Presentation on theme: "SOLUTION USING THEVENIN’S THEOREM"— Presentation transcript:
1SOLUTION USING THEVENIN’S THEOREM Working out resistor currents in unbalanced bridge circuits can be frustrating, especially when depending entirely on Kirchoff laws to do so. See Fig-1 below. However, one can employ other mathematical tools like Thevenin and Norton equivalents called “Source Transformations” to make the task easier. For example, Thevenin’s Theorem states that any complex linear circuit can be simplified to an equivalent voltage source and series resistor connected to a load resistor. (See Fig-1A below)Left click on this slide to see how this is done.
2Procedure:Call Rx the Load Resistor (See Fig-1 below)Remove Rx and calculate the Voltage potential between A & B This is the equivalent Voltage source. (See Fig-2 below)Place a short across 100V and calculate resistance between A & B This result is the equivalent series resistance. (See fig-3 below)Draw the series equivalent circuit and calculate the Rx current (See Fig.4 below) This result, 0.98A, is the current through Rx.Since the current through Rx (0.98A) is now known, the remaining currents in Fig-1 can be easily calculated using Kirchoff Laws.
3KIRCHOFF LAWSKirchoff’s Current Law: The sum of all currents entering a node must be equal currents leaving the node.Kirchoff Voltage Law: The sum of the voltages around a closed loop must equal to zero.To set up the problem draw the circuit and make notations as follows:Draw the circuit and enter Volts and resistance values (see Fig-5)Using conventional current flow (+ to -) enter arrows indicating direction of current flow in all resistors Label unknown currents I1, I2, & I3 through R1, R2, & R3 respectivelyLabel R4 current as (I1 - I3 ) & R5 current as (I2 + I3)
4KIRCHOFF LAWSSince there are 3 unknowns currents (I1, I2 & I3), select 3 loops in Fig-5 and from these, develop 3 equations to be used for working out I1, I2 & I3.Loop 1: 100V, R1 & R4 Loop 2: 100V, R2 & R5 Loop 3: R1, R2 & R3Eq. 1: – R1 I1 – R4(I1 – I3) = 0 Eq. 2: – R2 I2 – R5(I2 + I3) = 0 Eq. 3: R1 I1 + R3 I3 –R2 I2 = 0Substitute resistor values for R1, R2 & R3Eq. 1A: – 12 I1 – 24(I1 – I3) = 0 Eq. 2A: – 36 I2 – 18(I2 + I3) = 0 Eq. 3A: I I3 –36 I2 = 0
5KIRCHOFF LAWS From previous slide Eq. 1A: – 12 I1 – 24(I1 – I3) = 0 Eq. 2A: – 36 I2 – 18(I2 + I3) = 0 Eq. 3A: I I3 –36 I2 = 0Expand to get rid of parenthesesEq. 1B: – 12 I1 – 24I1 + 24I3 = 0 Eq. 2B: – 36 I2 – 18I2 – 18I3 = 0 Eq. 3B: I I3 –36 I2 = 0Consolidating we get:Eq. 1C: – 36 I I3 = Eq. 2C: – 54 I2 – 18 I3 = Eq. 3C: I I3 –36 I2 = 0We have 3 simultaneous equations with 3 unknowns to find I1, I2 & I3. Now the fun begins.
6From previous slideEq. 1C: – 36 I I3 = Eq. 2C: – 54 I2 – 18 I3 = Eq. 3C: I I3 –36 I2 = 0These 3 equations can be reduced to two equations with two unknown currents using traditional algebra. The procedure is as follows:Multiply Eq. 1C through by 18: 18(100 – 36 I I3 = 0) to get: Eq. 1D: – 648 I I3 =0Multiply Eq. 2C through by 24: 24(100 – 54 I2 – 18 I3 = 0) to get D: – 1296 I2 – 432 I3 =0By Adding Eq. 1D to Eq. 2D we get: – 648 I1 – 1296 I2 or Eq. 4A: 4200 = 648 I I2 (One of 2 equations with 2 unknowns)Repeat procedure using eq. 2C & 3C to get the other equation with 2 unknowns
7KIRCHOFF LAWSEq. 2C: – 54 I2 – 18 I3 = Eq. 3C: I I3 –36 I2 = 0Multiply Eq. 2C by 14 to get 14 (100 – 54 I2 – 18 I3 = 0) or: Eq. 2E: – 756 I2 – 252 I3 = 0Multiply Eq. 3C by 18 to get 18(12 I I3 –36 I2 = 0) or: Eq. 3E: I I3 – 648 I2 = 0Add Eq. 3E to Eq. 2E to get I1 – 1404 I2 or Eq. 4B: 1400 = – 216 I I2 (The 2nd Equation with 2 unknownsSo now we have:Eq. 4A: 4200 = 648 I I2 Eq. 4B: = – 216 I I2Multiply Eq. 4A by 1404 to get: Eq. 5A = – I I2 Multiply Eq. 4B by 1296 to get: Eq: 5B = I I2Subtract Eq.5A from Eq. 5B to get: = I1 or I1 = 3.43A
8KIRCHOFF LAWSWe calculated I1 = 3.43A going through R1. I3 Should be easy to find. Using the 100V, R1 R4 loop we get:100 = (12 x 3.43) + 24( I3) or 100 = –24 I3 or I3 =I3 = / 24 = A ~ 0.98AAs you can see from the above application of Kirchoff Laws, it takes a good working knowledge of Algebra and a close watch on the details. Even with these qualities, there are many pitfalls on the way to a solution: getting the signs wrong, arithmetic errors, misplacement of variables etc. I don’t think I want to ever repeat such an exercise again. When ever possible, choose the Thev enin Voltage equivalents or the Norton Current equivalents.