# Substitution Theorem : Steps Involved in Solving it, Example Problems & Its Applications

The fundamental network theorems used in network analysis are available in different types like the Thévenin’s, superposition, Norton’s, substitution, maximum power transfer, reciprocity & Millman’s theorems. Every theorem, they have its own application areas. So understanding each network theorem is very significant because these theorems can be used repeatedly in different circuits. These theorems help us in solving complex network circuits for a given condition. This article discusses one of the types of network theorem **substitution theorem** – examples.

## What is the Substitution Theorem?

Substitution Theorem statement is; that whenever the current throughout the branch or the voltage across any branch in a network is known, then the branch can be changed by the combination of different elements that will make the similar voltage & current throughout that branch. In other words, it can be defined as; the thermal voltage, as well as current, should be identical for equivalence of branch.

The substitution theorem concept mainly depends on the substitution of one element with another element. This theorem is also very helpful in proving some other theorems. Although this theorem is not applicable for solving the theorem which includes the above two sources which are connected neither in series nor parallel.

### Explanation of Substitution Theorem

### The steps involved in solving the substitution theorem mainly include the following.

**Step1:** First, we need to find the voltage & current of all the network elements. In general, the voltage & current can be calculated with the help of ohms law, Kirchoff Laws like KVL, or KCL.

**Step2:** Select the required branch that you desire to remove through a different element like voltage source/resistance and current source.

**Step3:** Find the right value of the substituted element provided that the voltage & current should not alter.

Step4: Check the new circuit by simply calculating the current & voltage of all elements and evaluate it by the original network.

### Substitution Theorem Circuit Diagram

Let us easily understand the substitution theorem by using the following circuit diagram. We know that the substitution theorem is the substitute of a single element with another equivalent element. If any element within a network is replaced/substituted with a current source or voltage source, whose current & voltage throughout or across the element will stay unchanged like the previous network.

The various resistances like R1, R2 & R3 are connected simply across the voltage source. The flow of current ‘I’ flowing throughout the circuit is separated into I1 & I2 where ‘I1’ is supplied throughout the ‘R1’ resistance & the ‘I2’ is flowing throughout the R2 resistance as shown in the circuit. Here, the voltage drops across the resistances R1, R2 & R3 are V1, V2 & V3 correspondingly.

Now if the ‘R3’ resistance is substituted by the ‘V3’ voltage source as shown in the following circuit diagram below:

In the following circuit diagram, the ‘R3’ resistance is replaced by the flow of current throughout that element ‘I1’.

From the above two cases, if the element is substituted with the current or voltage source then the circuit’s initial conditions do not change that means, the voltage supply across the resistance & current supply throughout the resistance is not changed even if they are replaced with other sources.

### Example Problems

Substitution theorem example problems are discussed below.

#### Example1:

Solve the following circuit with the substitution theorem to calculate the voltage & current within all the resistors.

**Step1:**

First, apply KVL to loop1 in the above circuit

14 = 6I1 – 4I2 ….(1)

Apply KVL to loop2 in the above circuit

0 = 12I2 – 4I1

12 I2 = 4I1 => I1 = 3I2 ……….(2)

Substitute this equation 2 in the above equation 1.

14 = 6(3I2) – 4I2

14 = 18I2 – 4I2 =>14I2 => 1A

I2 = 1A

From the above equation-(2)

I1 = 3I2

We know that I2 = 1A

I1 = 3A

**Step2:**

In this step, we need to remove the loop1 branches to make a single loop.

**Step3:**

We can place a current source/voltage source in place of the 4Ω resistor. Now, we will use a current source.

The flow of current throughout loop2 in the circuit is 1A. So, we substitute the branch with 1A current source. As a result, the residual circuit is shown below.

**Step4:**

In this step, need to check the voltage & current of all the elements. The above circuit includes a single loop i.e, a current source. Thus, the value of flowing current throughout the loop is similar to the current source value.

Here, the current source value is 1A. So, the flow of current throughout the 3Ω & 5Ω resister branches is 1A which is similar to the original network.

