# Chapter 4: Techniques of Circuit Analysis

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1 Chapter 4: Techniques of Circuit Analysis This chapter gies us many useful tools for soling and simplifying circuits. We saw a few simple tools in the last chapter (reduction of circuits ia series and parallel combinations of resistances, for example) but in this chapter we take circuit simplification much farther 4.1 Terminology Planar Circuits can be drawn on a plane with no crossing branches. NonPlanar Circuits cannot be drawn on a plane without branches crossing one another somewhere. Figure 4.1 (left) shows a circuit that has crossing branches but that can be redrawn without any branches crossing. This is a planar circuit. Figure 4.2 (right) shows a circuit that is not planar: there is no way to draw it without branches crossing. In Circuit Analysis we will consider planar circuits only. Many of the techniques we learn here cannot be applied to nonplanar circuits.

2 Definitions A node is a point in a circuit where two or more circuit elements meet. The number of nodes in a circuit is n. A path is formed when adjoining (connected) circuit elements are traced, in order, without passing through any node more than once. A closed path is a path whose starting and ending points are the same. A branch is a path connecting two nodes. A mesh is a closed path that does not contain any other closed paths The number of meshes in a circuit is m. Looking at the circuit in Figure 4.1b (shown again below), we can identify: 4 meshes; 8 closed paths; 6 nodes. Two of the four meshes are shown in green; two nodes are shown as red dots; one possible closed path which is not a mesh is shown in blue. The blue closed path contains two meshes. (The arrows on the paths are just for show: we can trace a path in either direction.) Essential Node: A useful kind of node is the essential node, which is a node at which at least three circuit elements meet. In the circuit aboe there are four of these; one of them is labeled as a green dot. One of the red dots is also an essential node; the other is not. An essential branch is a branch that connects two essential nodes. There are quite a few in the circuit aboe.

3 Where are we going with all this?? This terminology helps us figure out how many equations we will need to sole the circuit: We need as many equations as there are essential branches with unknown currents; call this number b e. Then we can write (n e 1) KCL equations so we need [b e (n e 1)] KVL equations. Here, n e is the number of essential nodes. We will shortly look at two ery important and ery powerful circuit analysis techniques: Node Voltage Method (NVM): allows us to sole a circuit with (n e 1) KCL equations. Mesh Current Method (MCM): allows us to sole a circuit with [b e (n e 1)] KVL equations. Using either method, we sae quite a few equations, since without them we would need (n e 1) KCL equations and [b e (n e 1)] KVL equations.

4 4.2 Introduction to the Node Voltage Method To use the Node Voltage Method (NVM) we define new circuit ariables called the node oltages. We find the node oltages by soling simultaneous equations called the node oltage equations. These are generally much fewer in number than we need with basic KVL, KCL, Ohm s Law techniques. If we hae the node oltages, we can find any other current or oltage in the circuit with a single equation (i.e., no more simultaneous equations will be necessary). Node Voltage Method Algorithm Basic steps in using NVM (we will clarify and expand on this shortly): Find and label all essential nodes. o There are n e essential nodes, and we will need to sole n e 1 node oltage equations. Choose one of the essential nodes as a reference and label it with the symbol. o This is where the 1 arises in ne 1 equations : we do not need a node oltage equation at the reference. o Typically we will choose the reference node to be the one with the largest number of circuit elements connected to it. But we do not hae to do this, and we will see there may be reasons for choosing a different node as the reference. Define the nonreference node oltages. o The nonreference nodes are the ones you did not choose as the reference node. The nonreference node oltages are the oltage drops from each nonreference node to the reference node (an example or two will clarify this). These are labeled with the positie sign at the node, and the negatie sign at the reference node. Apply Kirchhoff s Current Law (KCL) to each nonreference node, writing currents at each node in terms of the node oltages and any sources present. o Although there is more than one way to do this, in this class we will always set the sum of the currents leaing the node equal to zero. Sole the node oltage equations. The resulting node oltages are considered the solution to the circuit. If you hae those, you can find any current or oltage in the circuit with one equation (no more simultaneous equations needed).

