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Nodal Analysis

Discussion D2.3September 2006

Chapter 2Section 2-7

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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

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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 v1, v2, … 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 v1, v2, … vn-1.

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Example

Select a reference node as ground. Assign voltages v1, v2, and v3 to the remaining 3 nodes.

1v 2v 3v

2A 1r

2r

3

r 4

r 5rsi

5

Example

Apply KCL to each of the 3 non-reference nodes (sum of currents leaving node is zero).

2A 1r

2r

3

r 4

r 5rsi

1v 2v 3v

1i

2i

3i

4i

5i

1 22 0i i

2 3 4 0i i i

4 5 0si i i

Node 1:

Node 2:

Node 3:

6

Example

Now express i1, i2, …i5 in terms of v1, v2, v3 (the node voltages). Note that current flows from a higher to a lower potential.

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1

0vi

r

1 2

22

v vi

r

2

33

0vi

r

2 3

44

v vi

r

3

55

0vi

r

2A 1r

2r

3

r 4

r 5rsi

1v 2v 3v

1i

2i

3i

4i

5i

7

1 22 0i i

2 3 4 0i i i

4 5 0si i i

Node 1:

Node 2:

Node 3:

1 1 2

1 2 2

2v v v

r r r

31 2 2 2

2 2 3 4 4

vv v v v

r r r r r

3 32

4 4 5s

v vvi

r r r

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1

0vi

r

1 2

22

v vi

r

2

33

0vi

r

2 3

44

v vi

r

3

55

0vi

r

8

2A 1r

2r

3

r 4

r 5rsi

1v 2v 3v

1i

2i

3i

4i

5i

In MATLAB, if 1 2 3 4 5[ ]r r r r r r

then

1 2 3 4 51 2 3 4 5

1 1 1 1 11 . / r [ ] [ g g ]g g g g

r r r r r

is a row matrix of the five conductances

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Node 1:

Node 2:

Node 3:

1 2 1 2 2 30 2g g v g v v

31 2 2 2

2 2 3 4 4

0vv v v v

r r r r r

3 32

4 4 5

0 s

v vvi

r r r

1 1 2

1 2 2

0 2v v v

r r r

2 1 2 3 4 2 4 3 0g v g g g v g v

1 4 2 4 5 30 sv g v g g v i

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1 2 2 1

2 2 3 4 4 2

4 4 5 3

0 2

0

0 s

g g g v

g g g g g v

g g g v i

These three equations can be written in matrix form as

2 1 2 3 4 2 4 3 0g v g g g v g v

1 2 1 2 2 30 2g g v g v v

1 4 2 4 5 30 sv g v g g v i

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Gv k

Gv

k

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

1 2 2 1

2 2 3 4 4 2

4 4 5 3

0 2

0

0 s

g g g v

g g g g g v

g g g v i

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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 ki (the ith component of the vector k) = the algebraic sum of the independent currents connected to node i, with currents entering the node taken as positive.

The Gkj terms are the negative sum of the conductances connected to BOTH node k and node j.

The Gkk terms are the sum of all conductances connected to node k.

1 2 2 1

2 2 3 4 4 2

4 4 5 3

0 2

0

0 s

g g g v

g g g g g v

g g g v i

2A 1r

2r

3

r 4

r 5rsi

1v 2v 3v

1i

2i

3i

4i

5i

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Gv k

MATLAB Solution of Nodal Equations

1v G k

1 2 2 1

2 2 3 4 4 2

4 4 5 3

0 2

0

0 s

g g g v

g g g g g v

g g g v i

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1

2

3

1 1 2 1 0 2

1 1 1 4 1 3 1 3 0

0 1 3 1 3 1 2 1

v

v

v

v1 v2 v3

1

2

3

1.5 1 0 2

1 1.583 0.333 0

0 0.333 0.833 1

v

v

v

Test with numbers

2A 4

31

221A

1/3 S

1/4 S1/2 S

1/2 S1 S

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MATLAB Run

1

2

3

1.5 1 0 2

1 1.583 0.333 0

0 0.333 0.833 1

v

v

v

2A 4

31

221A

v1 v2 v3

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PSpice Simulation

MATLAB:

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Let's write a general MATLAB program to solve this problem

1 2 2 1

2 2 3 4 4 2

4 4 5 3

0 2

0

0 s

g g g v

g g g g g v

g g g v i

2A 1r

2r

3

r 4

r 5rsi

1v 2v 3v

1i

2i

3i

4i

5i

1 2 3 4 5[ ]r r r r r rInputs: [ 2; 0; ]sk i

Find all voltages and currents

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function nodal1(r,k)

