Phasor – Why Inductive or Capacitive Reactance Phasor is on the Imaginary Axis


As you can see there are electrical circuits of power supply, resistance, inductive reactance combination in one case and power supply resistance and capacitive reactance combination in another case are there.
Their respective phasor diagrams are also drawn.

But why is the inductive reactance or capacitive reactance phasor on imaginary axis while the resistance phasor is taken on the real axis? What will happen if we take resistance as the imaginary component and reactance as the real component?

Phasor diagram

Best Answer

In a resistive element, current and voltage are in phase with each other. However for an inductive element the voltage leads the current by \$90^\circ\$, and for a capacitive element voltage lags current by \$90^\circ\$.

So lets look at how we define impedance and why. We define impedance as:

$$Z = R + jX$$

Now an impedance respects Ohms law, so what we are saying is:

$$V = ZI=RI+jXI$$

When the reactance is zero, you can see we are left happily with the Ohms law we all know and love:

$$\begin{align} V_r&=RI+j0I\\\\ V_r&=IR\\ \end{align}$$

So that works. Now what about when resistance is zero. We get:

$$\begin{align} V_x&=0I+jXI=jXI\\\\ V_x&=|X|\angle90^\circ\times I\\ \end{align}$$

We can see now that the current and voltage must be \$90^\circ\$ out of phase in order to satisfy this equation. Great, that is what we needed too. So basically this formation of impedance matches what we require.

So lets look at what you said in a comment to @Barry. Why not define impedance as:

$$Z = X + jR$$

Well, lets go through the derivations again. From Ohms law:

$$V= ZI = XI + jRI$$

So, lets first look at what happens when reactance is zero:

$$\begin{align} V_r = 0I + jRI = jRI\\\\ V_r = R\angle90^\circ\times I \ne IR\\ \end{align}$$

Now we have a big problem. We have just said that current and voltage must be out of phase by \$90^\circ\$. But as we well know this is not the case. So clearly the impedance equation cannot correctly be expressed in this form.

If you want to put the resistive part on the imaginary axis, you simply rotate both the voltage and current by 90 degrees. You don't however change the impedance equation.

Ohms law in effect becomes:

$$jV = jIZ$$

Substituting in the correct impedance equation we get:

$$jV = jI(R + jX) = jIR - IX$$

This is now perfectly valid. The resistance remains a real number meaning that voltage and current remain in phase - we see this by again setting the reactance to 0, resulting in:

$$jV=jIR \rightarrow V=IR$$

In fact this shift doesn't have to be by 90 degrees - you can shift the Ohms law equation by any arbitrary angle and it still holds true:

$$V\angle35^\circ=(I\angle35^\circ\times R) \rightarrow V=IR$$