- Substitution Theorem Definition: The substitution theorem is defined as the process of replacing an element in a circuit with an equivalent voltage or current source without changing the initial conditions.
- Statement of Substitution Theorem: If an element is replaced by a voltage source with the same voltage or a current source with the same current, the rest of the circuit remains unchanged.
- Insight into Circuit Behavior: This theorem helps understand how circuits behave when elements are replaced with equivalent sources.
- Voltage Source Example: Replacing an impedance with a voltage source keeps the initial circuit conditions the same.
- Practical Example: In a circuit, replacing a resistor with a voltage or current source shows that the initial voltage and current conditions remain unchanged.
As the name implies, the substitution theorem is based on replacing one element in a circuit with another equivalent element. The Substitution theorem provides special insights into circuit behavior and is also used to prove other theorems.
Statement of Substitution Theorem
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 of Substitution Theorem
Let us take a circuit as shown in fig – a,
Let, V is supplied voltage and Z1, Z2 and Z3 is different circuit impedances. V1, V2 and V3 are the voltages across Z1, Z2 and 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.
If we replace Z3 with a V3 voltage source or an I2 current source, the initial conditions through other impedances and the source will remain unchanged, according to the substitution theorem.
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.
According to the voltage division rule, the voltage across 3Ω and 2Ω resistance is:
If we replace the 3Ω resistance with a voltage source of 6 V 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 V2Ω = 10 – 3× 2 = 4 V and voltage across 2A current source is V2A = 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.
