Nodal Analysis Discussion D 2 3 September 2006

Nodal Analysis Discussion D 2. 3 September 2006 Chapter 2 Section 2 -7 1

Nodal Analysis • Interested in finding the NODE VOLTAGES, which are taken as the variables to be determined • For simplicity we start with circuits containing only current sources 2

Nodal Analysis Steps 1. Select one of the n nodes as a reference node (that we define to be zero voltage, or ground). Assign voltages v 1, v 2, … vn-1 to the remaining n-1 nodes. These voltages are referenced with respect to the reference node. 2. Apply KCL to each of the n-1 non-reference nodes. Use Ohm’s law to express the branch currents in terms of the node voltages. 3. Solve the resulting simultaneous equations to obtain the node voltages v 1, v 2, … vn-1. 3

Example Select a reference node as ground. Assign voltages v 1, v 2, and v 3 to the remaining 3 nodes. 4

Example Apply KCL to each of the 3 non-reference nodes (sum of currents leaving node is zero). Node 1: Node 2: Node 3: 5

Example Now express i 1, i 2, …i 5 in terms of v 1, v 2, v 3 (the node voltages). Note that current flows from a higher to a lower potential. 6

Node 1: Node 2: Node 3: 7

In MATLAB, if then is a row matrix of the five conductances 8

Node 1: Node 2: Node 3: 9

These three equations can be written in matrix form as 10

is an (n – 1) x (n – 1) symmetric conductance matrix is a 1 x (n-1) vector of node voltages is a vector of currents representing “known” currents 11

Writing the Nodal Equations by Inspection • The matrix G is symmetric, Gkj = Gjk and all of the off-diagonal terms are negative or zero. The Gkk terms are the sum of all conductances connected to node k. The Gkj terms are the negative sum of the conductances connected to BOTH node k and node j. The ki (the ith component of the vector k) = the algebraic sum of the independent currents connected to node i, with currents entering the 12 node taken as positive.

MATLAB Solution of Nodal Equations 13

Test with numbers v 1 v 2 v 3 14

MATLAB Run v 1 v 2 v 3 15

PSpice Simulation MATLAB: 16

Let's write a general MATLAB program to solve this problem Inputs: Find all voltages and currents 17

function nodal 1(r, k) % Power. Point nodal example % Discussion D 2. 3 % r is a 1 x 5 vector of resistances % k is a 3 x 1 vector of known currents entering the three nodes % nodal 1(r, k) % g = 1. / r G = [g(1)+g(2) -g(2) 0; -g(2)+g(3)+g(4) -g(4); 0 -g(4)+g(5)] k v = inv(G)*k i(1) = v(1)*g(1); i(2) = (v(1) - v(2))*g(2); i(3) = v(2)*g(3); i(4) = (v(2) - v(3))*g(4); i(5) = v(3)*g(5); i kab = [i(1)+i(2) i(5)-i(4)] 18

Do same problem as before nodal 1(r, k) 19

MATLAB Run 20

Nodal Analysis for Circuits Containing Voltage Sources That Can’t be Transformed to Current Sources • Case 1. If a voltage source is connected between the reference node and a nonreference node, set the voltage at the nonreference node equal to the voltage of the source. • Case 2. If a voltage source is connected between two nonreference nodes, assume temporarily that the current through the voltage source is known and write the equations by inspection. 21

Example Assume temporarily that i 2 is known and write the equations by inspection. 22

There appears to be 4 unknowns (v 1, v 2, v 3, and i 2) and only 3 equations. However, from the circuit or so we can replace v 1 (we could also replace v 2) and write 23

Writing the above equation with the unknowns (v 2, v 3, i 2) on the LHS yields 24

Test with numbers v 1 v 2 v 3 Noting that 25

Test with numbers v 1 v 2 v 3 Unknowns: 26

MATLAB Run v 1 V V A v 2 v 3 i 2 v 3 27

PSpice Simulation MATLAB: v 2 v 3 i 2 28

Let's write a general MATLAB program to solve this problem Inputs: Find all voltages and currents 29

function nodal 2(g, V 0, is) % Power. Point nodal-2 example % Discussion D 2. 3 % g is a 1 x 4 vector of conductances % V 0 = the known dc voltage source % is = the known dc current source % nodal 2(g, V 0, Is) % G = [g(1) 0 1; g(2)+g(3) -1; -g(3)+g(4) 0] k = [-2+g(1)*V 0; 0; is] vvi = inv(G)*k v = zeros(1, 3); v(2) = vvi(1); v(3) = vvi(2); v(1) = v(2)-V 0; v i(1) = v(1)*g(1); i(2) = vvi(3); i(3) = v(2)*g(2); i(4) = (v(2) - v(3))*g(3); i(5) = v(3)*g(4); i kab = [i(1)+i(2) i(5)-i(4)] 30

Do same problem as before nodal 2(g, V 0, is) 31

MATLAB Run 32