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BME 205 L03 DC Circuits – Methods of Analysis Introduc>on • We now use Ohm’s Law and Kirchhoﬀ’s Laws to develop powerful techniques for circuit analysis: • Nodal Analysis – based on applica>on of Kirchhoﬀ’s Current Law • Mesh Analysis – based on applica>on of Kirchhoﬀ’s Voltage Law • For almost any circuit we can use these techniques to obtain a set of simultaneous equa>ons and solve for all voltages & currents. • We’ll use Cramer’s Rule to solve the simultaneous equa>ons; this is easy… Cramer’s Rule ! A # 11 # A21 # #" A31 x1 = A12 A22 A32 Δ1 Δ A13 $! x1 &# A23 &# x2 &# A33 &%#" x3 x2 = $ ! b $ & # 1 & & = # b2 & & # & &% #" b3 &% Δ2 Δ x3 = A11 A12 A13 Δ = A21 A22 A23 = A31 A32 A33 Δ3 Δ A11 ( A22 A33 − A32 A23 ) − A 12 ( A21 A33 − A31 A23 ) + A 13 ( A21 A32 − A31 A22 ) b1 A12 A13 Δ1 = b2 A22 A23 b3 A32 A33 A11 b1 A13 A11 A12 b1 Δ 2 = A21 b2 A23 Δ 3 = A21 A22 b2 A33 A31 A32 b3 A31 b3 Nodal Analysis • Nodal analysis (also known as the node-‐voltage method) provides a general procedure for analyzing circuits using node voltages as the circuit variables. • Choosing node voltages instead of element voltages as circuit variables is convenient and reduces the number of equa>ons one must solve simultaneously. • To simplify maXers, we’ll assume for now that circuits do not contain voltage sources. Circuits that contain voltage sources will be analyzed later on. • In this case, the steps to determine node voltages are... Steps to determine Node Voltages • Given a circuit with n nodes without voltage sources: 1. Select a node as the reference node. Assign voltages v1, v2, . . . , vn−1 to the remaining n − 1 nodes. The voltages are referenced with respect to the reference node. 2. Apply KCL to each of the n − 1 nonreference nodes. Use Ohm’s law to express the branch currents in terms of node voltages. 3. Solve the resul>ng simultaneous equa>ons to obtain the unknown node voltages. Reference / Ground • The ﬁrst step in nodal analysis is selec>ng a node as the reference or datum node. The reference node is commonly called the ground since it is assumed to have zero poten>al. A reference node is indicated by any of the three symbols below. The type of ground in (b) is called a chassis ground and is used in devices where the case, enclosure, or chassis acts as a reference point for all circuits. When the poten>al of the earth is used as reference, we use the earth ground in (a) or (c). We shall always use the symbol in (b). Assign nonreference nodes • Once we have selected a reference node, we assign voltage variable labels to nonreference nodes. Consider, for example, the circuit shown. Node 0 is the reference node (v = 0), while nodes 1 and 2 are assigned voltages v1 and v2, respec>vely. • Keep in mind that the node voltages are deﬁned with respect to the reference node. • Each node voltage is the voltage rise from the reference node to the corresponding nonreference node or simply the voltage of that node with respect to the reference node. Apply KCL • As the second step, we apply KCL to each nonreference node in the circuit. Label i1, i2, and i3 as the currents through resistors R1,R2, and R3, respec>vely. At node 1, applying KCL gives I 1 = I 2 + i1 + i2 I 2 + i2 = i3 • At node 2: • We now apply Ohm’s law to express the unknown currents i1 , i2 , and i3 in terms of node voltages. • Keep in mind the passive sign conven>on: in the equa>on v=iR, the “i” refers to current ﬂowing from the + to the – of v. • This gives us • We subs>tute these into our KCL equa>ons: ... or in terms of conductances: Solve for Node Voltages • The third step in nodal analysis is to solve for the node voltages. If we apply KCL to n−1 nonreference nodes, we obtain n−1 simultaneous equa>ons. For the current example circuit we could use any standard method, such as subs>tu>on, elimina>on, Cramer’s rule, or matrix inversion. • To use either of the last two methods, one must cast the simultaneous equa>ons in matrix form. For example, our conductance equa>ons can be cast in matrix form as which can be solved to get v1 and v2 . Example Calculate the node voltages in this circuit. Solu%on: First prepare the circuit for nodal analysis, labeling the node voltages and currents as shown below. No>ce how the currents are selected for the applica>on of KCL -‐ Except for the branches with current sources, the labeling is arbitrary but consistent, i.e. the currents entering a resistor are in the same direc>on when they go out the other end. The reference node is selected, and the node voltages v1 and v2 are now to be determined. • At node 1, applying KCL and Ohm’s law gives i1 = i2 + i3 ⇒ 5 = v1 − v2 v1 − 0 + 4 2 Mul>plying each term in the last equa>on by 4, we obtain 20 = v1 − v2 + 2v 1 or 3v1 − v2 = 20 At node 2, we do the same thing and get Mul>plying each term by 12 results in We could use elimina>on: add underlined equa>ons & subs>tute • • Alterna>vely, we could use Cramer’s rule: which is as we got before. • We can then get the currents easily if required. Examples Obtain the node voltages 1) 2) 3) Obtain the node voltages i2 i1 1) i3 1=i1+ i2 1=v1/2+ (v1-‐v2) /6 i2=i3+4 (v1-‐v2)/6=v2/7+ 4 6=3v1+ v1-‐v2 7v1-‐7v2=6v2+168 4v1-‐v2=6 7v1-‐13v2=168 " 4 −1 %" v1 % " 6 % '=$ ' $ '$ # 7 −13 &$# v2 '& # 168 & 4 −1 Δ= 7 −13 6 −1 Δ1 = 168 −13 Δ=-‐52+7=-‐45 V1=Δ1/Δ=-‐2 V Δ1=-‐78+168=90 4 6 Δ2 = 7 168 Δ2=4x168-‐7x6=670 V2=Δ2/Δ=-‐14 V Solu>on to problem 2) Label node voltages & currents: i2 3) v2 v1 i1 10=i1+ i2 10=(v1-‐v3) /2+ (v1-‐v2) /3 5v1-‐ 2v2-‐3v3=60 v3 i3 i1+ 4ix=ix (v1-‐v2) /3+ 4(v2/4)= v2/4 4v1+ 5v2=0 " 5 −2 −3 %" v1 % " 60 % ' $ $ '$ ' $ 4 5 0 '$ v2 ' = $ 0 ' $ 3 −6 −4 '$ v ' $ 0 ' # &$# 3 '& # & V1=Δ1/Δ=4.8 V i2=4ix+i3 (v1-‐v3) /2=v2+ v3 /6 3v1-‐ 6v2-‐4v3=0 5 −2 −3 Δ= 4 5 0 3 −6 −4 60 −2 −3 Δ1 = 0 5 0 0 −6 −4 5 60 −3 Δ2 = 4 0 0 3 0 −4 5 −2 60 Δ3 = 4 5 0 3 −6 0 V2=Δ2/Δ=2.4 V V3=Δ3/Δ=-‐2.4 V Nodal Analysis with Voltage Sources • CASE I. If a voltage source is connected between the reference node and a nonreference node, we simply set the voltage at the nonreference node equal to the voltage of the voltage source. In this example, v1 = 10 V. • CASE II. If the voltage source (dependent or independent) is connected between two nonreference nodes, the two nonreference nodes form a generalized node or supernode; we apply both KCL and KVL to A supernode is formed by enclosing determine the node voltages. a(dependent or independent) voltage source connected between two nonreference nodes and any elements connected in parallel with it. • So in this example, nodes 2 and 3 form a supernode. (We could have more than two). We analyze a circuit with supernodes using the same three steps men>oned in the previous sec>on except that the supernodes are treated diﬀerently. • Why? Because an essen>al component of nodal analysis is applying KCL, which requires knowing the current through each element. There is no way of knowing the current through a voltage source in advance. However, KCL must be sa>sﬁed at a supernode like any other node. Hence, at the supernode in this example, • To apply Kirchhoﬀ’s voltage law to the supernode, we redraw the circuit as shown. Going around the loop in the clockwise direc>on gives • So now we have 3 equa>ons in v1, v2, v3. • Note the following proper>es of a supernode: 1. The voltage source inside the supernode provides a constraint equa>on needed to solve for the node voltages. 