Equivalence Between Voltage Source and Current Source / Conversion of Voltage source into Current source and Vice Versa

 Equivalence Between Voltage and Current Source :

Equivalence Between Voltage and Current Source :

Practically, a voltage source is not different from a current source. In fact, a source can either work as a current source or as a voltage source.

 

It merely depends upon its working conditions. If the value of the load impedance is very large compared to the internal impedance of the source, It proves advantageous to treat the source as a voltage source.

On the other hand, if the value of the load impedance is very small compared to the internal impedance of the source, than it works like a current source, and it's better to represent the source as a current source.

 

From the circuit point of view, it does not matter at all whether the source is treated as a current or a voltage source. In fact, it is possible to convert a voltage source into a current source and vice versa.

Conversion of Voltage source into Current source and Vice Versa

Conversion of Voltage source into Current source and Vice Versa:

Consider an AC source connected to a load impedance Z_L. The source can either be treated as a voltage source or as a current source, as shown below:

The voltage source representation consists of an ideal voltage source V_S in series with a source impedance Z_S1. And the current source representation consists of an ideal current source I_S in parallel with the source impedance Z_S2.

 

There are two representations of the same. Both types of representations must appear the same to the externally connected load impedance Z_L. They, must give the same results.

In the following figure :

If the load impedance Z_L is reduced to zero (I.e., the terminals A and B are short-circuited), the current through

this short is given as below:

 

 I_S(short circuit) = V_S/Z_S1

We want both the representations to give the same results. This means that the current source in the following figure:

Must also give the same current ( as above equation) when terminals A and B are shorted. But the current obtained by shorting the terminals A and B (in the above figure) 

Is simply the source current I_S( the source impedance Z_S2 connected in parallel with a short circuit is as good as not being present). Therefore, we conclude that the current I_S of the equivalent current source must be the same as the given by the following equation :

 

I_L (Short circuit) = I_S= V_S/Z_S1

Again the two representations of the source must give the same terminal voltage when the load impedance Z_L is disconnected from the source (I.e., when the terminals A and B are open-circuited).

In the following figure:

 

The open circuit voltage is simply V_S . There is no voltage drop across the internal impedance Z_S1. Let us find out the open-circuit voltage in the current source representation of the following figure:

When the terminals A and B are open-circuited, the whole of the current I_S flows through the impedance Z_S2. The terminal voltage is then the voltage drop across this impedance. That is

 

V_T(open circuit)  = I_S Z_S2

 

Therefore, if the two representations of the source are to be equivalent, we must have

 

V_T = V_S

 

Using  both equation

 

I_L (Short circuit) = I_S= V_S/Z_S1

 

And

 

V_T(open circuit)  = I_S Z_S2

 

We get :

 

I_s Z_S1 = I_S Z_S2

 

Or we say   Z_S1 = Z_S2 = Z_S

 

Then the both equation reduce to

V_S = I_S Z_S

 

It may be noted then the both the representations of the source, the source impedance as faced by the load impedance at the terminals A and B, is the same (impedance Z_S).

 

Thus, we have established the equivalence between the voltage source representation and current source representation. For short circuits and for open circuits.

 

But, we are not sure that the equivalence is valid for any other value of load impedance.

 

To test this, let us check whether a given impedance Z_L draws the same amount of current when connected either to the voltage-source representation or to the current-source representation.

In the following figure:

The current through the  load impedance is :

 

I_L1 = V_S/ Z_S + Z_L

 

In following figure:

 

 

 

 

 

 

The current I_S  divides into two branches. Since the current divides itself onto two branches in inverse proportion of the impedance, the current through the load impedance Z_L is :

 

I_L2 = I_S * (Z_S / Z_S + Z_L )

 

=  I_S Z_S / Z_S + Z_L

 

By making use of equation  V_S = I_S Z_S  the above equation can be written as

 

I_L2 = V_S / Z_S + Z_L

 

We now see that currents I_L1 and I_L2 as given by equation  

 

I_L1 = V_S/ Z_S + Z_L

 And

 I_L2 = V_S / Z_S + Z_L

 

are exactly the same.