APPENDI B CIRCUIT ANALSIS TECHNIQUES The following methods can be used to combine impedances to simplify the topology of an electric circuit. Also, formulae are given for voltage and current division across/through impedances. SERIES IMPEDANCES + PARALLEL IMPEDANCES + DELTA-TO-WE TRANSFORMATION Z C + + Z C Z C + + ZC Z Z Z C + + ZC Industrial Power Distribution, Second Edition. Ralph E. Fehr, III. 2016 The Institute of Electrical and Electronics Engineers, Inc. Published 2016 by John Wiley & Sons, Inc. 361
362 APPENDI B CIRCUIT ANALSIS TECHNIQUES WE-TO-DELTA TRANSFORMATION Z Z + Z Z + Z Z A B B C C A Z Z + Z Z + Z Z A B B C C A Z C Z C Z Z Z Z + Z Z + Z Z A B B C C A (a) (b) VOLTAGE DIVIDER V V B V B = V + CURRENT DIVIDER I I B I B = I + MESH-CURRENT ANALSIS A mesh is defined as any closed path through a planar circuit that contains no other closed paths. A planar circuit is one where no conductors cross over each other. To apply mesh-current analysis to a nonplanar circuit, it must first be redrawn as a planar circuit. Example of a mesh
MESH-CURRENT ANALSIS 363 Closed path Closed path Not a mesh Per Kirchhoff s voltage law (KVL), the sum of the voltage rises around any mesh must equal the sum of the voltage drops around the same mesh. This is an extension of the law of conservation of energy. Mesh-current analysis leads to a system of n equations with n unknowns, where n is the number of meshes in the circuit. Matrix methods, such as Cramer s rule, are helpful to solve these systems of equations. Variables are assigned to each of the mesh currents. While the direction of the mesh current is arbitrary, clockwise currents are often assumed for uniformity. Mesh-current analysis can be applied to the following circuit to produce the equations shown. 12 78 15 84 14 62 + _ + _ 1.03 0 i 1 i 2 i 3 1.01 2.3 27 66 33 73 Writing the equation for the mesh defined by mesh current i 1, 1.03 0 = (12 78 )i 1 +(27 66 )(i 1 i 2 ) The algebraic sign of the voltage source is determined by the terminal from which one leaves the source while traversing the mesh. Traversing the i 1 mesh clockwise, one leaves the positive terminal of the source, so 1.03 0 is positive. Distributing the 27 66 and simplifying algebraically by combining like terms, Similarly the second mesh gives which simplifies to (38.8 70 )i 1 +( 27 66 )i 2 = 1.03 0 0 = (15 84 )i 2 + (33 73 )(i 2 i 3 ) + (27 66 )(i 2 i 1 ) The third mesh yields which simplifies to ( 27 66 )i 1 + (75 73 )i 2 + ( 33 73 )i 3 = 0 1.01 2.3 = (14 62 )i 3 + (33 73 )(i 3 i 2 ) ( 33 73 )i 2 + (47 70 )i 3 = 1.01 2.3.
364 APPENDI B CIRCUIT ANALSIS TECHNIQUES Note that while traversing the third mesh clockwise, the negative terminal of the source is exited, so 1.01 2.3 is used in the equation. These three equations can be written as a single matrix equation to facilitate implementation of a linear algebra solution method such as Cramer s Rule: (38.8 70 )i 1 + ( 27 66 )i 2 = 1.03 0 ( 27 66 )i 1 + (75 73 )i 2 + ( 33 73 )i 3 = 0 ( 33 73 )i 2 + (47 70 )i 3 = 1.01 2.3 38.8 70 27 66 0 27 66 42 72 14 62 0 1 0 1 0 i 1 i 2 i 3 1.03 0 = 1.01 2.3 3.6 31.7 When a current source is present, an expression for the voltage across the current source cannot be written. Defining a supermesh that avoids the current source avoids the problem of not being able to express the voltage across the current source. A supermesh is not a mesh since it contains at least one closed path, but KVL applies to all closed paths through a circuit, not just meshes. An additional equation must be written addressing the current source (and any other circuit elements bypassed by the supermesh). The following circuit illustrates the use of a supermesh. 12 78 15 84 14 62 + _ + _ 1.03 0 i 1 i 2 i 3 1.01 2.3 27 66 3.6 31.7 The equation for the first mesh can be written as usual: 1.03 0 = (12 78 )i 1 + (27 66 )(i 1 i 2 ) (38.8 70 )i 1 + ( 27 66 )i 2 = 1.03 0 Writing mesh equations for the second and third meshes would involve the current source, so the second two meshes are combined into a supermesh as follows: 1.01 2.3 = (15 84 )i 2 + (14 62 )i 3 + (27 66 )(i 2 i 1 ) ( 27 66 )i 1 + (42 72 )i 2 + (14 62 )i 3 = 1.01 2.3 The third equation is developed from the branch avoided by the supermesh the branch containing the current source. By examining that branch, we can see by inspection that i 3 i 2 = 3.6 31.7