5 To see how to express the current leaing a branch in terms of the node oltages, we look at the following example. i 12 The essential nodes are labeled 1, 2, and (reference node). The node oltages are defined as described aboe: positie at the nonreference node and negatie at the reference. The current i 12 is leaing node 1 and heading from node 2. It can be written in terms of node oltages as i [ ]. Where did we get that equation for i 12? From KVL: i (2[ ]) Soling this equation for i 12 gies the preious equation. We continue in this way to set up KCL at each nonreference node. So, at node 1, the sum of the currents leaing node 1 is The sum of the currents leaing node 2 is If the third term of the first equation is confusing, do a KVL around the loop containing the 10 [V] source, the 1 [] resistor, and the 5 [] resistor. Sole this KVL for the current leaing node 1, and you will get the third term in that equation. Eentually, you will get to the point where these terms can be written down quickly without haing to examine each KVL indiidually.

6 The third term in the second equation is easy, since we know the current in that branch: it s 2 [A] entering node 2, so the current leaing is 2 [A]. In this case, we don t try to express it in terms of the node oltages. Note also that we hae summed the currents leaing each node. This is an arbitrary choice, but we must choose something, and be consistent. We hae two equations for two unknowns. The solution is 1 = [V] 2 = [V] Now that we hae the node oltages, we can find any other oltage or current in the circuit. To do that we probably need to sole one equation, but we will not hae to sole simultaneous equations anymore. For example, the current in each branch is easy we hae already shown how to calculate i 12 in the equation aboe. Also, the oltage across any indiidual component can be found with one KVL. As one more example, the oltage across the current source in the circuit aboe is nothing but 2. Notation Rules (you knew we were going to hae this ) All node oltages must be labeled just as they are in the diagram aboe. Those node oltages hae a sign at the nonreference node, and a sign near the reference node. It is not sufficient to simply label 1, for example, near node 1. It is certainly not acceptable to leae out the node oltages from your diagram. If you do, significant credit will be subtracted from quizzes and exams.

7 4.3 The Node Voltage Method and Dependent Sources If we hae dependent sources, we will need one equation for each controlling ariable. We call these auxiliary equations. We will need to write these equations in terms of the node oltages. This is because we are trying to find the node oltages (which are the circuit ariables), and we don t want to introduce any extra ariables. Here is a circuit with a dependent source i Q 10 i Q 5V We hae identified the node oltages and the reference node. Our equations are i Q Just as in using Kirchhoff s Laws, we need another equation for the dependent ariable but be sure to express it in terms of the node oltages. i Q The Node Voltage Method: Some Special Cases Special cases arise when we hae a oltage source connected either (1) between an essential node and the reference node, or (2) between two nonreference essential nodes. (1) Voltage source between an essential node and the reference node. In the circuit below the 100 [V] source is connected from essential node 1 to the reference node.

8 s 100V A This is a simple case. Remember that we are trying to find the node oltages, which are the oltage alues between each of the essential nodes and the reference node. But we already know that in this case!! 1 s 100V This is the node oltage equation for node 1. We hae still hae the same number of node oltage equations, but one of them is triial. This works for dependent sources, too, although we of course need the auxiliary equation. The rest of the circuit is soled as follows: This gies 1 = 100 [V]; 2 = 125 [V] (2) Voltage source between two nonreference essential nodes. In the circuit below, a oltage source is connected between two nonreference essential nodes (nodes 2 and 3). We are using an independent source for illustration, but our remarks are alid for dependent sources as well. 10 [V] V i A

9 The node oltage equation for 1 is: But when we try to write a node oltage equation at node 2, we hae a problem: there is no way to express the current i as a /R term using the node oltages, as we hae been doing. Let s write a KCL anyway, using node oltages where we can. We will denote the unknown current as a ariable. Now at node 3 we hae the same situation: i i. What should we do about this unknown current? Let s add the last two equations together: Hmmm this last equation is what we hae got if we had treated node 2 and node 3, with the dependent source in between, as one big node it s as if it were a a Supernode!! V [V] A In the figure aboe, the red dashed line encircles the supernode, and the arrows show currents leaing it. Let s apply our usual node oltage technique to those currents. The currents leaing node 2 are

