Chapter 2. Circuit Analysis Techniques

Transcription

1 Chapter 2 Circuit Analysis Techniques 1

2 Objectives To formulate the node-voltage equations. To solve electric circuits using the node voltage method. To introduce the mesh current method. To formulate the mesh-current equations. To solve electric circuits using the mesh-current 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 end-to-end, 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 non-reference node. simply set the voltage at the non-reference 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 super-nodes 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 Super-node 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 Node-Voltage 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