Source Transformation Theorem

Source transformation is the process of replacing a voltage source Vs, in series with an impedance Z, by a current source I, in parallel with an impedance Z. This process can be applied to both independent and dependent (provided that the dependent variable is carefully handled) sources. It uses the very known formula in electric circuit analysis – Ohm’s Law – which is V = IZ or I = V/Z.

When you apply source transformation to a voltage source to transform it into a current source, all you have to do is to  is :

• use Is = V/Z to determine the value of the current source – where V is the voltage source and Z as the impedance value in series with the voltage source V.

•  After calculating the value of the current source, redraw the voltage source Vs into a current source Is and make the impedance Z, in series with the source, parallel to the transformed source.

Else, to transform a current source into a voltage source, you use Vs = I Z to determine the value of the voltage source – where I is the current source and Z as the impedance value in parallel with the current source Is. After the calculation, redraw the current source Is into a voltage source Vs and make the impedance Z, in parallel with the source, series to the transformed source.

Take note that the polarity of the not-yet-transformed source should be the same as the newly transformed source. For the voltage source, the positive terminal is the plus (+) sign (well, obviously) while the negative terminal is the minus (-) sign. For the current source, the arrow’s head is the positive terminal while the arrow’s tail is the negative terminal.

When using source transformation technique, there will be times when you’ll encounter two or more voltage sources connected in series, and two or more currents sources connected in parallel. You can actually combine those sources when it happens. That’s the whole point of using source transformation in simplifying circuits: to lessen the number of sources until you can no longer do so, then you throw the finishing blow with voltage and current divider, or nodal and mesh analysis. The question is how do you combine them. Simple.

If there are voltage sources connected in series with equal polarity direction, just add their values together and redraw the voltage sources into one voltage source with the same polarity direction as the multiple ones. If you encounter two voltage sources in series with different polarity direction, just compare their values from each other and determine which is the greater one. Then you subtract the greater value with the lesser value. After getting the difference of the two voltage sources, redraw them into one source having the same polarity direction as the greater one.

Take this one for example. Two voltage sources in series with equal polarity direction. Combine them by adding their values together (V = V1 + V2) and redraw. The polarity of the new voltage source should be the same as the two that were fused.

For this case, determine if V1 is greater than or less than V2. If V1 is greater, then do Vs = V1 – V2 with the polarity of the new voltage source the same as V1. Else, if V2 is greater, then do Vs = V2 – V1 with the polarity of the new voltage source equal to V2. In the picture above, it is shown that V2 is assumed to be greater than V1 which is why the polarity of Vs is of the same as V2.

The same principle applies to current sources but in parallel, instead of in series. Multiple current sources in parallel can be combined into one current source by simply adding their values together, as long as they have the same direction of polarity. If you were to combine two current sources in parallel with different polarity direction, just do the same for voltages sources in series but with opposite polarity directions. Determine which source has the greater value, then subtract the source’s value with the lesser value. Redraw the two into one source having the same polarity direction as the greater one.

Just because they are in parallel, doesn’t mean you combine them like impedances in parallel. Simply add them like they are in series: Is = I1 + I2. Remember that you only do this if they are both pointing at the same direction. Which means the polarity of the new current source should be the same as the two that were fused. Similar to what we did to the voltage sources.

if they are not of the same direction of polarity, find out if I1 is greater than or less than I2. Then use Is = I1 – I2if it is greater, and redraw with the same polarity of I1. Otherwise, Is = I2 – I1 with the same polarity of I2. It is also assumed in this illustration that I2 is greater than I1, resulting in the polarity of Is and I2 being identical.