Source Transformation

In this article, we will study about the source transformation, i.e. voltage source to current source transformation and current source to voltage source transformation in detail.

An energy source is a type of active circuit element that can provide excitation to the circuit. In simple words, an energy source is a circuit element that can deliver power to the circuit.

In electrical engineering, we come across two types of energy sources namely,

• Voltage sources
• Current sources

A voltage source is a type of energy source that maintains a constant voltage across its terminals whatever current may be drawn from it. Therefore, for a voltage, the terminal voltage remains constant and it is independent of the current taken from the source.

On the other hand, a current source is one which delivers a constant current to the circuit and this current is independent of the voltage across its terminals.

However, based on the internal resistance, the voltage and current sources are further classified into the following types:

• Ideal voltage source
• Practical/real voltage source
• Ideal current source
• Practical/real voltage source

An ideal voltage source is one whose internal resistance is zero, while a practical voltage source has an internal resistance of finite value in series with the ideal voltage source.

An ideal current source is the type of current source for which the internal resistance is infinite, whereas a practical current source has some finite resistance that is connected in parallel with an ideal current source.

Although, a practical voltage source can be converted into a practical current source and a practical current source can be converted into a practical voltage source by the source transformation technique. The source transformation is sometimes very helpful in simplifying the electrical circuit during circuit analysis.

Voltage Source to Current Source Transformation

According to source transformation theory, a practical or real voltage source of voltage V volts in series with an internal resistance R_i­v can be replaced with a current source of current I_s in parallel with an internal resistance R_ic.

Where,

`\I_s=V/R_{iv}"     and   "R_{ic}=R_{iv}`

It always should be kept in mind that the direction of current of the current source should not be changed at the load terminals, i.e. the current should be leaving from the positive terminal.

Proof:

It can be easily proved that the above two parameters (I_s and R_ic) are technically valid. For this consider a transformation circuit of a voltage source into a current source (refer to the above figure).

Here, we can follow the following step to prove the conversion of a real voltage source into a real current source:

Step 1 – In order to find the value of I_s, short circuit the load terminals (x-y) of the voltage source (refer to the left figure shown below)  and calculate the short circuit current I_vsc as:

`\I_{vsc}=V/R_{iv}`

Step 2 – Short circuit the load terminals of the current source (refer to the right figure shown above) and calculate the short circuit current I_csc as:

`\I_{csc}=I_s`

It is because the current always follows the path of low resistance. Hence, the total source current ideally flows through the short circuit.

Step 3 – Two sources are identical if currents at the load terminals are the same, i.e.

`\I_{vsc}=I_{csc}`

`\⟹V/R_{iv} =I_s`

Step 4 – Now to prove that R_iv = R_ic, open the load terminals of both the voltage source and current source and deactivate the sources (i.e. short-circuit the voltage source and open circuit the current source) refer to the below figures. Then, calculate the resistance of both the circuit viewing from the load terminals x-y. 

Therefore, we have,

`\R_{vxy}=R_{iv}`

And

`\R_{cxy}=R_{ic}`

Step 5 – Again two sources are identical if their resistances viewing from the load terminals are equal, i.e.

`\R_{vxy}=R_{cxy}`

`\∴R_{iv}=R_{ic}`

Hence, in this way, we can prove that a real voltage source can be transformed into a real current source.

Current Source to Voltage Source Transformation

By using the source transformation technique, we can also convert a practical current source into a practical voltage source.

According to this, a real current source with source current I and a parallel internal resistance R_ic can be converted into a real voltage source of voltage V and a series internal resistance R_iv, where,

`\V=IR_{ic}`

And

`\R_{iv}=R_{ic}`

This can also be proved in the same way as the proof of voltage-to-current source transformation.

Numerical example – A voltage source of 12 V having an internal resistance of 2 ohms, it is required to transform it into an equivalent current source. Calculate the current and internal resistance of the equivalent current source.

Solution –

For the voltage source,

`\V=12" V"`

`\R_iv=2" Ω"`

Now, the current and internal resistance of the equivalent current source will be,

`\I_S=V/R_{iv} =12/2=6" A"`

And

`\R_{ic}=R_{iv}=2" Ω"`

Hence, I_s and R_ic will be connected in parallel to obtain the current source equivalent to the given voltage source.

Therefore, in this article, we have discussed everything about source transformation along with proof of statements and solved numerical examples.