2. A supernode has no voltage of its own. 3. A supernode requires the applica>on of both KCL and KVL. Example Examples Find v and i Find node voltages i1 v2=v v1=7 v3=2i i2 i3 i1=i2+i+i3 (7-‐v)/3=v/3+i+2i/6 -‐v-‐3+2i=0 (7-‐v)/3=v/3+4i/3 2i=v+3 21-‐3v=4v+16i 16i=8v+24 21-‐16i=7v 21-‐8v-‐24=7v -‐3=15v à v=-‐0.2 V 16i=8(-‐0.2)+24à i=1.4 A i2 ix iy i4 i3 -‐v1+10+v2=0àv1-‐v2=10 v1=2i i+i2+i3=iy -‐v2-‐5i+v3=0à-‐v2-‐2.5v1+v3=0à2.5v1+v2-‐v3=0 i3+i4=ix+i2 v1/2+(v1-‐v3)/6+v2/4=iy i3=ix+iy V2/4+v3/3=ix+(v1-‐v3)/6 v1/2+v2/2+v3/3=v2/4à v1/2+v1/4+v3/3=0 " 1 −1 0 $ −1 $ 5/2 1 $ 1/ 2 1/ 4 1/ 3 # %" v1 % " 10 % ' $ '$ ' '$ v2 ' = $ 0 ' '$ v ' $ 0 ' &$# 3 '& # & v1=3.043 V v2=-‐6.96 V v3=0.65 V Mesh Analysis • Mesh analysis provides another general procedure for analyzing circuits, using mesh currents as the circuit variables. • Using mesh currents instead of element currents as circuit variables is convenient and reduces the number of equa>ons that must be solved simultaneously. • Recall that a loop is a closed path with no node passed more than once. A mesh is a loop that does not contain any other loop within it. • Nodal analysis applies KCL to ﬁnd unknown voltages in a given circuit, while mesh analysis applies KVL to ﬁnd unknown currents. Mesh analysis is not quite as general as nodal analysis because it is only applicable to a circuit that is planar. • A planar circuit is one that can be drawn in a plane with no branches crossing one another; otherwise it is nonplanar. A circuit may have crossing branches and s>ll be planar if it can be redrawn such that it has no crossing branches. • For Example, (a) has two crossing branches, but it can be redrawn as in (b), so it is planar. • However, circuit (c) is nonplanar, because there is no way to redraw it and avoid the branches crossing. Nonplanar circuits can be handled using nodal analysis, but they will not be considered in this course. (c) A mesh is a loop which does not contain any other loops within it. • For example, paths abefa and bcdeb are meshes, but path abcdefa is not a mesh. The current through a mesh is known as mesh current. • In mesh analysis, we are interested in applying KVL to ﬁnd the mesh currents in a given circuit. In the mesh analysis of a circuit with n meshes and no current sources, we take the following three steps: 1. Assign mesh currents i1, i2, . . . , in to the n meshes. 2. Apply KVL to each of the n meshes. Use Ohm’s law to express the voltages in terms of the mesh currents. 3. Solve the resul>ng n simultaneous equa>ons to get the mesh currents. • For this example, the ﬁrst step requires that mesh currents i1 and i2 are assigned to meshes 1 and 2. Although a mesh current may be assigned to each mesh in an arbitrary direc>on, it is conven>onal to assume that each mesh current ﬂows clockwise. As the second step, we apply KVL to each mesh. Applying KVL to mesh 1, we obtain or For mesh 2, applying KVL gives or Note that in the 1st equa>on the coeﬃcient of i1 is the sum of the resistances in the ﬁrst mesh, while the coeﬃcient of i2 is the nega>ve of the resistance common to meshes 1 and 2. The same is true for the 2nd equa>on. The third step is to solve for the mesh currents. In matrix form: • If a circuit has n nodes, b branches, and l