Chapter 08. Methods of Analysis

Transcription

1 Chapter 08 Methods of Analysis Source: Circuit Analysis: Theory and Practice Delmar Cengage Learning C-C Tsai Outline Source Conversion Mesh Analysis Nodal Analysis Delta-Wye ( -Y) Conversion Bridge Networks C-C Tsai 2 1

2 Linear and Nonlinear V-I Curves Ohm Law I = V / R R is fixed I V / R R may be thermistor or photocell C-C Tsai 3 Constant Current Sources Maintains same current in branch of circuit Regardless of how components are connected external to the source Direction of current source indicates direction of current flow in branch For example: Calculate the voltage V s across current source I if the resistor is 100 Ω V s = I *R = 2 * 100 = 200 V C-C Tsai 4 2

3 Example: Constant Current Sources Determine V S C-C Tsai 5 Example: Constant Current Sources Determine the voltage V S and currents I 1 and I 2 C-C Tsai 6 3

4 Source Conversions Ideal current source I Infinite shunt (parallel) resistance Rs = Real current source I Some shunt (parallel) resistance Rs C-C Tsai 7 Source Conversions If internal resistance of a source is considered: Voltage source may be converted to current source Calculate current from E/R S, R S does not change, and place current source and resistor in parallel Current source may be converted to voltage source E = I R S and place voltage source in series with resistor C-C Tsai 8 4

5 Current and Voltage Sources Exchange A load connected to a voltage source or its equivalent current Should have same voltage and current for either source Although sources are equivalent Currents and voltages within sources may differ Sources are only equivalent external to terminals C-C Tsai 9 Voltage Source Current Source Determine I L C-C Tsai 10 5

6 Current Source Voltage Source Determine I L C-C Tsai 11 Current Sources in Parallel and Series Current sources in parallel Simply add together algebraically Add magnitude currents in one direction Subtract magnitude currents in opposite direction C-C Tsai 12 6

7 Noted: Current Sources in Parallel and Series Current sources with different values Never place in series and This violates KCL C-C Tsai 13 Example1: Current Sources in Parallel and Series C-C Tsai 14 7

8 Example2: Current Sources in Parallel and Series C-C Tsai 15 Branch Current Analysis Used for circuits having more than one source Use different methods of analysis Begin by arbitrarily assigning current directions in each branch Label polarities of the voltage drops across all resistors Step0: Assume all the current I 1, I 2, Step1: Write KVL around all loops Step2: Apply KCL at enough nodes so all branches have been included Step3: Solve resulting equations C-C Tsai 16 8

9 Example1: Branch Current Analysis From KVL: 6-2I 1 + 2I 2-4 = 0 4-2I 2-4I = 0 From KCL: I 3 = I 1 + I 2 Solve simultaneous equations C-C Tsai 17 Example2: Branch Current Analysis Loop badb: - 2I 2 + 3I 3-8 = 0 Loop bacb: - 2I 2 + I 4-6 = 0 Node a: I 3 + I 4 = 5 + I 2 Solve simultaneous equations C-C Tsai 18 9

10 Ex0 using Source Conversions R1 R3 2Ω V1 6 V R2 2Ω V2 4 V 4Ω V3 2 V + U1 3.6 V - DC 10MOhm I1 3 A R4 2Ω I2 2 A R6 2Ω I3 0.5 A R5 4Ω V U2 DC 10MOhm C-C Tsai 19 Mesh Analysis Step0: Arbitrarily assign a clockwise current to each interior closed loop (Mesh) Step1: Indicate voltage polarities across all resistors Step2: Write KVL equations Step3: Solve resulting simultaneous equations C-C Tsai 20 10

11 Example1: Mesh Analysis Assign loop currents and voltage polarities Using KVL: 6-2I 1-2I 1 + 2I 2-4 = 0 4-2I 2 + 2I 1-4I = 0 Simplify and solve equations C-C Tsai 21 Example2: Mesh Analysis C-C Tsai 22 11

12 Example3: Mesh Analysis C-C Tsai 23 Example4: Mesh Analysis C-C Tsai 24 12

13 Example5: Mesh Analysis C-C Tsai 25 Nodal Analysis Step0: Assign a reference node within circuit and indicate node as ground Convert voltage sources to current sources Arbitrarily assign a current direction to each branch where there is no current source Step1: Assign voltages V 1, V 2, etc. to remaining nodes Step2: Apply KCL to all nodes except reference node Rewrite each current in terms of voltage Step3: Solve resulting equations for voltages C-C Tsai 26 13

