Lecture 10

Node voltage method

Node voltage method

• The unknowns are the voltages (relative to a reference) at the circuit nodes.

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• It is suitable for analyzing circuit with medium number of nodes.

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• KVL equation is not necessary. Only KCL equation is required.

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• The branch current and voltage is the linear combination of node voltages.

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Node voltage method

• The number of independent KCL equations is (n-1), where n is the number of nodes. In contrast, the number of equations in branch current method is b, the number of branches.

• Select any node as the reference, and the voltage of a node, a, is the difference between the potential at node a and the potential at the reference point.

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Procedure of Node voltage method

• Select a reference node and label the other (n-1) nodes.
• For each of the (n-1) nodes, list the KCL equation with unknown node voltages.
• Solve the (n-1) equations to get the (n-1) node voltages.
• Solve the branch current.

Node voltage method

• Example:

1

2

3

i₁ + i₂ = i_s1 + i_s2 (1)

i₂ – i₃ - i₄ = 0 (2)

i₃ – i₅ = i_s2 (3)

KCL equations at node 1, 2, and 3

To represent branch currents with node voltages.

u_n1/R₁ + (u_n1-u_n2)/R₂ = i_s1 + i_s2 (1)

(u_n1-u_n2)/R₂ - (u_n2-u_n3)/R₃ – u_n2/R₄ = 0 (2)

(u_n2-u_n3)/R₃ - (u_n3-u_s)/R₅ = i_s2 (3)

Node voltage method

• Example:

The above equations are reformulated as

(1/R₁ + 1/R₂)*u_n1 + (-1/R₂)*u_n2 = i_s1 + i_s2 (1)

(-1/R₂)*u_n1 + (1/R₂ + 1/R₃ + 1/R₄)* u_n2 + (-1/R₃)* u_n2 = 0 (2)

(-1/R₃)* u_n2 + (1/R₃ + 1/R₅)* u_n3 = -i_s2 + u_s/R₅ (3)

1

2

3

where G₁₁=(1/R₁ + 1/R₂), G₁₂=(-1/R₂), G₁₃=0

G₂₁=(-1/R₂), G₂₂= (1/R₂ + 1/R₃ + 1/R₄), G₂₃=(-1/R₃)

G₃₁= 0, G₃₂=-(1/R₃), G₃₃=(1/R₃ + 1/R₅)

Self-conductance at a node

Mutual-conductance of two neighboring nodes

Node voltage method

Note

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• In node voltage method, the self-conductance at a node is the sum of conductance which connects to it, and it is always positive.

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• The mutual-conductance between a node and its neighbor is a negative conductance, which is associated with the node and its neighbor.

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• The right-side of each node voltage equation is the difference of currents (the in-flowing current – the out-flowing current).

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• Once knowing the voltage at each node, the branch current can be readily calculated.

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Node voltage method

• Example:

Node voltage method

Special case: what if the circuit model has an ideal voltage source (unaccompanied voltage source)?

Procedure of Node voltage method

• Add the current flowing through the ideal voltage source as one extra unknown. Add extra constraint between node voltage and the ideal voltage source

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• Select a reference node and label the other (n-1) nodes.

i

u₁ - u₃ = u_s (4)

KCL equations at node 1, 2, and 3

Extra constraint

Node voltage method

• Example:

i_s1

u_R2

R₁

R₂

R₃

i₁

+

-

gu_R2

1

2

(1/R₁ + 1/R₂)*u_n1 + (-1/R₂)*u_n2 = i_s1 (1)

(-1/R₁)*u_n1 + (1/R₁ + 1/R₃)* u_n2 = - i_s1 - gu_R2 (2)

The dependent current source is viewed as an independent current source.

KCL equations at node 1, 2

Extra constraint

u_n1 = u_R2

Node voltage method

• Example:

(1/R₁ + 1/R₅)*u_n1 + (-1/R₅)*u_n3 = i_s1 + u_s/R₅ + i’ (1)

• The dependent voltage source is viewed as an independent one.

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• Add the current, i’ flowing through the unaccompanied voltage source as one extra unknown.

KCL equations at node 1, 2 and 3

(1/R₂ + 1/R₃)*u_n2 + (-1/R₃)*u_n3 = -i’ (2)

(-1/R₅)*u_n1 + (-1/R₃)*u_n2 + (1/R₃ + 1/R₄ + 1/R₅)*u_n3 = -gu₃ - u_s/R₅ (3)

Extra constraint

i’= -u_n2/R₂ - (u_n2 – u_n3)/R₃ (4)

u₃ = u_n2 – u_n3 (5)