Principles of Electric Machines and Power Electronics Chapter 2-1 Transformers Third Edition P. C. Sen
Transformer application 1: power transmission
Ideal Transformer Assumptions: 1. Negligible winding resistance 2. No leakage flux 3. Infinite permeability of core 4. Zero core loss v 1 Voltage relation: d v1 e1 N1 dt Excitation current: a a v2 e2 N2 dt v v N N 1 1 2 2 a d 1, step-down transformer 1, step-up transformer V 1 ω = jωl I 1 (ω) I 1 ω = V 1 ω jωl L = N 1 2 μ 0 μ r A l I 1 ω = 0
Basic equations- general definition N 1 i 1 N 2 i 2 R = l Aμ Current relationship: N i N i 1 1 2 2 1 1 2 2 i1 N2 1 i N a 2 1 N i N i net mmf=0 Instantaneous power: v i v i 1 1 2 2
Polarity marking: Like polarity: two entering currents produce the same direction of magnetic flux
Voltage polarity 2 Parallel operation of two single-phase transformer
Impedance Reflection: Z 2 = V 1 I 1 = av 2 I 2a = a 2 V 2 I 2 = a 2 Z 2 Z Z a ' 1 1 2
Example: Determine the primary and secondary currents for the ideal transformer below if Z s = (18-j4) Ωand Z 2 = (2+j1) Ω.
Other single-phase transformers Single primary multiple secondary windings V i = V 1 a i I 1 = a i I i
Non- ideal transformer: model development 1 Ideal transformer assumptions: 1. Negligible winding resistance 2. No leakage flux Practical transformer: ideal transformer+ external impedance
Non- ideal transformer: model development 2 Ideal transformer assumptions: 3. Infinite permeability of core 4. No core loss Practical transformer: Magnetizing reactance Core resistance
Non- ideal transformer: model development 3 V 2 = av 2 I 2 = I 2 a = I 1 X l2 = a 2 X l2 R w2 = a 2 R w2 Z 2 = a 2 Z 2
Approximate equivalent circuit Voltage drops across primary winding resistance and reactance is quite small Neglect excitation branch
Determine equivalent circuit parameters- No load test Procedure: Apply rated voltage to either high-voltage or low-voltage side Primary current: exciting current Loss: core loss (the same for applying either high-voltage or low-voltage side) Parameters obtained: Magnetizing reactance Core resistance
Determine equivalent circuit parameters short circuit test Procedure: Short-circuiting one winding Apply rated current to the other winding Parameters obtained: Primary and secondary resistance Primary and secondary leakage reactance Current source
Transformer ratings: power, primary/secondary voltages turn ratio, current ratings Example: single-phase transformer, 10 kva, 2200/220V, 60 Hz (1) Determine core loss resistance and magnetizing inductance from no load test. (2) Derive the parameters for the approximate equivalent circuits referred to the High voltage side (3) Derive the parameter when refer to low voltage side
Solution (practice): single-phase transformer, 10 kva, 2200/220V, 60 Hz
Voltage regulation Basic definition VR V 2 NL 2 V 2 L V L Refer to the primary VR V V ' ' 2 NL 2 ' V2 L L
Voltage regulation Load voltage is normally taken as rated voltage VR V V ' 1 2 rated 100% ' V2 rated
Example: single-phase transformer, 10 kva, 2200/220V, 60 Hz Determine voltage regulation in percent for (a) 75% full load, 0.6 power factor lagging (b) 75% full load, 0.6 power factor leading (c) Draw the phasor diagram for (a) and (b)
Solution (practice)
Efficiency of a transformer Loss of transformer Core loss (hysteresis and eddy current) Winding (copper) resistive loss P P P P P P P P P out out out in out loss out c cu 100% P I R I R I R 2 2 2 cu 1 w1 2 w2 2 eq2 Core loss Almost constant Obtain from no-load test
Example (practice) Example (practice): single-phase transformer, 10 kva, 2200/220V, 60 Hz Determine Efficiency at 75% rated output and 0.6 PF