10 which is the equation we got immediately aboe. This last equation, and the one aboe that was written for node 1, gie us two equations, but we hae three unknowns: 1, 2, and 3. We need another equation. Note that we hae not included any information about the 10 [V] source. We include this by simply doing a KVL around a loop that includes the dependent source: [ V ] This equation comes from the fact that the oltage between node 2 and node 3 is constrained by the oltage source. This is the constraint equation. Now we hae three equations in three unknowns and we can sole for the node oltages. Important Note The supernode equation (the first of our four equations aboe) contains 2 AND 3, so we do not hae any fewer node oltages to sole for. But the kind of equations is different from the case where we do not hae a supernode: Instead of two node oltage equations, we hae a supernode equation and a constraint equation. Counting Equations It is good practice to count essential nodes and determine how many equations, and of what type, you are going to need. We will do this in the examples in class. You will need one node oltage equation for each nonreference essential node. one constraint equation for each supernode one auxiliary equation for each dependent source. Choosing the Reference Node You can choose any essential node as the reference node. Typically this will be the one with the largest number of branches connected to it, since then you will aoid haing to write an equation with a lot of terms. One reason you might not want to choose the node with the largest number of branches is that you might be able to simplify the problem by choosing the reference node so that one or more of them is triial. If you choose the reference so that a oltage source (dependent or independent) is connected to it, with the other end of the oltage source at another essential node, then that node oltage is triial.

11 4.5 Introduction to the Mesh Current Method To use the Mesh Current Method (MCM) we define new circuit ariables called the mesh currents. We find the mesh currents by soling simultaneous equations called the mesh current equations. These are generally much fewer in number than we need with basic KVL, KCL techniques. If we hae the mesh currents, we can find any other current or oltage in the circuit with a single equation (i.e., no more simultaneous equations will be necessary). Consider the circuit below, where KCL is applied to the node at the top. R 1 R 2 s1 i 2 i 1 i 3 R 3 s2 KCL: i i i KVL around the left and right loops gies S1 = i 1 R 1 i 3 R 3 S2 = i 2 R 2 i 3 R 3 If we sole the KCL equation for i 3 and substitute the result into the KVL equations we get: S1 = i 1 (R 1 R 3 ) i 2 R 3 S2 = i 2 (R 1 R 3 ) i 1 R 3 We now hae two equations, in the form of KVLs, in two unknowns. This substitution, and the resulting equations, can be done automatically using the mesh current method To use the mesh current method to get the last two equations directly, we do the following.

12 Mesh Current Method Algorithm 1. Find the meshes and label each with a current. Remember that we learned how to count meshes m, essential nodes n e, and essential branches b e. Then the number of meshes will be m = b e (n e 1). o The two mesh currents in the circuit below are labeled i a, i b. The direction shown (clockwise for both) is arbitrary we could hae chosen either one (or both) to go in the other direction. We will refer to the two meshes as mesh a and mesh b. o Note that the mesh currents are defined as the currents going around the perimeter of the loops. It is important to realize that these are not necessarily the same as branch currents i 1, i 2, i 3. R 1 R 2 s1 i a R3 i b s2 2. Apply KVL around each mesh, writing oltages across resistors in terms of the mesh currents. The result for the circuit aboe is: i R i i R Mesh a 1 a 1 a b 3 0 Mesh b i i R 3 i R a b b 3. Sole for the mesh currents. Then, find the branch currents in terms of the mesh currents. Going back to the original circuit where we defined the branch currents i 1, i 2, i 3, is should be clear that i 1 = i a and i 2 = i b. What about i 3? Looking at the circuit below you should be able to conince yourself that i 3 = i a i b. So we hae found the branch currents in terms of the mesh currents. This is the reason for the statement aboe that the mesh currents are not necessarily equal to the branch currents.

13 R 1 R 2 s1 i 3 i a R 3 i b s2 If we substitute our results for the branch currents in terms of the mesh currents into the mesh current equations, we can show that the mesh current equations are equialent to the two equations we got earlier: Specifically, the equations i R R i R 0 1 a 1 3 b 3 i R _ i R R 2 a 3 b 2 3 are the same as the preious KVL equations if we make the assignments i 1 = i a, i 2 = i b, and i 3 = i a i b.

