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: Superposition theorem�

In a linear circuit, the current or voltage of any branch can be regarded as the algebraic sum of the current or voltage generated on the branch when each independent power source in the circuit acts on the circuit alone.

Proof:

Based on node voltage method, the KCL equation at node 1 is

(G₂ + G₃)*u_n1 = G₂*u_s2 + G₃*u_s3 + i_s1 (1)

u_n1 = u_s2*G₂/(G₂ + G₃) + u_s3*G₃ /(G₂ + G₃) + i_s1/(G₂ + G₃)

u_n1 = i_s1*a₁ + u_s2*a₂ + u_s3*a₃

or u_n1 = u_n1⁽¹⁾ + u_n1⁽²⁾ + u_n1⁽³⁾

Circuit theorem: Superposition theorem

The branch current can be

i₂ = (u_n1 -u_s2)*G₂

=-(G₂*G₃)*u_s2 /(G₂+G₃) + (G₂*G₃)*u_s3/(G₂+G₃)

+ G₂*i_s1/(G₂+ G₃)

= i_s1*b₁ + u_s2*b₂ + u_s3*b₃

= i₂⁽¹⁾ + i₂⁽²⁾ + i₂⁽³⁾

i₃ = (u_n1 –u_s3)*G₃

=(G₂*G₃)*u_s2 /(G₂+G₃) + (-G₂*G₃)*u_s3/(G₂+G₃)

+ G₃*i_s1/(G₂+ G₃)

= i₃⁽¹⁾ + i₃⁽²⁾ + i₃⁽³⁾

Note:

• The superposition theorem can only be applicable to linear circuit (i.e., circuit with linear elements).
• “One independent power source in the circuit acts on the circuit alone” meaning that the other power sources (if any) are either short-circuited (for voltage source) or open-circuited (for current source).

Circuit theorem: Superposition theorem

=

+

+

Circuit theorem: Superposition theorem

Note:

• The superposition theorem can only be applicable to linear circuit (i.e., circuit with linear elements).

​

• “One independent power source in the circuit acts on the circuit alone” meaning that the other power sources (if any) are either short-circuited (for voltage source) or open-circuited (for current source).

​

• The superposition theorem cannot be applied to power calculation (voltage x current).

​

• The superposition theorem can be applicable to the circuit containing dependent source, but the dependent source should be reserved.

Circuit theorem: Superposition theorem

Example: to solve the current i of the voltage source and its power

=

+

=

+

Circuit theorem: Superposition theorem

Wheatstone bridge circuit

If R_A*R_D = R_B*R_C 🡪 v_bc=0 🡪 no current between b and c

Circuit theorem: Superposition theorem

Example: to solve the current i and voltage u (This example is used as an assignment as well)

• The superposition theorem can be applicable to the circuit containing dependent source, but the dependent source should always be reserved.

+

+

=

KVL

Circuit theorem: Superposition theorem

Homogeneity principle:

​

In a linear circuit, if all excitation (independent power supply) increases or decreases by the same factor, the response (current or voltage) in the circuit also increases or decreases by the same factor.

u_n1 = i_s1*a₁ + u_s2*a₂ + u_s3*a₃

or u_n1 = u_n1⁽¹⁾ + u_n1⁽²⁾ + u_n1⁽³⁾

k*u_n1 = (k*i_s1)*a₁ + (k*u_s2)*a₂ + (k*u_s3)*a₃

Circuit theorem: Superposition theorem

i

Circuit theorem: Superposition theorem

If u_s=1V, i_s=1A, we have i=2A

If u_s=-1V, i_s=2A, we have i=1A

What if u_s=-3V, i_s=5A, i=?

Assignment

Circuit theorem: Superposition theorem