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 vs. Here the voltage source vs acts as input. The circuit ends with a load resistance R. we can take the current I through R as the output.
Suppose vs = 5V and i = 1A. According to linearity property if the voltage is multiplied by 2 then the voltage vs = 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.