14 4.6 The Mesh Currents and Dependent Sources For each dependent source, we need an additional equation defining the controlling ariable. This is always the case, but now we want to write the defining equations in terms of the mesh currents. If we use, say, branch currents to define the controlling ariables, we will hae introduced additional unknowns. 4.7 The Mesh Currents Method: Some Special Cases Two special cases arise when current sources are present. (Note that these cases are analogous to the case of oltage sources in the node oltage method.) (1) A branch contains a current (dependent or independent) R 1 R 2 i s1 = 3 [A] i a R 3 i b s1 In the circuit shown aboe, it should be clear thatia 3[A], which means that we hae one less mesh current equation to sole because one of the unknowns (i a ) is now known. So there is only one mesh current equation for this circuit, which is i i R i R s1 b a 3 b 2 0 So a current source in a branch means that the number of mesh current equations is reduced by 1.

15 (2) A current source (dependent or independent) is being shared by two meshes R 1 i b R 2 R 3 S1 i a i S1 i c S2 R 4 R 5 The mesh current equations for the circuit aboe are as follows. Mesh b) i R i i R i i R b 1 b c 3 b a 2 0 KVL around either mesh a or c includes the current source. We cannot assume that the oltage across the current source is 0, but we don t know what it is, so we will simply label it. Then we hae Adding these last two equations gies i i R i R S1 a b 2 a 4 0 i i R 3 2 i R 5 0 c b S c i i R i i R i R i R S1 a b 2 c b 3 S 2 c 5 a 4 0 But this equation is just what we would hae obtained if we had done a KVL around the path shown in red below. Why, it s almost as if we had a a

16 Supermesh!! R 1 i b R 2 R 3 S1 i a i S1 i c S2 R 4 R 5 Aboe we hae drawn a dashed line around the supermesh. We can write the equation aboe in one step using this mesh: i i R i i R i R i R S1 a b 2 c b 3 S 2 c 5 a 4 0 But we hae to get the alue of the current source inoled, which we do by recognizing that there is a constraint imposed on i a and i c : i i 5 a So we hae still hae three equations: one regular mesh current equation, and one super mesh equation, and one constraint equation (the constraint equation is the one we just wrote). c

17 Let s do an example we will find the power absorbed by the 10 [ resistor and the power deliered by the 3 [A] source in the circuit below. 25 [] 30 [] 3 [A] i b 5 [] 7 [V] i a 10 [] i c 10 [V] 6 [] There are 3 meshes, two of which share a current source. So we will hae three mesh current equations: one regular mesh current equation one supermesh equation one constraint equation The regular mesh current equation is The super mesh equation is i i i i c b c a 7 25i 5 i i 10 i i 6i 0 b b c a c a The constraint equation is i a i 3 b Soling these equations together gies i a = [A]; i b = [A]; i c = [A]. Now that we hae found the currents, we can find anything else we might need using only one equation. Since we want power for the resistor and the current source, we hae labeled the current in the resistor and the oltage across the current source in the next figure.

18 25 [] 30 [] 3 [A] i b 5 [] 7 [V] s i x i a 10 [] i c 10 [V] 6 [] The current i x is i x = i a i c = [A] Then p abs,10[ω] = i x 2 (10) = 27.8 [W] How did we know that i x was i a i c and not i c i a? Because at the 10 [] resistor, i x is defined to be going in the direction of i a and opposite the direction of i c. To find s we need a KVL: This gies S = [V]. Then 7 30 i i 10 i i 6i 0 a b S a c a pdel,3[ A] [ W]

19 4.8 The Node Voltage s. the Mesh Current Method We want to think carefully in each case about whether the node oltage or the mesh current method is easier to implement. Some factors to consider: One of the methods may require fewer equations to sole simultaneously. For any circuit we will hae n e 1 node oltage equations and m = b e (n e 1) mesh current equations. Voltage sources may reduce the number of node oltage equations by introducing triial equations. Current sources may reduce the number of mesh current equations by introducing triial equations. This of course depends on where we put the reference. The ariable we are looking for may already be a mesh current or a node oltage, in which case soling by the appropriate method will yield the answer directly. An obseration: Most beginning students prefer mesh currents, but in fact the node oltage method almost always inoles fewer equations. Also, node oltage method is extremely useful in many electronics applications. I would strongly suggest getting familiar with the node oltage method.

