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Linearity in Circuits

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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

${{V}_{1}}={{I}_{1}}R$

Similarly, if I2 is applied to R, then V2=I2R results. But if I=I1+I2 is applied, then the response

$V=IR=({{I}_{1}}+{{I}_{2}})R={{I}_{1}}R+{{I}_{2}}R={{V}_{1}}+{{V}_{2}}$

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

$R*(KI)=K*(RI)=K*V$

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

${{P}_{1}}=I_{1}^{2}R$

Whereas if the current is I2, then the power absorbed is

${{P}_{2}}=I_{2}^{2}R$

However, the power absorbed due to the current I1+I2 is

${{P}_{3}}={{({{I}_{1}}+{{I}_{2}})}^{2}}R=I_{1}^{2}R+I_{1}^{2}R+2{{I}_{1}}{{I}_{2}}R\ne {{P}_{1}}+{{P}_{2}}$

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.

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