Transformers for Electrical Energy Conversion
Explore the functioning of transformers in converting electrical energy at different voltage levels using coils and magnetic fields. Learn about ideal transformers, voltage relationships, impedance, current/voltage rules, power transfer, and practical examples.
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Presentation Transcript
Lecture # 10 TRANSFORMERS 1
TRANSFORMER A transformer can change electrical energy of a given voltage into electrical energy at a different voltage level. It consists of two coils arranged in such a way that the magnetic field surrounding one coil cuts through the other coil. When an alternating voltage is applied to (across) one coil, the varying magnetic field set up around that coil induces an alternating voltage in the other coil. Transformers will not work with direct current, since no changing magnetic field is produced, and therefore no current can be induced. 2
Ideal Transformer -Voltage First we review the voltage/current relationships for an ideal transformer no real power losses magnetic core has infinite permeability( =0) no leakage flux. We ll define the primary side of the transformer as the side that usually receives power from a line etc, and the secondary as the side that usually delivers power to a loadetc: primary is usually the side with the higher voltage, but may be the low voltage side on a generator step-up transformer. 3
Ideal Transformer -Voltage i2 i1 + v1 - + v2 - N1 N2 Load AC v 1 =N1 v2 V1=N1 V2 = a = turns ratio N2 N2 4
Ideal Transformer - Impedance i2 i1 + v1 - + v2 - N1 N2 Load AC Turns ratio, a =N1 Input impedance: Z'=V1 L N2 2V N Z'= 1 L 2 I2 N 2 1I 2 N N N = 1 N V = 1V Z' Z L L 1 2 2 2 I =N2I Z' = a2Z 1 2 L L N 1 5
Current/Voltage Relationships The two simple rules to follow are: 1.If V1 and V2 are both positive or both negative at the dotted terminals, use +a . Otherwise, use a. 2. If I1 and I2 both enter into or both leave the dottedterminals, use a. Otherwise, use+a. 6
Ideal Transformer - Power i2 i1 + v - + P = vi v2 - N1 N2 Load AC 1 Power delivered by the primary P1 = v1i1 Power delivered to load P2 = v2i2 =N2v N and i =N 1i but v 2 1 2 1 N Power delivered to an ideal transformer by the source is transferred to the load. 1 2 P2=v2i2=v1i1= P1 7
Analysis Of Ideal Transformers EXAMPLE: An ideal transformer, connected to a 240 V mains, supplies a 12 V, 150 W lamp. Calculate the transformer turns ratio and thecurrent taken from the supply. Solution: 8
Analysis Of Ideal Transformers EXAMPLE: For the circuit shown, find the average power delivered tothe 4 k resistor. a2 = 0.4 a1 = 0.25 Solution: Zr1 = (a1)2 4k = 250 Zr2 = (a2)2 Zr1 = 40 I1=100 / (10+40) = 2A and P4k = P40 = (I1)2 Zr2 = (2)2 40 = 160W or, I2 = (-a2) I1 =- 0.8A Thus, P4k = (I3)2 ZL = (-0.2)2 4000 =160W and I3= (a1) I2 = - 0.2A 9
Analysis Of Ideal Transformers EXAMPLE: Determine all indicated voltages and currents. a = 4 Solution: V1= ZL I1 = (32+ j16) 2.33 13.5 V1 = 35.78 26.57 2.33 13.5 =83.36 13.07 Z'= a2Z Strategy: Reflect impedance into the primary side and make transformer transparent to user. Z'= 32 +j16 ' L L L ZL Z L Careful With Polarities And Current Directions! I2 = a I1 = 4I1 (current into dot) V V = = 0.25V ( + is opposite todot) a Zi = 50 +j12 120 0 120 0 51.42 13.5 1 = 2.33 13.5 = I1 =50+ j12 2 1 10
Analysis Of Ideal Transformers a = 1/2 EXAMPLE: Find the current I1, andVo. I2 = a I1 Solution: Strategy: Reflect impedance into the primary side and make transformer transparent to user. 1I =12 0 = 3 j2.5 =3.07 39.81 (A) 12 0 3.91 39.81 =4 j2 = 1 j0.5 Z' = a2Z Z' L L L 4 Strategy: Find current in secondary and then use Ohm s Law Z' I1 L I2 = a I1 =2 I 2 1 V = 2 =3.07 39.81 V Z = 3 j2.5 i o 11
Voltage and Frequency Effects Iron-core transformer characteristics vary with frequency and voltage. To determine why, we start with Faraday s law, e(t) = N d /dt. Specializing this to the sinusoidal ac case, it can be shown that the rms value of e.m.f. induced in primary, E1 = 4.44 f mN1 Vrms and rms value of e.m.f. induced in secondary, E2 = 4.44 f mN2 Vrms where m is the mutual coreflux. 12
Voltage and Frequency Effects Example: A 100 kVA, 4000 V/200 V, 50 Hz single-phase transformer has 100 secondary turns.Determine (a) the primary and secondarycurrent, (b) the number of primary turns,and (c) the maximum value of the flux. Solution: 13
Voltage and Frequency Effects (b) (c) or 14 Dr. Esam Jaragh
Real Transformers Real transformers have Losses Leakage flux Finite permeability of magnetic core Real power losses 1. Resistance in windings (I2R) 2. Core losses due to eddy currents and hysteresis 15
