Substitution Theorem

As name implies, the main concept of this theorem which is based upon substitution of one element by another equivalent element. Substitution theorem gives us some special insights in circuit behavior. This theorem is also used to prove several other theorems.

Statement

Substitution theorem states that "if an element in a network is replaced by a voltage source whose voltage at any instant of time is equals to the voltage across the element in the previous network then the initial condition in the rest of the network will be unaltered". Or alternately "if an element in a network is replaced by a current source whose current at any instant of time is equal to the current through the element in the previous network then the initial condition in the rest of the network will be unaltered".

Explanation

Let us take a circuit as shown in fig - a,
 
 Let, V is supplied voltage and Z₁, Z₂ & Z₃ is different circuit impedances. V₁, V₂ and V₃ are the voltages across Z₁, Z₂ & Z₃ impedance respectively and I is the supplied current whose I₁ part is flowing through the Z₁ impedance whereas I₂ part is flowing through the Z₂ and Z₃ impedance.


Now if we replace Z₃ impedance with V₃ voltage source as shown in fig-b or with I₂ current source as shown in fig-c then according to Substitution Theorem all initial condition through other impedances and source will remain unchanged.
i.e. - current through source will be I, voltage across Z₁ impedance will be V₁, current through Z₂ will be I₂ etc.

Example of Substitution Theorem

For more efficient and clear understanding let us go through a simple practical example: Let us take a circuit as shown in fig - d

 

 As per voltage division rule voltage across 3Ω and 2Ω resistance are




If we replace the 3Ω resistance with a voltage source of 6V as shown in fig - e, then

According to Ohm’s law the voltage across 2Ω resistance and current through the circuit is
 
Alternately if we replace 3Ω resistance with a current source of 2A as shown in fig - f, then
 Voltage across 2Ω is V_2Ω = 10 - 3× 2 = 4 V and voltage across 2A current source is V_2A = 10 - 4 = 6 V We can see the voltage across 2Ω resistance and current through the circuit is unaltered i.e all initial condition of the circuit is intact.