Chapter 2. Circuit Analysis Techniques


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1 Chapter 2 Circuit Analysis Techniques 1
2 Objectives To formulate the nodevoltage equations. To solve electric circuits using the node voltage method. To introduce the mesh current method. To formulate the meshcurrent equations. To solve electric circuits using the meshcurrent method. 2
3 Two Powerful Techniques for Circuit Analysis Nodal Analysis, based on a systematic application of Kirchhoff s current law (KCL Mesh Analysis, based on a systematic application of Kirchhoff s voltage law (KVL) we can analyze almost any circuit by obtaining a set of simultaneous equations that are then solved to obtain the required values of current or voltage. 3
4 Nodal Analysis So far, we have been applying KVL and KCL as needed to find voltages and currents in a circuit. Good for developing intuition, finding things quickly but what if the circuit is complicated? What if you get stuck? Systematic way to find all voltages in a circuit by repeatedly applying KCL: Node Voltage Method (Nodal Analysis). 4
5 Branches and Nodes ( reminder from last part) Branch: elements connected endtoend, nothing coming off in between (in series) Node: place where elements are joined includes entire wire 5
6 Steps to Determine Node Voltages Method : Step 1 Select a node as the reference node. This is the reference Node The reference node is commonly called the ground since it is assumed to have zero potential. 6
7 Steps to Determine Node Voltages Method ( Cont.) Step 2 Assign voltages v 1, v 2,..., vn 1 to the remaining n 1 nodes. The voltages are referenced with respect to the reference node. Node 0 is the reference node (v = 0), while nodes 1 and 2 are assigned voltages v 1 and v 2. 7
8 Steps to Determine Node Voltages Method ( Cont.) Step 3 Apply KCL to each of the n 1 non reference nodes add i 1, i 2, and i 3 as the currents through resistors R 1,R 2, and R 3, respectively. At by applying KCL gives node 1 node 2 8
9 Steps to Determine Node Voltages Method ( Cont.) Step 4 Use Ohm s law to express the branch currents in terms of node voltages. The key idea to bear in mind is that, since resistance is a passive element, by the passive sign convention, current must always flow from a higher potential to a lower potential. 9
10 Steps to Determine Node Voltages Method ( Cont.) Step 5 Solve the resulting simultaneous equations to obtain the unknown node voltages. 10
11 Node Voltage Equations (Resistors) 11
12 Example: The voltage drop from node X to a reference node (ground) is called the node voltage V x. The current through resistors can be expressed as Iab Va = Vb R 12
13 Example Calculate the node voltages Solution 13
14 At node 1 Multiply by 4 14
15 At node 2 Multiply by 12 15
16 Example Find the voltage at node 1, 2 & 3 Solve this example and handle it to me 16
17 Conclusion for Nodal Analysis Nodal analysis is simply writing KCL equations in a systematic way assuming all currents leaving. Nodes voltages are the circuit variables. Currents are expressed in terms of nodes voltages. The number of variables = Number of nodes
18 Nodal Analysis with Voltage Sources note that: A current source produces constant current in a give direction. I s is leaving Va I s Va Vb  I s is leaving Vb A Voltage source maintains the voltage constant between its terminals. Va = Vs No need to consider Va a circuit variable. Vs Va
19 If a voltage source is connected between the reference node and a nonreference node. simply set the voltage at the nonreference node equal to the voltage of the voltage source v1 = 10 V 19
20 If the voltage source (dependent or independent) is connected between two nonreference nodes. we apply both KCL and KVL to determine the node voltages. V 1 and v 2 are called a supernodes as they enclose a (dependent or independent) voltage source connected between them and any elements connected in parallel. 20
21 Steps to Determine Nodal Analysis with Voltage Sources. Step 1 Choose a reference node (ground, node 0) Step 2 Define unknown node voltages (those not connected to ground by voltage sources). Va, Vb, Step 3 Write KCL equation at each unknown node. How? Each current involved in the KCL equation will either come from a current source (giving you the current value) or through a device like a resistor. If the current comes through a device, relate the current to the node voltages using I V relationship (like Ohm s law). 21
22 Step 4 Apply KCL to the supper node Supernode 22
23 Step 5 Apply KVL to the supper node Step 6 Solve the set of equations (N linear KCL equations for N unknown node voltages). 23
24 Example node voltage set R 1 V a R 3 V b +  I S V 1 R 2 R 4 Choose a reference node. reference node Define the node voltages (except reference node and the one set by the voltage source). Apply KCL at the nodes Va and Vb with unknown voltage. V V R V R Va V R a 1 + a + b = Vb V R 3 a + V R b 4 = I S Solve for Va and Vb in terms of circuit parameters. 24
25 Example Find the node voltage Solution Apply KCL at the super node Multiply by 4 25
26 .1 Apply KVL to the loop From 1 and
27 Example Calculate the power absorbed by the 6 ohm resistor using nodal analysis Apply nodal KCL at V 1 v 2 v1 v = 0 v1 v2 v1 + 1 =
28 At Node V 2 v1 v2 v2 4 = 6 7 v2 v1 v = 6 7 Solve for 1 & P 6 = i R6 2 v1 v2 144 P = 6 = = 24W
29 Example Find I o using the NodeVoltage At node V 1 4 = i + i 1 2 v = 3 v v 2 v1 v1 v2 + 4 =
30 At node V 2 I o = i 3 + i 4 v 6 1 v2 v v v 6 v2 = 4 v2 + 4 Solve for 1 and 2 V 1 = V V 2 = V = v
31 Example Use nodal analysis, find v o Solution At Node V 1 5 i 1 = i2 + v v1 v0 = v v =
32 40 v v = Multiply by 2 ( ) = ( v v ) ( v v ) = 32
33 At Node V 0 i = i 3 v 2 1 v0 v0 + v0 5 = 4 + ( 20) 8 Multiply by 8 4v 7v0 1 = 20 Solving for v 0 and v 1 V0 = 30v 33
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