% PowerPoint nodal example

% Discussion D2.3

% r is a 1 x 5 vector of resistances

% k is a 3 x 1 vector of known currents entering the three nodes

% nodal1(r,k)

%

g = 1 ./ r

G = [g(1)+g(2) -g(2) 0; -g(2) g(2)+g(3)+g(4) -g(4); 0 -g(4) 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)]

2A 1r

2r

3

r 4

r 5rsi

1v 2v 3v

1i

2i

3i

4i

5i

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2A 4

31

221A

1/3 S

1/4 S1/2 S

1/2 S1 S

Do same problem as before

[2 1 4 3 2]r

[ 2; 0; ]sk i

nodal1(r,k)

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MATLAB Run

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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.

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Example

2A1g

0V 3g

2g 4gsi

DC1v 2v 3v

1i

2i

3i

4i

5i

Assume temporarily that i2 is known and write the equations by inspection.

1 1 2

2 3 3 2 2

3 3 4 3

0 0 2

0

0 s

g v i

g g g v i

g g g v i

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There appears to be 4 unknowns (v1, v2, v3, and i2) and only 3 equations. However, from the circuit

0 2 1V v v 1 2 0v v V or

so we can replace v1 (we could also replace v2) and write

1 1 2

2 3 3 2 2

3 3 4 3

0 0 2

0

0 s

g v i

g g g v i

g g g v i

1 2 0 2

2 3 3 2 2

3 3 4 3

0 0 2

0

0 s

g v V i

g g g v i

g g g v i

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Writing the above equation with the unknowns (v2, v3, i2) on the LHS yields

1 2 1 0

2 3 3 3

3 3 4 2

0 1 2

1 0

0 s

g v g V

g g g v

g g g i i

1 2 0 2

2 3 3 2 2

3 3 4 3

0 0 2

0

0 s

g v V i

g g g v i

g g g v i

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v1 v2 v3

Test with numbers

1 2

2 2

3

1 2 0 0 2

0 1 3 1 4 1 3

0 1 3 1 3 1 2 1

v i

v i

v

Noting that 1 2 2v v

2 2

2 2

3

1 2 0 0 2 2

0 1 3 1 4 1 3

0 1 3 1 3 1 2 1

v i

v i

v

2V

DC

2A g

g

g g1A

1/3 S

1/4 S1/2 S

1/2 S

i2

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v1 v2 v3

Test with numbers

Unknowns: 2 3 2 1 2, , ( 2)v v i v v

2 2

2 2

3

1 2 0 0 2 2

0 0.5833 1 3

0 1 3 0.8333 1

v i

v i

v

2

3

2

0.5 0 0 1

0.5833 0.3333 1 0

0.3333 0.8333 0 1

v

v

i

2V

DC

2A g

g

g g1A

1/3 S

1/4 S1/2 S

1/2 S

i2

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MATLAB Run

2

3

2

0.5 0 1 1

0.5833 0.3333 1 0

0.3333 0.8333 0 1

v

v

i

VVA

v1 v2 v3

1 2 2 2.6316Vv v v2v3

2V

DC

2A G

G

G G1A

1/3 S

1/4 S1/2 S

1/2 S

i2

i2

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PSpice Simulation

MATLAB: 1 2 2 2.6316Vv v v2v3i2

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Let's write a general MATLAB program to solve this problem

1 2 3 4[ ]g g g g gInputs: 1 0[ 2 ; 0; ]sk g V i

Find all voltages and currents

1 2 1 0

2 3 3 3

3 3 4 2

0 1 2

1 0

0 s

g v g V

g g g v

g g g i i

2A1g

0V 3g

2g 4gsi

DC1v 2v 3v

1i

2i

3i

4i

5i

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function nodal2(g,V0,is)% PowerPoint nodal-2 example% Discussion D2.3% g is a 1 x 4 vector of conductances% V0 = the known dc voltage source% is = the known dc current source% nodal2(g,V0,Is)%

G = [g(1) 0 1; g(2)+g(3) -g(3) -1; -g(3) g(3)+g(4) 0]k = [-2+g(1)*V0; 0; is]vvi = inv(G)*kv = zeros(1,3);v(2) = vvi(1);v(3) = vvi(2);v(1) = v(2)-V0;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);ikab = [i(1)+i(2) i(5)-i(4)]

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Do same problem as before

[1/ 2 1/ 4 1/ 3 1/ 2]g

0 2V

nodal2(g,V0,is)

2V

DC

2A g

g

g g1A

1/3 S

1/4 S1/2 S

1/2 S

i2

1is

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MATLAB Run