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##### ECE 3336 Introduction to Circuits & Electronics

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**ECE 3336 Introduction to Circuits & Electronics**Set #16 Transformers Fall 2011, TUE&TH 4-5:30 pm Dr. Wanda Wosik**Inductors**Inductors model the interaction between magnetic fields and voltage and current. - Magnetic flux in webers [Wb]**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 coilwilltry to respond to eliminate these changes (the second coil wants to maintain the original magnetic field). • This occurs by producing a voltage in the secondcoil that would result in a currentopposing 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 changeinmagnetic field will be a basis for transformers. Coupled Inductors Produces Constant Magnetic Field Produces Decreasing Magnetic Field Produces Increasing Magnetic Field Lenz’s law http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/indmut.html#c1**Mutual Inductance**Magnetomotive force 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” vp(t) vs(t) Voltage induced in the secondary coil in response to current changes in the primary coil. Current direction = polarity important v(t)=N•d/dt turns vp(t) vs(t) http://www.allaboutcircuits.com/vol_2/chpt_9/7.html**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, M12 = M21 = M. Sometimes we will use L11 and L22 for the self inductance and L12 and L21 for mutual inductance. We have i1 current in the primary coil We have i2 current in the secondary coil Remember that transformers will operate in AC conditions. The DC input voltagewill not induce any voltage in the secondary winding (e.i. electrical isolation will give an isolation transformer). http://www.allaboutcircuits.com/vol_2/chpt_9/7.html**We can derive equations for voltage and current ratios in**the ideal transformers both in the time domain and phasor domain Ideal Transformers time domain The same flux Hambley Voltage and current ratios**If v1 and v2 are both defined as the dotted terminal with**respect to the undotted terminal, then n2 / n1 = v2 / v1 = N. If v1 and v2 are both defined as the undotted terminal with respect to the dotted terminal, then n2 / n1 = v2 / v1 = N. Otherwise, n2 / n1 = -v2 / v1 = N. Dot Conventions for Ideal Transformers**The Dot Convention**Phases of the voltages in the primary and secondary windings are identified by dots. The samephase is obtained for both instantaneous voltages v1(t) and v2(t) 180° phase shift between instantaneous voltages**Dot Conventions for Ideal Transformers**Analogous relations are for currents: If i1 and i2 are both defined as entering the dotted terminal, then n2 / n1 = -i1 / i2 = N. If i1 and i2 are both defined as leaving the dotted terminal, then n2 / n1 = -i1 / i2 = N. Otherwise, n2 / n1 = i1 / i2 = N.**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. Center-tapped transformer 240 V We can tap the secondary voltage at two (or more) points 120V line 120V line**Figure 7.31, 7.32**Examples of transformers Other configurations of transformers Center-tapped transformer Power transformers can be huge or small http://www.allaboutcircuits.com/vol_2/chpt_9/7.html**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**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 reflectedimpedance.**Figure 7.35**The maximum power transfer in AC circuits Maximum power transfer to Rload (in DC) occurs when Rload=Rsource. In AC circuits we will need very similar impedance matching with the source:Zs=Rs+jXs 2) Now, from the complex power 1) The real power absorbed by Rload We calculate the real power (again) Where: Maximum power transfer if: ZL=ZS* RL=RS and XL=-XS When will PLMAX? PL=PLMAX 0**Impedance transformation improves power delivery - example**The heaters (62.5Ω and 15.625Ω) 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 a) b) Now if we use a step-up (N=2) transformer: the current delivered to Rload is again I=4A and the power is restored to P=1,000 W http://www.allaboutcircuits.com/vol_2/chpt_9/7.html**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 http://www.allaboutcircuits.com/vol_2/chpt_9/7.html**Figure 7.37a, b**Electric power transmission (a) direct power transmission is affected by the line resistance (b) power transmission with transformers Reflected load here M=1/N What will be =? We will use impedance transformation for Figure 7.37a, b, Rizzoni**Figure 7.37c, d**Electric power transmission - reduction of Rline by 1/N2 (c) equivalent circuit seen by generator (d) equivalent circuit seen by load Figure 7.37c, d**Figure 7.40, 7.41**Balanced three-phase Power (AC circuit) Positive, or abc, sequence for balanced three-phase voltages (“-” acb) Wye-wye (Y-Y) connection neutral Line voltages ab, bc, ca All line voltages Phase voltages Figure 7.41 Rizzoni**Figure 7.42**Balanced three-phase AC circuit (redrawn) Three circuits are in parallel. Can be eliminated Advantage of the 3 phase also in less wiring (3) Compared to single phase (6 wires). Constant! power Figure 7.42**Figure 7.43**Loads can be also in a delta connection Delta-connected generators Line (-to-line) voltage V=0 I=0 Currents drawn by wye- and by delta connected loads For both currents to be the same we have to have Z=3Zy Delta draws 3 times more current than a wye load does. Rizzoni, Figure 7.43**Figure 7.50**Line voltage convention for residential circuits A 3-wire AC system supplied by the power company red Higher line loss will be from the 120V source. To reduce power loss (I2R) thick wires are used. white (earth ground) 83.3A Power loss=69.4 W black Rs=0.02Ω 41.7.A Power loss=34.7 W This is the voltage (rms) between the hot wires**Figure 7.52**A typical residential wiring arrangement • Limit power dissipation by appropriate connections fro various loads. • Avoid heat generation (safety aspects) Figure 7.52**Figure 7.58**Structure of an AC power distribution network (just for your curiosity) That reduces power losses in transmission lines An electric power network =the Power grid allows for redistribution of power to various substations (various V levels obtained after stepping-down). Step-up transformer Substations Your house is carefully wired! Figure 7.58