1 ECE 3336 Introduction to Circuits & Electronics Set #16 Transformers Fall 2011, TUE&TH 4-5:30 pm Dr. Wanda Wosik

2 Inductors Inductors model the interaction between magnetic fields and voltage and current.  - Magnetic flux in webers [Wb]

3 Coupled Inductors Produces Constant Magnetic Field Produces Decreasing Magnetic Field Produces Increasing Magnetic Field Lenz’s law A current flowing in one inductor induces magnetic field that is felt by the second coil If this field changes due to changes in the original current, the second coil will try to respond to eliminate these changes (the second coil wants to maintain the original magnetic field). This occurs by producing a voltage in the second coil that would result in a current opposing these changes (through the magnetic field). Such interaction of both coils regarding their currents and voltages is referred to as mutual inductance. Such voltage generation that opposes the change in magnetic field will be a basis for transformers.

4 Mutual Inductance We model this effect with what we call “mutual inductance”, and we call it M [H] We also rename L [H], as the “self inductance” Magnetomotive force v p (t) v s (t) v p (t) v s (t) v(t)=Nd  /dt turns Voltage induced in the secondary coil in response to current changes in the primary coil. Current direction = polarity important

5 Mutual Inductance “Mutual inductance”, the influence of one coil on the other, is assumed in this course to be the same as the second on the first. That is, M 12 = M 21 = M. We have i 1 current in the primary coilWe have i 2 current in the secondary coil Remember that transformers will operate in AC conditions. The DC input voltage will not induce any voltage in the secondary winding (e.i. electrical isolation will give an isolation transformer). Sometimes we will use L 11 and L 22 for the self inductance and L 12 and L 21 for mutual inductance.

6 Ideal Transformers We can derive equations for voltage and current ratios in the ideal transformers both in the time domain and phasor domain The same flux time domain Hambley Voltage and current ratios

7 Dot Conventions for Ideal Transformers If v 1 and v 2 are both defined as the dotted terminal with respect to the undotted terminal, then n 2 / n 1 = v 2 / v 1 = N. If v 1 and v 2 are both defined as the undotted terminal with respect to the dotted terminal, then n 2 / n 1 = v 2 / v 1 = N. Otherwise, n 2 / n 1 = -v 2 / v 1 = N.

8 The Dot Convention Phases of the voltages in the primary and secondary windings are identified by dots. 180° phase shift between instantaneous voltages The same phase is obtained for both instantaneous voltages v 1 (t) and v 2 (t)

9 Dot Conventions for Ideal Transformers Analogous relations are for currents: If i 1 and i 2 are both defined as entering the dotted terminal, then n 2 / n 1 = -i 1 / i 2 = N. If i 1 and i 2 are both defined as leaving the dotted terminal, then n 2 / n 1 = -i 1 / i 2 = N. Otherwise, n 2 / n 1 = i 1 / i 2 = N.

10 Power in Ideal Transformers Note that if the voltage increases going from one side of a transformer to the other, the current decreases, and by the same factor. There is no power gain. The factor is the ratio of the number of turns. We named this as the turns ratio, N. We can tap the secondary voltage at two (or more) points Center-tapped transformer 240 V 120V line

11 Figure 7.31, 7.32 Center-tapped transformer Power transformers can be huge Other configurations of transformers Examples of transformers or small

12 Impedance Reflection with Transformers Transformers can be used to match loads (impedances). Note that If we divide the first equation by the second, we get

13 Impedance Reflection with Transformers Transformers can be used to match loads (impedances). This means that if we look at the apparent impedance seen at the primary side of a transformer (Z’) we will see the impedance at the secondary side divided by the turns ratio squared. This can be very useful. It is often referred to as the reflected impedance.

14 The maximum power transfer in AC circuits Figure 7.35 Maximum power transfer to R load (in DC) occurs when R load =R source. In AC circuits we will need very similar impedance matching with the source: Z s =R s +jX s 1) The real power absorbed by R load Where: 2) Now, from the complex power We calculate the real power (again) Z L =Z S * R L =R S and X L =-X S Maximum power transfer if: When will P LMAX ? 0 P L =P LMAX

15 Impedance transformation improves power delivery - example The heaters (62.5Ω and Ω) of 1,000W power rating operate in two circuits a) and b). If we use the heater a) with 125 V source the power will decrease P=250 W e.i. [(125V/62.5Ω)x125V] because I=2A Now if we use a step-up (N=2) transformer: the current delivered to R load is again I=4A and the power is restored to P=1,000 W a) b)

16 What happens with power? How to play it loud? Maximum power would be delivered to the load of 500 Ω. But we have 8Ω. AC Thevenin circuit We want to supply power from a high impedance (V high I low) amplifier to a low impedance (low V high I) speaker. A transformer will give impedance transformation ratio 500:8 so that the delivered power will reach its maximum

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18 Figure 7.37a, b Electric power transmission Figure 7.37a, b, Rizzoni (a) direct power transmission is affected by the line resistance What will be  =? We will use impedance transformation for  Reflected load hereM=1/N (b) power transmission with transformers

19 Figure 7.37c, d Electric power transmission - reduction of R line by 1/N 2 Figure 7.37c, d (c) equivalent circuit seen by generator (d) equivalent circuit seen by load

20 Figure 7.40, 7.41 Balanced three-phase Power (AC circuit) Figure 7.41 Rizzoni neutral Phase voltages Line voltages ab, bc, ca Positive, or abc, sequence for balanced three-phase voltages ( “ - ” acb) All line voltages Wye-wye (Y-Y) connection

21 Figure 7.42 Balanced three-phase AC circuit (redrawn) Figure 7.42 Three circuits are in parallel. Constant! power Can be eliminated Advantage of the 3 phase also in less wiring (3) Compared to single phase (6 wires).

22 Figure 7.43 Delta-connected generators Rizzoni, Figure 7.43 Currents drawn by wye- and by delta connected loads For both currents to be the same we have to have    y Delta draws 3 times more current than a wye load does.  V=0 I=0 Loads can be also in a delta connection Line (-to-line) voltage

23 Figure 7.50 Line voltage convention for residential circuits red black white (earth ground) This is the voltage (rms) between the hot wires A 3-wire AC system supplied by the power company Higher line loss will be from the 120V source. To reduce power loss (I 2 R) thick wires are used. 83.3A 41.7.A R s =0.02Ω Power loss=69.4 W Power loss=34.7 W

24 Figure 7.52 A typical residential wiring arrangement Figure 7.52 Limit power dissipation by appropriate connections fro various loads. Avoid heat generation (safety aspects)

25 Figure 7.58 Structure of an AC power distribution network (just for your curiosity) Figure 7.58 Step-up transformer That reduces power losses in transmission lines Substations Your house is carefully wired! An electric power network =the Power grid allows for redistribution of power to various substations (various V levels obtained after stepping-down).