By using the ohms law, find the voltage value across the 3Ω resistor

V = IR

V = I x R

V = 1 x 3 => 3V.

Similarly, by using the ohms law, we need to find the voltage value across 5Ω resistor.

V = IR

V = I x 5

V = 1 x 5 => 5V.

Thus, the current & voltage are similar to the original network. So, this is how this theorem works.

Now, if we choose the voltage source in place of the current source within step 3. So in this condition, the voltage source value is similar to the 4Ω resister branch value.

The flow of current throughout the 4Ω resister branch within the original network is

I1 – I2 => 3 – 1 => 2A

According to Ohm’s law;

The voltage at 4Ω resistor is V = 2 x 4 = 8V

So, we need to connect the voltage source with 8V in the network & the residual circuit is shown in the below diagram.

V= 2 x 4 = 8V

So, we need to connect the 8V voltage source with the network and the remaining circuit is as shown in the below figure.

Apply KVL to the above loop to verify the voltage & current.

8 = 3I + 5I => 8I

I = 1A.

By using the ohms law, the voltage across resistor 3Ω can be calculated as;

V = 1 × 3 => 3V

Similarly, the voltage across resistor 5Ω is;

V= 1 × 5 => 5V

Thus, the voltage & current are the same after substitution as the original network.

#### Example2:

Let us take the following circuit to apply the substitution theorem.

According to the voltage division ruler, voltage across 2Ω & 3Ω resistors is;

The voltage at the 3Ω resistor is

V = 10×3/3+2 = 6V

The voltage at the 2Ω resistor is

V = 10×2/3+2 = 4V

The flow of current throughout the circuit is calculated as I = 10/3+2 = 2A.

In the above circuit, if we substitute a 6Vvoltage source in place of the 3Ω resistor then the circuit will become like the following.

Based on Ohm’s law, the voltage across the 2Ω resistor & flow of current throughout the circuit is

V = 10-6 => 4V

I = 10-6/2 = 2A

If we substitute a 2A current source in place of a 3Ω resistor then the circuit will become like the following.

Voltage across 2Ω resistor is V = 10 – 3* 2 => 4 V & voltage across ‘2A’ current source is V = 10 – 4 => 6 V. So the voltage across 2Ω resistor & current throughout the circuit is not changed.

### Advantages

The **advantages of the substitution theorem** include the following.

- This theorem concept mainly depends on the substitution of a single element from another element.
- This theorem provides intuition on the circuit behavior & also assists in verifying various other network theorems.
- The advantage of using this theorem is that this theorem provides the correct values for the variables like X & Y which correspond to the intersection point.

### Limitations

The **limitations of the substitution theorem** include the following.

- This theorem cannot be used for solving a network that includes a minimum of two or above sources that are not within series/parallel.
- In this theorem, when replacing the element, the circuit behavior should not change.

### Applications

The **applications of the substitution theorem** include the following.

- The substitution theorem is used to prove numerous other theorems.
- This theorem is helpful in solving the system of equations in mathematics.
- This theorem replaces the circuit’s one element with one more element.
- This theorem is used to analyze the circuits with dependent sources.

**On which circuit’s substitution theorem is not applicable?**

The circuit which has the above two sources that are connected either in parallel or series, then this substitution theorem is not applicable.

**Why compensation theorem is called substitution?**

Both the theorems like compensation and substitution are identical in terms of procedure & reduction. So this theorem is applicable for antennas and is also called the substitution theorem.

**How do you use the substitution theorem?**

This theorem can be used by substituting any branch with a different branch within a network without troubling the voltages & currents in the entire network. So this theorem is used in both linear & nonlinear circuits.

**What is substitution property?**

The substitution property states that, if a variable ‘a’ is equivalent to another variable ‘b’, then ‘a’ can be replaced in place of ‘b’ in any expression or equation & ‘b’ can be replaced in place of ‘a’ in any expression or equation.

Thus, this is all about an overview of a substitution theorem – circuit with examples. Here is a question for you, what is the compensation theorem?