NODE-VOLTAGE ANALSIS 365 These three equations can be written as a single matrix equation as follows: (38.8 70 )i 1 + ( 27 66 )i 2 = 1.03 0 ( 27 66 )i 1 + (42 72 )i 2 + (14 62 )i 3 = 1.01 2.3 i 2 + i 3 = 3.6 31.7 38.8 70 27 66 0 27 66 42 72 14 62 0 1 0 1 0 i 1 i 2 i 3 1.03 0 1.01 2.3 3.6 31.7 NODE-VOLTAGE ANALSIS A node is defined as any closed path enclosing part of a circuit. When the size of a node approaches zero, it becomes a single point. Nodes Examples of nodes Per Kirchhoff s current law (KCL), the sum of the currents entering a node must equal the sum of the currents exiting that node. This is an extension of the law of conservation of charge. Node-voltage analysis leads to a system of n equations with n unknowns, where n + 1 is the number of nodes in the circuit. The equation for the last node (the reference node) is not linearly independent of the first n equations, so it is not necessary. Matrix methods, such as Cramer s rule, are helpful to solve these systems of equations. Variables are assigned to represent the node voltages. Then, KCL is applied to each labeled node, assuming a zero voltage reference at the reference node at the bottom of the circuit. 12 78 v 15 84 14 62 1 3 0 27 66 33 73 3.2 1.6
366 APPENDI B CIRCUIT ANALSIS TECHNIQUES At the node labeled v 1, a current of 3 0 enters from the left. Assuming the other two currents exit the node, the following node equation can be written: 3 0 = v 1 27 66 + v 1 15 84. Combining the v 1 terms and taking the reciprocals of the denominators, 0.103 77.6 v 1 +0.0667 96 = 3 0. A current of 3.2 1.6 enters node from the right. Assuming the other two currents exit the node, the following node equation can be written: 3.2 1.6 = 33 73 + v 1 15 84. Combining the terms and taking the reciprocals of the denominators, 0.0667 84 v 1 +0.0966 81 = 3.2 1.6. These two equations can be written as a single matrix equation: 0.103 77.6 v 1 +0.0667 96 = 3 0 0.0667 84 v 1 +0.0966 81 = 3.2 1.6 [ 0.103 77.6 0.0667 96 ] [ ] [ ] v1 3 0 = 0.0667 84 0.0966 81 3.2 1.6 When a voltage source is encountered during node-voltage analysis, the current through the voltage source cannot be expressed. 12 78 1.02 1 v 1 _ + 14 62 3 0 27 66 33 73 3.2 1.6 Creating a supernode that encompasses the voltage source will allow the nodevoltage process to be used. 12 78 1.02 1 v 1 _ + 14 62 3 0 27 66 33 73 3.2 1.6
EVALUATING DETERMINANTS 367 Now, a single node equation can be written for the supernode, assuming the currents through the 27 66 and 33 73 impedances leave the supernode. 3 0 + 3.2 1.6 v = 1 27 66 + 33 73 Simplifying the equation above, 0.0370 66 v 1 + 0.0303 73 = 6.2 0.8. Since the supernode equation contains two unknowns, a second equation must be written. By examining the supernode, it can be seen that v 1 = 1.02 1 These two equations can be written as a single matrix equation: 0.0370 66 v 1 + 0.0303 73 = 6.2 0.8 v 1 + = 1.02 1 [ 0.0370 66 0.0303 73 ][ ] [ v1 6.2 0.8 1 0 1 0 = 1.02 1 ]. EVALUATING DETERMINANTS The determinant of any square matrix can be evaluated in a number of ways. The determinant of a 2 2 matrix is defined as [ ] a b det = a b c d c d = ad bc. For a 3 3 matrix, the pattern that defines the 2 2 determinant can be expanded as follows: a b c d e f = (aek + bfg + cdh) (ceg + bdk + afh). g h k An alternative method is to decompose the larger determinant to 2 2 determinants using a process called minoring. Minoring must be used to evaluate determinants larger than 3 3. One way to apply minoring to find a 3 3 determinant is a b c d e f g h k = a e f h k b d f g k + c d e g h. Minoring can be done on any row or column. In the example above, minoring was done on the first row. It is advantageous to minor on a row or column that contains zeroes, since a zero entry will eliminate a term in the expansion. A minor has a dimension one less than the original determinant. Minoring can be repeated until every determinant is a 2 2 determinant. In the example above, when the a element is minored, the resulting minor is the 2 2 determinant remaining
368 APPENDI B CIRCUIT ANALSIS TECHNIQUES when the row and column containing the a element is eliminated. The minor must be multiplied by ( 1) i+j, where i and j are the row and column of the minored element in the original determinant. This term causes the algebraic sign of each term in the minor expansion to alternate. CRAMER S RULE This powerful linear algebra technique is useful for solving systems of equations, particularly when the coefficients are complex. For example, a 3 3 system of equations can be written as a matrix equation: ax + by + cz = m a b c x m dx + ey + fz = n d e f y = n gx + hy + kz = p g h k x p Four determinants can be defined: the first being the determinant of the coefficient matrix, and the next three being the determinant of the coefficient matrix with the constant vector substituted for one of the columns: a b c m b c a m c a b m D = d e f A = n e f B = d n f C = d e n g h k p h k g p k g h p The solution for the system of equations is x = A D, y = B D, z = C D. Cramer s rule can be extended to handle any size system of equations, and evaluation of the determinants can be done easily with software.