independent loops or meshes, then l = b−n+1. Hence, l independent simultaneous equa>ons are required to solve the circuit using mesh analysis. • No>ce that the branch currents are diﬀerent from the mesh currents unless the mesh is isolated. To dis>nguish between the two types of currents, we use i for a mesh current and I for a branch current. The current elements I1, I2, and I3 are algebraic sums of the mesh currents. • In the last example it is evident that Example −i1 + 2i2 = 1 We can use Cramer’s Rule: We obtain the determinants More examples 1) Find io 2) 3) 1) KVL at mesh 1: -‐12+2i1+12(i1-‐i2)+4i1=0à18i1-‐12i2=12 KVL at mesh 2: 12(i2-‐i1) +9i2 + 8 +3i2=0 à-‐12i1+24i2=-‐8 " 18 −12 %" i1 % " 12 % '=$ $ '$ ' −8 i −12 24 $ ' # &# 2 & # & 2 3 i2 = 0 i1 = 3) KVL at mesh 1: -‐20+4(i1-‐i3)+2(i1-‐i2)=0 à 6i1-‐2i2-‐4i3=20 KVL at mesh 2: 2(i2-‐i1)+8(i2-‐i3)-‐10io=0à -‐2i1+10i2-‐18i3=0 KVL at mesh 3: 6i3+8(i3-‐i2)+4(i3-‐i1)=0à -‐4i1-‐8i2+18i3=0 " 6 −2 −4 %" i1 % " 20 % ' $ $ '$ ' $ −2 10 −18 '$ i2 ' = $ 0 ' $ −4 −8 18 '$ i ' $ 0 ' # &$# 3 '& # & io = i3 = −5 (io = i3 ) Mesh Analysis with Current Sources • CASE I. When a current source exists only in one mesh. Consider top circuit for example. We set i2 = −5 A and write a mesh equa>on for the other mesh in the usual way, that is, • CASE II. When a current source exists between two meshes. Consider circuit (a), for example. We create a supermesh by excluding the current source and any elements connected in series with it, as shown in (b). Thus, (a) (b) • We create a supermesh as the periphery of the two meshes and treat it diﬀerently. (If a circuit has two or more supermeshes that intersect, they should be combined to form a larger supermesh.) • Why do we need supermeshes that “go over the head” of a current source? Because mesh analysis applies KVL—which requires that we know the voltage across each branch—and we do not know the voltage across a current source in advance. • However, a supermesh must sa>sfy KVL like any other mesh or loop. Therefore, applying KVL to the supermesh in (b) gives or (b) • We apply KCL to a node in the branch where the two meshes intersect. Applying KCL to node 0 gives • Solving the two equa>ons in i1 and i2 we get • Note the following proper>es of a supermesh: 1. The current source in the supermesh is not completely ignored; it provides the constraint equa>on necessary to solve for the mesh currents. 2. A supermesh has no current of its own. 3. A supermesh requires the applica>on of both KVL and KCL. Example Find i1 to i4 using mesh analysis. Solu%on: Note that meshes 1 and 2 form a supermesh since they have an independent current source in common. Also, meshes 2 and 3 form another supermesh because they have a dependent current source in common. The two supermeshes intersect and form a larger supermesh as shown. Applying KVL to the larger supermesh, or Solve to get: Prac>ce Problem • Use mesh analysis to determine i1, i2, and i3 “L-‐shaped” loop: -‐6+2(i1-‐i3)+4(i2 – i3)+8i2=0 à2i1+12i2-‐6i3=6 “Corner” loop: 2(i3-‐i1)+2i3+4(i3-‐i2)=0à=-‐2i1-‐4i2+8i3=0 At supernode: 3+i2=i1ài1-‐i2=-‐3 " 2 12 −6 %" i1 % " 6 % ' $ $ '$ ' $ −2 −4 8 '$ i2 ' = $ 0 ' $ 1 −1 0 '$ i ' $ 3 ' # &$# 3 '& # & i1 = 3.474 i2 = 0.4737 i3 = 1.1052 Nodal and Mesh Analysis by Inspec>on • When all sources in a circuit are independent current sources, we do not need to apply KCL to each node to obtain the node-‐voltage equa>ons as we did in last sec>on. We can obtain the equa>ons by mere inspec>on of the circuit. As an example, let us reexamine the circuit shown. The