The Node-Voltage Method Topic 10 The Node-Voltage Method (4.1, 4.2 & 4.3)

Some Terminology (table 4.1 from text) Name Definition node A point where two or more circuit elements join essential node A node where three or more circuit elements join path A trace of adjoining basic elements with no element included more than once branch A path that connects two nodes essential branch A path which connects two essential nodes without passing through an essential node loop A path whose last node is the same as the starting node mesh A loop that does not enclose any other loops planar circuit A circuit than can be drawn on a plane with no crossing branches 12/26/2018 Node Method

(a) identify all nodes R1 a b v1 R5 R2 R3 d c e I R7 R6 v2 R4 f g 12/26/2018 Node Method

(b) identify all essential nodes 3 or more circuit elements join R1 a b v1 R5 R2 R3 d c e R7 I R6 v2 R4 f g b, c, e & g are essential 12/26/2018 Node Method

(c) Identify all branches A path that connects 2 nodes R1 a b v1 R5 R2 R3 d c e R7 I R6 v2 R4 f g Branches R1 v1 R5 v2 R2 R3 R4 R6 R7 I 12/26/2018 Node Method

(d) Identify all essential branches A path that connects 2 essential nodes R1 Remove the non-essential nodes a b v1 R5 R2 R3 d c e R7 I R6 v2 R4 f g Essential Branches V1-R1 R5 R2-R3 V2-R4 R6 R7 I 12/26/2018 Node Method

(e) identify all meshes A loop that does not enclose any other loops R1 a b M1 M3 M4 v1 R5 R2 R3 d c e R7 I M2 R6 v2 R4 f g R2-R3-R6-R4-V2 M2 R5-R7-R6 M3 R7-I M4 R1-R5-R3-R2-V1 M1 12/26/2018 Node Method

Yes, it can be drawn on a plane with no crossovers Is this circuit planar? R1 R4 R7 vs R3 R6 R8 R8 R1 R2 R3 R4 R5 R6 R7 vs R2 R5 Yes, it can be drawn on a plane with no crossovers 12/26/2018 Node Method

A Systemeatic Approach To Solving Circuits If we have be essential branches in which the current is unknown R2 R3 R4 R5 R6 R7 I v1 v2 b c d e f g be=6 in this example a R1 ne=4 We need be equations to find the currents And we have ne essential nodes So we need to write KVL around 3 loops We can get ne-1 independent KCL equations Leaving be-ne+1 loops to write KVL 12/26/2018 Node Method

v1, v2, I and all R’s are known b c d e f g R1 i3 Let’s define the branch currents KVL around 3 meshes Now KCL at 3 of the 4 essential nodes at b top left at c btm left at e centre 12/26/2018 Node Method

Organizing equations to see structure 12/26/2018 Node Method

The Node-Voltage Method We want to “solve” this circuit 1Ω 2Ω 10v 5Ω 10Ω 2A That is, find all unknown voltages and currents Choose a convenient ground Often the node with the most connections Number the other essential nodes & label their node voltages At each node, we have written KCL in the following organized manner: the sum of the currents leaving the node is zero Now write KCL at the essential nodes (except the ground reference node) At n1 At n2 12/26/2018 Node Method

At this point we have two equations in two unknowns 1Ω 2Ω 10v 5Ω 10Ω 2A At this point we have two equations in two unknowns Multiply each by 10 to get rid of the fractions Gather like terms 12/26/2018 Node Method

Since we can now compute any current 9.09 10.91 Knowing v1 and v2 means that we have effectively solved the entire circuit i2 v1 v2 1Ω 2Ω 10v 5Ω i5 10Ω 2A Since we can now compute any current For example: 12/26/2018 Node Method

Problem 3.27 ig v1 (using node-voltage approach) 2Ω 15Ω 12Ω 6Ω 13Ω 20Ω 125v (using node-voltage approach) We are asked to find ig and io io How do we deal with the third current? We didn’t define a node voltage Choose the ground node Label the voltages on the other essential nodes Write KCL at the two essential nodes Once again, two equations in two unknowns Node 1 The rest is just algebra! Node 2 12/26/2018 Node Method

(using node-voltage approach) Problem 3.27 ig v1 v2 2Ω 6Ω 5Ω 15Ω 12Ω 13Ω 20Ω 125v (using node-voltage approach) 1 io 2 1 1 2 12/26/2018 Node Method

Assesment 4.3 6Ω 8Ω 4Ω 2Ω 50v 5A 3iφ iφ (dependent sources) v1 v2 50 Use the node-voltage method to find the power associated with each source & whether the power is delivered or extracted We have 3 essential nodes—choose one as the reference -iφ n1: And label the other two voltages n2: Now write KCL at the two nodes Write it in terms of the node voltages! How do we deal with iφ? 12/26/2018 Node Method

The Algebra n1: 1 3 n2: 2 Put 3 into 1 X 24 4 5 Put 3 into 2 X 4 6 7 8 9 8 7 into 12/26/2018 Node Method

From the point of view of each source, current is being driven uphill 6Ω 8Ω 4Ω 2Ω 50v 5A 3iφ iφ Assesment 4.3 (dependent sources) 32 16 v1 v2 50 Use the node-voltage method to find the power associated with each source & whether the power is delivered or extracted From the point of view of each source, current is being driven uphill So all three sources are delivering power 12/26/2018 Node Method