Lecture 11

Circuit theorem

Circuit theorem

• Superposition theorem
• Substitution theorem
• Thevenin theorem and Norton theorem
• Maximum power transmission theorem
• Tellegen's theorem (*)
• Reciprocity theorem (*)
• Duality theorem (*)

Circuit theorem: Substitution theorem�

Given any branch, if its voltage is u and current is i, it can be replaced by an independent voltage source with voltage u, an independent current source with current i, or a resistor with resistance u/i. After replacement, the voltage or current of all branches in the circuit remain unchanged.

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Proof:

Circuit theorem: Substitution theorem�

By adding two extra voltage sources, the terminal voltage of A circuit does not change.

The potential at node 1 and 2 is equal. Thus, node 1 and node 2 can be short-circuited.

Circuit theorem: Substitution theorem�

Example: to solve the current i₁ i₂ i₃ and voltage u

i₁ = 110V / (5 + (5+10)*10/(5+10+10) )=10A

i₂ = i₁ *0.6 = 6A, i₃ = i₁ *0.4 = 4A

u = 60V

We can easily solve

Suppose we only know the voltage of the 10Ω-resistor, we can replace the resistor with a 60V voltage source. The other quantities can be solved very easily.

Circuit theorem: Substitution theorem�

Note:

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After the substitution, there should be no voltage-source loop in the circuit.

5V

What is i₂ and i₃?

Circuit theorem: Substitution theorem�

Note:

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After the substitution, there should be no joint in the circuit where the current of current-sources meets.

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u

What is u?

Circuit theorem: Substitution theorem�

Example: if we want i_x = i₁/8, what is the value of the substitute resistance R_x ?

To get the equivalent resistance, it is necessary to solve the voltage u on the resistor.

Circuit theorem: Substitution theorem�

u^(a) = u^(b) + u⁽¹⁾

So u⁽¹⁾ = u^(a) - u^(b)

=1Ω * i₁ *(1.0/2.5) – 0.5Ω * i₁ *(1.5/2.5)

= 0.1 * i₁

u^(a) + u^(b) + u⁽²⁾ =0

So u⁽²⁾ = -(u^(a) + u^(b))

= -(1+0.5)Ω * i_x *(1/2.5)

= -0.075 * i₁

u = u⁽¹⁾ + u⁽²⁾ = 0.025 * i₁

R_x = u/i_x = 0.025 * i₁ / 0.125i₁ = 0.2Ω

Circuit theorem: Substitution theorem�

Assignment: to prove that after the branch is replaced by the current source with current i_s, the voltage or current of all branches in the circuit remain unchanged.

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Circuit theorem: Substitution theorem�

Assignment: if u_ab=0，what is the value of R?

Circuit theorem: Substitution theorem�

Assignment: If u_a= is replaced by a resistor R，what is the resistance value of R to keep the other quantities unchanged?

4Ω

2Ω

2Ω

10Ω

10Ω

4V

u_a=2V

3A

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Circuit theorem: Substitution theorem�

Assignment: if u_ab=0，what is the value of R?