The substitution theorem states that any branch within a circuit can be substituted by an equivalent branch that has the same current and same voltage through it.
Illustration:
Consider a series circuit containing two resistors: 10 ohms and 5 ohms connected to a 12 v source. From voltage divider rule, the Resistor R1 has 4 V drop across it while resistor R2 has a drop of 8 Volts across it:
![](https://electricalandelectronicsengineering.com/wp-content/uploads/2019/06/voltage-divider-rule-solved-example.jpg)
From Ohm’s law, the current passing through 10 ohms resistor is:
I = V/R = 8 V / 10 Ω = 0.8 A
According to substitution theorem, the resistor R2 can be replaced by any combination of components that fulfills our condition of 0.8 A current and a voltage drop of 8 volts across it.
For instance, the resistor R2 which is now 10 ohms can be replaced by a resistor of 4 ohms provided and a series voltage drop of 4.8 v. The current through the circuit is still 0.8 Amps
![](https://electricalandelectronicsengineering.com/wp-content/uploads/2019/07/substitution-circuit-01-300x245.jpg)
Example: Replace the resistor R2 in the same circuit with 200 mA current source pointing upward and a shunt resistor.
Solution: To maintain the same current in the circuit, the branch current should stay the same. However, we need to apply Kirchhoff’s current law to find the current across the shunt resistor.
![](https://electricalandelectronicsengineering.com/wp-content/uploads/2019/07/sub-t-1-2-300x195.jpg)
Now unknown current (???),
I = 800 mA + 200 mA = 1 A
Voltage across 5 ohms resistor = 5 ohms * 800 mA = 4000 mA = 4 V
From Kirchhoff’s Voltage law rest all voltage (8 V) will be dissipated across the branch is 8 volts.
To maintain 8 volts @ 1 Amps we need to have a resistance of 8 ohms:
![](https://electricalandelectronicsengineering.com/wp-content/uploads/2019/07/s-t-e-12.jpg)
So we need a 200 mA current source pointing upward and a shunt source of 8 ohms to obtain substituted circuit.