LINEARITY PROPERTY 

Linear property is the linear relationship between cause and effect of an element. This property gives linear and nonlinear circuit definition. The property can be applied in various circuit elements. The homogeneity (scaling) property and the additivity property are both the combination of linearity property.

The homogeneity property is that if the input is multiplied by a constant k then the output is also multiplied by the constant k. Input is called excitation and output is called response here. As an example if we consider ohm’s law. Here the law relates the input i to the output v. 

mathematically , 

V= IR

If we multiply the input current  i by a constant k then the output voltage also increases correspondingly by the constant k. The equation stands,   

  kiR = kv

The additivity property is that the response to a sum of inputs is the sum of the responses to each input applied separately.

Using voltage-current relationship of a resistor if

                                     

  v₁ = i₁R       and   v₂ = i₂R

Applying (i₁ + i₂)gives

V = (i₁ + i₂)R = i₁R+ i₂R = v₁ + v₂

We can say that a resistor is a linear element. Because the voltage-current relationship satisfies both the additivity and the homogeneity properties.

We can tell a circuit is linear if the circuit both the additive and the homogeneous. A linear circuit always consists of linear elements, linear independent and dependent sources.

What is linear circuit?

A circuit is linear if the output is linearly related with its input.

The relation between power and voltage is nonlinear. So this theorem cannot be applied in power.

See a circuit in figure 1. The box is linear circuit. We cannot see any independent source inside the linear circuit.

 

The linear circuit is excited by another outer voltage source v_s. Here the voltage source v_s acts as input. The circuit ends with a load resistance R. we can take the current I through R as the output.

Suppose v_s = 5V and i = 1A. According to linearity property if the voltage is multiplied by 2 then the voltage v_s = 10V and then the current also will be multiplied by 2 hence i = 2A.

The power relation is nonlinear. For example, if the current i₁ flows through the resistor R, the power p₁ = i₁²R and when current i₂ flows through the resistor R then power p₂ = i₂²R.

If the current (i₁ + i₂) flows through R resistor the power absorbed

   P₃ = R(i₁ + i₂)² = Ri₁² + Ri₂² + 2Ri₁i₂ ≠ p₁ + p₂

So the power relation is nonlinear. Circuit solution method superposition is based on linearity property.