20 4.9 Source Transformations We can sometimes simplify a circuit using equialent circuits Two circuits are equialent if one can be replaced with the other without changing circuit ariables (oltages and currents) in the rest of the circuit. We hae already seen how to use equialent circuits to simplify using series and parallel resistor combinations. In that case, we are replacing a group of resistors by a single equialent resistor. Source transformation is another such simplification technique. The circuits 1 and 2 below are equialent with respect to terminals a and b, proided that s, R s, i s, and R p are related to one another in a particular way. If they are, then a resistor R L connected to terminals a and b will hae the same oltage across it (and the same current through it) whether it is connected to circuit 1 or to circuit 2. In fact, anything connected to terminals a and b of either circuit will hae the same oltage across it and current through it. That is what equialent means. 1 S R S a b i L R L L 2 a i L i S R P R L b L Note that it is ery important to include the qualifier with respect to terminals a and b, because two circuits are not necessarily equialent at just any two terminals. Bottom Line: If the parameters are related correctly, a oltage source in series with a resistor can be replaced with a current source in parallel with a resistor.

21 But what relationship must exist among s, R s, i s, and R p in order for these circuits to be equialent? We find this relationship by requiring that equialence must hold for any load resistor R L, and in particular it must hold for R L = 0 and R L =. For R L = 0, we ask what current i L flows through R L. (By the way, R L = 0 means that a short circuit exists at terminals a and b, and the resulting current is the shortcircuit current). If it is connected to circuit 1, then If it is connected to circuit 2, then i L S R S i L i S For these to be equialent, we must hae i S R S. S Now consider R L =. In that case, terminals a and b are open circuit, and the oltage L is the opencircuit oltage. Then when R L is connected to circuit 1 we hae and when it is connected to circuit 2, L S So that means i R. L S P i R S S P Comparing this equation to the one for i S aboe shows that we also need R S = R P. Summary: A oltage source S in series with a resistor R S will be equialent to a current source i S in parallel with a resistor R P if i R and RS RP. S S P This is the source transformation theorem.

22 4.10 Theenin and Norton Equialents We are often interested in the oltage and/or current in a load that is connected to a particular pair of terminals in a circuit. For example, we may want to connect a load resistor, or maybe seeral different load resistors, to the terminals labeled a), b) in the circuit shown below. 25 [] 7 [V] 30 [] 5 [] a) 3 [A] b) 10 [V] 6 [] The following idea is ery powerful, and may help in analyzing a case like this: Theenin Equialent Circuit: The behaior of any linear circuit at a specific pair of terminals in a circuit may be modeled by a oltage source TH in series with a resistor R TH. We will look only at linear circuits in this course; linear circuit is defined later in the section on superposition. What we are saying is that the circuit below on the right can be modeled by the circuit on the left. Linear Circuit a b Model TH R TH a b

23 Important Notes: We are modeling the circuit at two particular terminals with TH and R TH we are not suggesting that the only things inside the box are a resistor and a oltage source. The model holds for any load but only at terminals a), b). If we different terminals in the original circuit, the alues of TH and R TH will change. The circuit must be linear, but it can contain any of the basic circuit elements: oltage sources, current sources (dependent and independent), resistors, capacitors, and inductors. Finding TH and R TH : The box in the figure below contains an arbitrary linear circuit. We hae labeled terminals a) and b). On the right, we hae an open circuit at a), b), resulting in an opencircuit oltage OC. (We can think of this as an infinite load resistance.) On the left, we hae connected a short to the terminals, resulting in a shortcircuit current i sc. a a any linear circuit b OC any linear circuit b i SC By comparing the drawing on the left with the Theenin Equialent drawing aboe, it should be clear that OC. TH By comparing the drawing on the left with the Theenin Equialent, we can see also that i SC R TH. TH So we already hae an algorithm for finding a Theenin Equialent: If we know the opencircuit oltage and the shortcircuit current at the terminals a), b), we can find the Theenin Equialent: TH and OC R TH TH. isc

24 If this were an experiment, we could measure OC and i SC. If it is an analytical problem, we can calculate them using our knowledge of circuit theory. Is this useful?? Wow, yeah! This idea is used a lot. What it means is that we can talk about a lot of complicated circuits without haing to know anything about those circuits except their Theenin Equialents. This is extremely useful. It also means that if we need to analyze how seeral different load resistors behae when connected to a circuit at two particular terminals, we only need the Theenin Equialent, and we can make the calculations much simpler. This idea is shown below. a R TH a complicated circuit b R L TH b R L Generally it is a lot easier to handle the Theenin Equialent circuit on the right than it is to analyze the complicated circuit on the left.

