Preamble Circuit Analysis II Physics, 8 th Edition Custom Edition Cutnell & Johnson When circuits get really complex methods learned so far will still work, but they can take a long time to do. A particularly bad case is using the Superposition Theorem with a large number of sources. We need to have some convenient way of analyzing these circuits. it (R. Bolton - 2012) Circuit Analysis II 1 (R. Bolton - 2012) Circuit Analysis II 2 Physics 155.3: Introduction to Electricity and Magnetism 1 Physics 155.3: Introduction to Electricity and Magnetism 2 Mesh Analysis Mesh analysis defines a unique array of currents to a circuit. This is done by applying Kirchoff s Voltage Law (KVL) to each loop (window) in the circuit diagram. Note that in applying KVL to the network, currents are determined. The number of unique mesh currents in a circuit is equal to the number of loops (windows) in the circuit. it Consider the following circuit: Loop 1 Loop 2 R1 E1 It has two loops (windows). I1 R3 I2 R2 E2 (R. Bolton - 2012) Circuit Analysis II 3 (R. Bolton - 2012) Circuit Analysis II 4 Physics 155.3: Introduction to Electricity and Magnetism 3 Physics 155.3: Introduction to Electricity and Magnetism 4
Note that it appears that there are 2 currents defined to be flowing through resistor R 3. This is not true; there is only one current. It is the difference between current I 1 and current I 2. Once currents have been determined, voltage across any component or any point in the circuit it can be calculated. l The procedure is as follows: Assign a distinct current in the clockwise direction to each independent closed loop in the circuit. Indicate the polarities within each loop for each resistor as determined d by the assumed direction of the loop current for that loop. (R. Bolton - 2012) Circuit Analysis II 5 (R. Bolton - 2012) Circuit Analysis II 6 Physics 155.3: Introduction to Electricity and Magnetism 5 Physics 155.3: Introduction to Electricity and Magnetism 6 Current Sources Apply Kirchoff s Voltage Law (KVL) around each closed loop in the clockwise direction. If a resistor has two or more currents through it make sure all currents (and their directions) are included. Loops without sources must be included in the analysis. They are handled the same. Solve the simultaneous linear equations for the assumed loop currents. Note that in mesh analysis only voltage sources are considered (i.e., KVL is used). Any current sources in the circuit would be source tranformed into equivalent voltage sources (if possible). This applies to real current sources (i.e., with an internal resistance). (R. Bolton - 2012) Circuit Analysis II 7 (R. Bolton - 2012) Circuit Analysis II 8 Physics 155.3: Introduction to Electricity and Magnetism 7 Physics 155.3: Introduction to Electricity and Magnetism 8
Example 1 Mesh Analysis Consider the following circuit Determine the current through R 3. Example 1 Solution: (1A ) Step 1: Assign a distinct current in the clockwise direction to each independent closed loop in the circuit. (R. Bolton - 2012) Circuit Analysis II 9 (R. Bolton - 2012) Circuit Analysis II 10 Physics 155.3: Introduction to Electricity and Magnetism 9 Physics 155.3: Introduction to Electricity and Magnetism 10 Step 2: Indicate the polarities within each loop for each resistor as determined by the assumed direction of the loop current for that loop. Step 3: Apply Kirchoff s Voltage Law (KVL) around each closed loop in the clockwise direction. If a resistor has two or more currents through it make sure all currents (and their directions) are included. (R. Bolton - 2012) Circuit Analysis II 11 (R. Bolton - 2012) Circuit Analysis II 12 Physics 155.3: Introduction to Electricity and Magnetism 11 Physics 155.3: Introduction to Electricity and Magnetism 12
Loop 1: (clockwise start at bottom left) +E 1 V 1 V 3 = 0 +2V (2Ω)I 1 (4Ω)(I 1 I 2 )=0 Loop 2: (clockwise start at bottom middle) V 3 V 2 -E 2 = 0 (4Ω)(I 2 I 1 ) (1Ω)I 2 6V= 0 Step 4: Solve the simultaneous linear equations for the assumed loop currents. Rewrite equations without units (to make is less cluttered) and simplify. (R. Bolton - 2012) Circuit Analysis II 13 (R. Bolton - 2012) Circuit Analysis II 14 Physics 155.3: Introduction to Electricity and Magnetism 13 Physics 155.3: Introduction to Electricity and Magnetism 14 Loop 1: or +2 2I 1 4(I 1 I 2 ) = 0 +2 2I 1 4I 1 +4I 2 =0 6I 1 +4I 2 =-2 Loop 2: or 4(I 2 I 1 ) 1I 2 6V= 0 4I 2 +4I 1 1I 2 6V= 0 +4I 1 5I 2 = +6 (R. Bolton - 2012) Circuit Analysis II 15 (R. Bolton - 2012) Circuit Analysis II 16 Physics 155.3: Introduction to Electricity and Magnetism 15 Physics 155.3: Introduction to Electricity and Magnetism 16
Nodal Analysis Solving these equations: I 1 = -1A I 2 = -2A The minus signs indicate that the currents are (in fact) in the opposite directions to that assumed. What is the net current through the 4Ω resistor? Pick a loop equation and substitute for I 1 and I 2 : If we pick Loop 1 (i.e., in the direction i of I 1 ) I 4Ω = I 1 I 2 = -1A (-2A) = 1A (i.e., in direction of I 1 ) If we pick Loop 2 (i.e., in the direction of I 2 ) I 4Ω = I 2 I 1 = -2A (-1A) = -1A (i.e., opposite to direction of I 2 ) It should come as no surprise that there is a complementary analysis technique to mesh analysis. Nodal analysis defines a unique array of voltages to a circuit. This is done by applying Kirchoff s Current Law (KCL) to each loop (window) in the circuit diagram. Note that in applying KCL to the network, voltages are determined. Once voltages have been determined, current through any component or at any node in the circuit can be calculated. Note that in nodal analysis only current sources are considered (i.e., KCL is used). Any voltage sources in the circuit would be source transformed into equivalent current sources. This applies to real voltage sources (i.e., with an internal resistance). We will not be developing nodal analysis as part of Physics 155.3. This analysis technique will be used in upper year engineering classes. (R. Bolton - 2012) Circuit Analysis II 17 (R. Bolton - 2012) Circuit Analysis II 18 Physics 155.3: Introduction to Electricity and Magnetism 17 Physics 155.3: Introduction to Electricity and Magnetism 18