circuit has two nonreference nodes and the node equa>ons were derived before as Observe that each of the diagonal terms is the sum of the conductances connected directly to node 1 or 2, while the oﬀ-‐diagonal terms are the nega>ves of the conductances connected between the nodes. • Also, each term on the right-‐hand side is the algebraic sum of the current sources entering the node. • In general, if a circuit with independent current sources has N nonreference nodes, the node-‐voltage equa>ons can be wriXen in terms of the conductances as or simply where Gkk = sum of the conductances connected to node k Gkj = Gjk = Nega>ve of the sum of the conductances directly connec>ng nodes k and j, k ≠ j vk = Unknown voltage at node k; ik = Sum of all independent current sources directly connected to node k, with currents entering the node treated as posi>ve. G is called the conductance matrix, v is the output vector; and i is the input vector. This equa>on can be solved to obtain the unknown node voltages. Keep in mind that this is valid for circuits with only independent current sources and linear resistors. • Similarly, we can obtain mesh-‐current equa>ons by inspec>on when a linear resis>ve circuit has only independent voltage sources. • Consider again the circuit shown. The circuit has two nonreference nodes and the node equa>ons were derived before as • " R +R $ 1 3 $# −R3 • %" i % " V '$ 1 ' = $ 1 R2 + R3 '&$# i2 '& $# −V2 −R3 % ' '& We no>ce that each of the diagonal terms is the sum of the resistances in the related mesh, while each of the oﬀ-‐diagonal terms is the nega>ve of the resistance common to meshes 1 and 2. Each term on the right-‐hand side is the algebraic sum taken clockwise of all independent voltage sources in the related mesh. • In general, if the circuit has N meshes, the mesh-‐current equa>ons can be expressed in terms of the resistances as or simply where Rkk = Sum of the resistances in mesh k Rkj = Rjk = Nega>ve of the sum of the resistances in common with meshes k and j, k ≠ j ik = Unknown mesh current for mesh k in the clockwise direc>on vk = Sum taken clockwise of all independent voltage sources in mesh k, with voltage rise treated as posi>ve R is called the resistance matrix, i is the output vector; and v is the input vector. We can solve this matrix equa>on to obtain the unknown mesh currents. Examples Write the node-‐voltage matrix equa>ons by inspec>on. 1) 2) ! # # # # " ! 1.3 $!# # &# 0.2 # &# # &# 1.25 # 0.75 &%# " " v1 $ ! & v2 & # & = ## v3 & # v4 &% " $ & & & & % ! $# &# &# &# &# %" v1 $ ! & v2 & # & = ## v3 & # v4 &% " $ & & & & % " v % " 1.3 −0.2 % " 1 −1 0 $ ' $ '$ v ' $ −0.2 0.2 0 0 $ '$ 2 ' = $ $ −1 0 1.25 −0.25 '$ v3 ' $ $ 0 0 −0.25 0.75 '&$ v4 ' $# # # & " v % " " 1.3 −0.2 % 1 −1 0 $ '$$ v '' $ 0 0 ' 2 $ −0.2 0.2 =$ $ ' $ −1 0 1.25 −0.25 '$ v3 ' $ $ 0 0 −0.25 0.75 '&$ v4 ' $# # # & 0 3 −1 3 % ' ' ' ' & % ' ' ' ' & Example write the mesh-‐current equa>ons by inspec>on 3) 4) ! # # # # # #" $!# &# &# &# &# &# &%# " i1 $ ! & i2 & # & ## i3 & = # i4 & # & i5 &% #" " 170 −40 0 −80 0 %"$ $ '$ −40 80 −30 −10 0 $ '$ $ 0 −30 50 0 −20 '$ $ −80 −10 0 90 0 '$ $ ' 0 −20 0 80 '&$ $# 0 $# ! 170 $!# # &# 80 # &# # &# 50 # &# 90 # & 80 &%# #" #" $ & & & & & &% i1 % " ' i2 ' $ ' $ i3 ' = $ $ i4 ' $ ' i5 '& $# % ' ' ' ' ' '& i1 $ ! & # i2 & & # i3 & = # # i4 & # & i5 &% #" " 170 −40 0 −80 0 %"$ i1 %' " $ ' i $ $ −40 80 −30 −10 0 '$$ 2 '' $ $ 0 −30 50 0 −20 '$ i3 ' = $ $ −80 −10 0 90 0 '$ i4 ' $ $ '$ $ ' 0 0 −20 0 80 $# '&$ i ' $# # 5 & $ & & & & & &% 24 0 −12 10 −10 % ' ' ' ' ' '&