The article discusses the concept of linearity in circuits, focusing on the relationship between voltage and current in resistors based on Ohm’s Law. It explains the conditions for linearity, highlights how power relationships are non-linear, and defines linear circuit elements, including dependent sources.
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.
- You May Also Read: Ohm’s Law with Examples
Linearity in Circuits Key Takeaways
Understanding the concept of linearity in circuits is essential for designing and analyzing electrical systems efficiently. Linear circuits simplify calculations, allowing engineers to apply principles like superposition and Thevenin’s theorem to predict circuit behavior accurately. These concepts are widely used in signal processing, control systems, and electronic circuit design, where predictable and stable responses are crucial. By recognizing the limitations of non-linear power relationships, engineers can make informed decisions when designing circuits for various applications, ensuring optimal performance and reliability.