Consider the relationship between voltage and current for a resistor (Ohm’s Law). Suppose that c current I1 (the excitation or input) is applied to a resistor, R. then the resulting voltage V1 (the response or output) is
Similarly, if I2 is applied to R, then V2=I2R results. But if I=I1+I2 is applied, then the response
In other words, the response to a sum of inputs is equal to the sum of the individual responses (Condition 1).
In addition, if V is the response to I (that is V=IR), then the response to KI is
In other words, if the excitation is scaled by the constant K, then the response is also scaled by K, (Condition 2).
Because conditions 1 and 2 are satisfied, we say that the relationship between current (input) and voltage (output) is linear for a resistor. Similarly, by using the alternate form of Ohm’s law I=V/R, we can show that the relationship between voltage (excitation) and current (response) is also linear for a resistor.
Although the relationships between voltage and current for a resistor are linear, the power relationships P=I2R and P=V2/R are not.
For instance, if the current through a resistor is I1, then the power absorbed by the resistor R is
Whereas if the current is I2, then the power absorbed is
However, the power absorbed due to the current I1+I2 is
Hence the relationship P=I2R is non-linear.
Since the relationships between voltage and current are linear for resistors, we say that a resistor is a linear element.
A dependent source (either current or voltage) whose value is directly proportional to some voltage and current is also a linear element. Because of this, we say that a circuit consisting of independent sources, resistors, and linear dependent sources is a linear circuit.