25 4.11 More on Deriing a Theenin Equialent There is another method for finding R TH : The Test Source Suppose we had a circuit that could be modeled using only a resistor. Then if we were to apply a oltage source T and measure or calculate the current i T through it, we could find R TH as: a R TH i T T R TH it T b Circuit modeled by R TH only Test source Here, T is known as the test source. If our circuit is simply a resistance, we can use the test source to find that resistance. This will be the Theenin Equialent resistance R TH. Be careful: the polarity of T and the current i T that we calculate hae to be in the actie sign conention. Otherwise, we will get the wrong sign for R TH. This can be seen just by noting that if in the circuit aboe, R TH is positie and T is positie, i T will be positie. If we were to reerse the direction of the current and then take the ratio, we would get the wrong sign for R TH. But is this useful? If we are finding the Theenin Equialent of a circuit that is just resistances, like the one aboe, we can simply combine these into one resistance (series, parallel, deltatowye), and we hae R TH. We don t need a test source. But what if our circuit is not just a resistance, and contains sources as well? We can use the test source idea as follows. 1. Deactiate all independent sources. To deactiate an independent oltage source, we replace it with a short. Note that a short is a oltage source of alue 0 [V]. deactiate

26 To deactiate an independent current source, we replace it with an open circuit. Note that an open circuit is a current source of alue 0 [A]. deactiate 2. Apply a test source of known alue; it doesn t matter what the alue is. You can een leae the alue out and just call it T. 3. Calculate the current i T. Then Notes R T TH. If you hae not gien your test source a alue, just calculate the ratio T /i T. it You cannot deactiate dependent sources. You need to leae them in; they affect the equialent resistance of the circuit. If you hae nothing but resistances and independent sources, you don t need the test source: you can simply deactiate the independent sources and find R TH by resistor combinations. See the example on the next page. If there are dependent sources but no independent sources, you hae to use a test source, because the open circuit oltage and short circuit current will both be 0, so you can t take the ratio. See the example two pages forward.

27 Example: Resistances and independent sources. Find the Theenin Equialent resistance of the circuit below at terminals a), b). 5 [] 4 [] a) 25 [V] 20 [] 3 [A] b) Since we hae only independent sources, we can simply deactiate them: 5 [] 4 [] a) 20 [] b) Now simple resistor combinations gie us R [ ] TH

28 Example: Resistances and dependent sources. Find the Theenin Equialent resistance of the circuit below at terminals a), b). We will not do this problem here in the notes. Here we just note that at terminals a), b), both the open circuit oltage and the short circuit current are 0 because there are no independent sources. So this circuit is modeled by a resistance only. But we cannot find R TH from OC and i SC we must apply a test source. R1 a) 6i x y 4 y i x R 3 R 2 b)

29 On Finding the Theenin Equialent If the circuit contains independent sources, you can find an open circuit oltage and a short circuit current, or you can use a test source to find R TH. We only need to choose two of these things to find the Theenin Equialent. It is a smart idea to check to see which of these methods is easier to use: the short circuit current may remoe components in parallel to the terminals of interest, for example. The test source method is useful if we want to deactiate independent sources. If the circuit consists only of resistances, these can be combined into one to find R TH. In that case, the open circuit oltage and short circuit current will be both be 0, which means the Theenin oltage is 0. If the circuit contains only resistances and dependent sources (or only dependent sources), the open circuit oltage and short circuit current will again be 0. In that case, there is no choice but to use a test source. On a Negatie Theenin Equialent We assume that there are no negatiealued resistors (of the type you find in your lab kit, for example). Howeer, when modeling a circuit that contains dependent sources, it is possible that the Theenin Equialent resistance is negatie. This does not mean that we can hae negatie alued resistors. It means that the circuit model includes a negatie resistance. That resistance is simply part of the model; it is not an actual circuit component. Only circuits with dependent sources can hae negatie R TH. But just because a circuit has a dependent source does not mean it will hae a negatie R TH.

