Chapter 4 Circuit Analysis Techniques A) Node Voltage B) Mesh Current

Node Voltage and Mesh Current methods derive a minimal set of equations solve all circuit variables Node Voltage uses KCL Mesh Current uses KVL NV more powerful than MC –MC only solves "planar circuits"

Non Planar Circuit Node Voltage CAN solve, MC cannot

Node Voltage Method Basic idea—What is the current i ?

Node Voltage Method What if unknown voltages

Node Voltage Method v1,v2 label unknown node voltages

NV solves unknown node voltages Node: point where two or more components connect Essential node: point where 3 or more components connect all currents can then be derived

CleoCleo: determine the number of nodes in each circuit. Draw closed contours around each node

Now encircle the "Essential Nodes"

It can be proven... The number of NV equations needed to fully solve a circuit is Ne – 1 –Ne is number of essential node voltages (including the reference node)

Encircle the "Essential Nodes" in this circuit

Node Voltage Method 1.establish a ground node 2.label unknown essential node voltages (v1 is voltage relative to ground) 3.write KCL for each unknown essential node 4.solve system of eqns

Find v1, v2

Remember Matlab/Freemat To solve systems of equations such as 9v1 – v2 = 36 -3v1 + 11v2 = 180 use the following A = [ 9 -1 ; -3 11]; b = [36; 180] A\b ans = 6 18

Cleo: Problem 2Cleo: Problem 2 Find all the node voltages in the circuit. V50k = 2.86V (Watch video 4:42)

Do handout problems 1,2

NV Special Cases Dependent sources Voltage source applied between 1 essential node and ground Voltage source applied between 2 essential nodes (not ground)

Voltage Source between Essential Node and ground How to figure current down that branch? –you don't have to! –don't write node equation at v1 –v1 = 15V by inspection –one less unknown node to solve

Problem 3Problem 3 Find the indicated currents; use NV I A = -27 μA, I B = -133 μA, I C = 17.1 μA Only 1 unknown node

Can't write NV at v1 (current thru source?) nor V2 (same reason) SOLN: –SUPERNODE! –KCL applies to regions as well as nodes 1) Circle V1+V2 2) write KCL for Supernode 3) KVL v1, v2 and 10V Floating Voltage Source (Supernode)

Use Supernode NV to Find v1,v2

Dependent Sources all types possible adds another unknown (ix) but ix can be expressed in terms of node voltages eventually removed from node equations not difficult to handle

Cleo: Problem 1 Use nodal analysis to determine Ix (.027A) Video 6:47Problem 1

Do Handout problems 3, 4

Handout Problems

1) Cleo:Problem 1 Write 1 NV eq at V2 and solveProblem 1

2) Use NV to find v1, v2 and i1 (60,10,10) b) how much power delivered by 15A src? (-900W, del) c) how about 5A source? (50W absorb)

3) Write a Supernode NV to find V1, V2(6,18)

4) Find v1, v2, ix (16,10,6/5)