Linearity Property and Source Transformation

Linearity property

This property gives linear and nonlinear circuit definition. The property can be applied in various circuit elements. The homogeneity (scaling) property and the additive 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 additive 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 additive 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 a 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.

source transformation

Source transformation is simplifying a circuit solution, especially with mixed sources, by transforming a voltage into a current source, and vice-versa.  Finding a solution to a circuit can be difficult without using methods such as this to make the circuit appear simpler. Source transformation is another tool for simplifying circuits. Basic to these tools is the concept of equivalence.

 A Source Transformation is the process of replacing a voltage source Vs in series with a resistor R by a current source Is in parallel with a resistor R, or vice-versa.

Source transformation requires that,

Vs = IsR     or     Is = Vs/R

Source transformation also applies to dependent sources, provided we carefully handle the dependent variable. As shown in below, a dependent voltage source in series with a resistor can be transformed to a dependent current source in parallel with the resistor or vice versa.

OVERVIEW AND INSIGHTS

Linearity Property talks about the voltage (v) and current (i), whereas voltage is directly proportional to the current, that is, when the voltage is increasing the current also increasing and vice-versa. I also learned that Source Transformation is only applicable to simple circuits and not in a complicated circuits.

Source Transformation can be applied if and only if the voltage is in series with the resistor and/or the current is in parallel with the resistor.

In Source Transformation you need to check the polarity of the source, when the polarity of the voltage source has a positive on top therefore the resulting current source is pointing upward and vice-versa. Also when the voltage source is dependent therefore the resulting current is also dependent, same calculations with independent sources.