14 Example0: Nodal Analysis Assign voltage at node v 1, then using KVL (V 1-6)/2 + (V 1-4)/2 + (V 1 -(-2))/4= 0 C-C Tsai 27 Example1: Nodal Analysis Using KCL for nodes V 1 and V 2 200mA+50mA = I 1 +I 2 200mA+I 2 = 50mA+I 3 C-C Tsai 28 14

15 Example2: Nodal Analysis Using KCL for nodes V 1 and V 2 I 1 +I 2 = 2A 3A+I 2 = I 3 +I 4 C-C Tsai 29 Example3: Nodal Analysis Using KCL for nodes V 1 and V 2 V 1 /3+(V 1 -V 2 )/5+6 = 1 V 2 /4+(V 2 -V 1 )/5+2+1 = 0 C-C Tsai 30 15

16 Example4: Nodal Analysis Using KCL for nodes V 1 and V 2 V 1 /5K+V 1 /3K+(V 1 -V 2 )/4K+3mA = 2mA V 2 /2K+(V 2 -V 1 )/4K = 2mA C-C Tsai 31 Example5: Nodal Analysis Determine voltages V 1 and V 2 C-C Tsai 32 16

17 Delta-Wye ( -Y) Conversion Resistors connected to a point of Y Obtained by finding product of resistors connected to same point in Delta Divided by sum of all Delta resistors R 1 =(R C *R B ) / (R A +R B +R C ) C-C Tsai 33 Example: Y Conversion Given a Delta circuit with resistors of 30, 60, and 90 Resulting Y circuit will have resistors of 10, 15, and 30 R 1 =(30*60) / ( ) = 10 C-C Tsai 34 17

18 Wye-Delta Conversions A Delta resistor is found: Taking sum of all two-product combinations of Y resistor values Divided by resistance of Y directly opposite resistor being calculated R A =(R 1 R 2 +R 2 R 3 +R 1 R 3 ) /R 1 C-C Tsai 35 Example1: Y Conversions For a Y circuit having resistances of 2.4, 3.6, and 4.8 K Resulting Delta resistors will be 7.8, 10.4, and 15.6 K R A =(3.6K*2.4K+2.4K*4.8K+4.8K*3.6K) /4.8K = 7.8K C-C Tsai 36 18

19 Example2: Y- Conversions C-C Tsai 37 Bridge Networks Three same equivalent bridge networks Balanced bridge: R 1 R 4 = R 2 R 3 and I R5 =0 Unbalanced bridge: R 1 R 4 R 2 R 3 and I R5 0 C-C Tsai 38 19

20 Examples: Bridge Networks Balanced bridge: 30*240 = 60*120 R 1 R 4 = R 2 R 3 and I R5 =0 Unbalanced bridge: 20*80 40*60 R 1 R 4 R 2 R 3 and I R5 0 C-C Tsai 39 Example: Bridge Networks Balanced bridge: 3*24 = 6*12 R 1 R 4 = R 2 R 3 and I R5 =0 R 5 can be replaced with an open circuit or a short circuit. C-C Tsai 40 20

21 Example: Bridge Networks Unbalanced bridge: 6*3 12*3 R 1 R 4 R 2 R 3 and I R5 0 Mathod1:Using mesh analysis with KVL C-C Tsai 41 Example: Bridge Networks Unbalanced bridge: 6*3 12*3 R 1 R 4 R 2 R 3 and I R5 0 Mathod2: Using node analysis with KCL C-C Tsai 42 21

22 Example: Bridge Networks Unbalanced bridge: 6*3 12*3 Mathod3: Using Y conversion C-C Tsai 43 Kernel abilities 1. Can use Mesh Analysis for solving the unknown voltage and current of a circuit. 2. Can use Nodal Analysis for solving the unknown voltage and current of a circuit. 3. Can use Delta-Wye ( -Y) Conversion for solving the unknown voltage and current of a circuit. 4. Can recognize a Bridge circuit whether is balance or unbalance and solve the unknown voltage and current. C-C Tsai 44 22

23 Problem 14 Determine the voltage V ab C-C Tsai 45 Problem 21 Determine the current I 2 C-C Tsai 46 23

24 Problem 47 Determine the current I C-C Tsai 47 24