30 Two interesting cases Consider the circuit below, where we are interested in terminals a), b). R 2 R 1) 3 a) i s R 1 S1 S2 R 5 R 4 b) 2) Any circuit components to the left of the source S2 cannot hae any effect on what happens at terminals a), b), because S2 fixes the oltage across those components. So no matter what alues R 1 and R 2 or S1 hae, the oltage across them is S2, and something connected to a), b) will see S2 but not those components. What that means is that as long as we are interested only in what happens at a), b), which is to the right of terminals 1) and 2), we can redraw the circuit as follows. 1) R 3 a) i s S2 R 5 R 4 b) 2) Now think about the current source. Nothing outside of R 4 and the current source can see R 4, because the current through it is fixed by i S. In other words, if R 4 doubled in alue, nothing different would happen at terminals a), b) or terminals 1), 2) for that matter. So as long as we are interested in something outside the branch with i S and R 4, we can remoe R 4 as well.

31 Bottom line Circuit components in parallel with a oltage source can be replaced by just the oltage source, proided we are interested only in what is happening outside of those components. Circuit components in series with a current source can be replaced by just the current source, proided we are interested only in what is happening outside of those components.

32 4.12 Maximum Power Transfer Sometimes we are interested in maximizing the power transferred to a load: we want to get as much power to our stereo speakers as possible, for example. The power transferred to a load from a circuit depends on the circuit as well as on the load. The circuit below shows a Theenin Equialent of anything maybe a stereo system. The resistance connected a), b) represents the load a speaker, for example. We want to analyze how much power is deliered to the load. a R TH TH i L RL Theenin Equialent of any circuit b The power deliered to R L is i R 2 TH del, RL L L L RL RTH 2 R Analysis: If R L is ery small (approaching 0), no power is deliered to R L. If R L is ery large (approaching infinity), again no power is deliered. So there is a maximum at some finite alue of R L, which we can find by differentiating with respect to R L and equating to 0: dp del, RL 2 RTH RL RL RTH R L 0 TH drl RTH RL 2 R R R R R 2 0 TH L L TH L R R So the power deliered to the load is a maximum (i.e., we are getting as much power to the load as possible) if the load resistance is equal to the Theenin resistance. The amount of power deliered in that case is L TH

33 p del, R max l 4R 2 TH L We may not always be able to choose the load resistance, but if we can, it should be as close as possible to the Theenin resistance. Audio speakers are made to hae resistances equal to the output resistance (Theenin resistance at the output) of typical stereo amplifiers.

34 4.13 Superposition Idea: For some kinds of systems, the response of the system to seeral sources is equal to the sum of the responses to each source indiidually. Definitions: System: For our purposes, a system is any circuit we may be interested in. In mechanical engineering, it could be a collection of parts connected by springs, for example. Source: For our purposes, a source is any independent oltage or current source. Response: For our purposes, the response of the system will be any oltage or current generated by the sources. Superposition: The idea that the system response to seeral sources is the same as the sum of the responses to the indiidual sources is the Superposition Principle. Linear systems: A system for which superposition holds is a linear system. How can we use this idea to sole circuits? Here is the algorithm: Application of the Superposition Principle: 1. Deactiate all but one of the independent sources. We do not consider dependent sources here; those are always left in the circuit. (Deactie means to replace oltage sources with a short (0 oltage) and current sources with an open circuit (0 current), as we did for finding Theenin Equialent Resistances.) 2. Find the response of the system (a oltage or current we are looking for) to the remaining source. 3. Repeat steps 1 and 2 for each source. 4. Add the responses to each source to get the total response to all sources. Superposition may make some circuits easier to sole, but it will usually be more trouble than it s worth for the kind of circuits we hae been dealing with so far. But when we go on to ac sources and the phasor domain later in the course, we will hae to use it if the sources hae different ac frequencies. We return to this idea later

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