coupling between two coils can never be greater than 1. • Coils with low coefficients of coupling are termed loosely coupled.

Mutual Inductance Coefficient of coupling, k very close to 1

Mutual Inductance

Mutual Inductance • Mutual inductance between two coils is proportional to the instantaneous change in flux linking one coil due to an instantaneous change in the current through the other coil. • In terms of inductance of each coil and the coefficient of coupling, the mutual inductance is:

The Iron-core Transformer

The Iron-core Transformer • The iron core will serve to increase the coefficient of coupling, kbetween the coils by increasing the mutual flux m.. • The magnetic flux lines will always take the path of least reluctance, which in this case is the iron core. • When the current Ip through the primary circuit of the iron-core transformer is a maximum, the flux mlinking both coils is also a maximum. • The flux is directly proportional to the current through the winding.

The Iron-core Transformer If; then; By Faraday’s law; Then by differentiation;

The Iron-core Transformer The effective value of Epis; Since the flux linking the secondary flux equals that of the primary;

The Iron-core Transformer Dividing; and For and ideal transformer; Hence;

The Iron-core Transformer • The ratio of the magnitudes of the induced voltages is the same as the ratio of the corresponding turns: • The ratio Np/Ns , usually represented by the lowercase letter a, is referred to as the transformation ratio. • If a < 1, the transformer is a step-up transformer. • If a > 1, the transformer is a step-down transformer.

The Iron-core Transformer Example 1 (a) Find the maximum flux, m. (b) Find the secondary turns, Ns.

The Iron-core Transformer Example 1 – Solution (a)

The Iron-core Transformer Example 1 – Solution (cont’d) (b)

The Iron-core Transformer • The primary and secondary current of a transformer are related by the inverse ratio of the turns:

Reflected Impedance and Power • The impedance of the primary circuit of an ideal transformer is related to the impedance load by the transformation ratio • If the load is capacitive or inductive, the reflected impedance will also be capacitive or inductive. • For an ideal iron-core transformer;

Reflected Impedance and Power Example 2 (a) Find Ip and Vg. (b) Find Zp.

Reflected Impedance and Power Example 2– Solution (a) Primary current, Ip

Reflected Impedance and Power Example 2 – Solution (cont’d) (b) Input impedance, Zp

Reflected Impedance and Power Example 22.4 For the residential supply, determine (assuming a totally resistive load) the following : • The value of R to ensure a balanced load • the magnitude of I1and I2 • The line voltage VL • The turn ratio a=Np/Ns

Reflected Impedance and Power Example22.4 - Solution • PT = (10) (60) W+ 200 W + 2000 W • = 600 W + 200 W + 2000 W = 2800 W Pin = Pout VpIp=VsIs=2800 W (purely resistive load) (2400 V) Ip= 2800 W and Ip=1.17 A R=Vphase/Ip=2400 V/1.17 A = 2051.28 ohms

Reflected Impedance and Power Example 22.4 – Solution (cont’d) b. P1=600 W = VI1= (120V) I1 And I1= 5 A P2= 2000 W = VI2= (240W) I2 And I2= 8.33 A c. d.

Equivalent Circuit (Iron- core Transformer) • For the non ideal or practical iron-core transformer, the equivalent circuit appears below. • Part of the equivalent circuit includes an ideal transformer.

Equivalent Circuit (Iron- core Transformer)

Equivalent Circuit (Iron-core Transformer) • All elements in the iron-core transformer other than the ideal transformer are the elements of the transformer that contribute to the non-ideal characteristics of the device. • Resistance Rp and Rs are simply the geometric resistance of the primary and secondary windings. • Leakage flux Lp and Ls is a small amount of flux that links each coil but does not pass through the core.

Equivalent Circuit (Iron-core Transformer) • The resistance RC represents the hysteresis and eddy current losses (core losses) within the core due to an ac flux through the core. • The inductance Lm (magnetizing inductance) is the inductance associated with the magnetization of the core, that is, the establishing of the flux m in the core. • The capacitances CP and CS and the lumped capacitances of the primary and secondary circuits, respectively, and Cw represent the equivalent lumped capacitance between the windings of the transformer.

Equivalent Circuit (Iron-core Transformer) For power transformer; • Winding capacitances have negligible effects because of the low operating frequency. • Under normal operating condition, normally i’p >> im. Hence, Cp, Cs, Cw, RC and Lm may be omitted. The approximate equivalent circuit becomes as follows;

Equivalent Circuit (Iron-core Transformer) Reduced equivalent circuit for the non-ideal iron-core transformer.

Equivalent Circuit (Iron-core Transformer) Normally, Lp and Ls are represented by their equivalent reactances; and

Equivalent Circuit (Iron-core Transformer) Xs and Rs may be reflected into the primary circuit using the relationship; and

Equivalent Circuit (Iron-core Transformer) The equivalent circuit now becomes;

Equivalent Circuit (Iron-core Transformer) Combining the series resistances and reactances, results in the following circuit;

Equivalent Circuit (Iron-core Transformer) The load resistance RL may also be reflected into the primary circuit;

Equivalent Circuit (Iron-core Transformer) By voltage divider rule; is the load resistance reflected into the primary circuit of the transformer

Equivalent Circuit (Iron-core Transformer) Example 3 (a) Determine Re and Xe. (b) Determine the magnitude VL and Vg.

Equivalent Circuit (Iron-core Transformer) Example 3 – Solution (a)

Equivalent Circuit (Iron-core Transformer) Example 3 – Solution (b) Magnitude VL ,Vg

ELECTRICAL TECHNOLOGY ERT 105/3