Why is the inductive reactance or capacitive reactance phasor on the imaginary axis?

Question

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?

Answer 1

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:

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

Answer 2

• First of all I suggest that you review what a phasor actually is. Reactances (or impedances in general) are never phasors. Just because they are complex quantities doesn't make them phasors.

  Now what is a phasor? A phasor is a complex quantity that has a \$e^{j\omega t}\$ term cancelled as a result of the phasor transformation, e.g. a voltage \$V(t) = V e^{j(\phi + \omega t)} = V e^{j\phi} e^{j\omega t}\$ → \$V e^{j\phi} = V_{re} + j V_{im}\$.
  The "→" is the phasor transformation that drops the \$e^{j\omega t}\$ resulting in a complex quantity that is time independent (which makes further handling easier).

  Although impedances \$Z\$ may be complex quatities they are not the result of a phasor transformation and therfore are not phasors.

• Now back to your original question that I assume should just be
  "Why are the inductive impedances or capacitive impedances imaginary?".

  Answer: That is just the result when you expose
  an inductor (with voltage/current relationship \$V(t) = L \frac{d}{dt}I(t)\$) or
  a capacitor (with voltage/current relationship \$V(t) = \frac{1}{C}\int I(t) dt\$)
  to a sinusoidal source (i.e. voltage source of the form \$V(t) = V_{src} e^{j(\phi_{src} + \omega t)}\$ or similar current source) and apply KCL and/or KVL.

  Only then you can apply phasor analysis and miraculously all the \$e^{j\omega t}\$ terms can be canceled and the terms containing \$L\$ and \$C\$ "automatically" become pure imaginary constants \$Z_L = jX_L = j\omega L\$ because \$L\frac{d}{dt}Ie^{j\omega t} = j\omega LIe^{j\omega t}\$ i.e. the operator \$L\frac{d}{dt}\$ is the same as multiplication by \$j\omega L\$ (and similarly for capacitances).

  That way you get a simple time independend equation for voltage and current \$V = Z I\$ (where \$V\$ and \$I\$ are phasors, and \$Z\$ is a complex quantity) that looks the same as the equation for a circuit with a resistor and a DC source: Ohm's Law \$V = RI\$ (where all three \$V\$, \$I\$ and \$R\$ are real quantities).

Answer 3

In a resistor voltage and current are in phase so to imply that a resistor's impedance is totally imaginery is missing the point. In capacitors and inductors, the voltage and current are 90 degrees apart so this naturally means that the impedance expressed in polar co-ordinates is totally imaginery.

I can't begin to see a counter argument to this straightforward way of looking at things.

Answer 4

You might be able to construct such a system and make it work.

But the fact that the physically real power dissipated as heat in a resistance appears on the real axis, makes the system we currently use, vastly more convenient and simpler to use.

Andy's point is more fundamental : the phase between voltage and current is zero in a resistance, so V=I*R (Ohm's Law) works and is useful in situations where you can neglect reactances altogether. Then, P = V*I = V^2/R = I^2*R describes this real power directly.

Then, taking reactance on the imaginary axis allows it to be described and calculated in a manner completely consistent and backwards compatible with basic Ohm's Law.

So there is nothing to be gained from changing axes when moving from resistive to reactive calculations, and a lot of simplicity to be lost.

Answer 5

I don't know your level of understanding of the concept of impedance. In any case, the properties of capacitors and inductors are such that, for both of them, the current through them and the voltage across them are 90 degrees out of phase. Since the real and imaginary axes represent quantities that are 90 degrees out of phase, it is natural to express reactance on the imaginary axis and resistance on the real axis.

Answer 6

You need to consider what the phasor represents. For a general AC circuit, the phasor is a way of representing a sinusoidally (sine wave) varying input signal. The phasor has fixed length in phasor space, and rotates counterclockwise at a set frequency. The actual voltage or current is the real component of the phasor, or the projection of the phasor onto the real axis. We use the phasor as a tool to tell us how the RLC circuit responds to the specific input voltage/current and frequency.

So when you ask why is the resistive component on the real axis and the capacitive/inductive component on the imaginary axis, that's just because we choose an instant in time when they are aligned that way. Look at the circuit a moment later, and the voltages and currents that we measure will change as the phasor rotates.

Consider the comparison between circular motion and simple harmonic motion. Take the projection of the circular motion onto a line, and the result is simple harmonic motion. The phasor is like the circular motion, and the output voltage/current is like the simple harmonic motion.

Now, for a resistor, the voltage and current are in phase, so V = I R, as Ohm's law tells us. For the capacitor, the current leads the voltage by 90 degrees, and for an inductor, the voltage leads the current by 90 degrees.

All that said, the answer to your question is the reason we take the resistance on the real axis is convenience. The phasor rotates over time, and we're looking at a snapshot when we assign the resistance to the real part.

A classic physics question is to take a RLC circuit with a fixed AC voltage (or current) applied, and then ask what the current (voltage) is when the instantaneous voltage (current) is some given value. The first real step in the problem is to determine the instantaneous angle of the phasor to get the given voltage (current) value. I've assigned this kind of problem regularly on homeworks and tests (and the students just LOVE me for it).

Answer 7

In addition to the more mathematically rigorous comments regarding the phase angles between the voltage and current in reactive and non-reactive components, it helps to remember that an ideal inductor and an ideal capacitor do not dissipate energy (they accumulate and release energy), while an ideal resistor does dissipate energy.

In my own mind, I like think of the imaginary values on the Y-axis as representing the energy that's just "sloshing around" in the circuit's reactive components and is not being consumed—i.e., electric field energy or magnetic field energy that is accumulated and released by capacitors and inductors, respectively—while the real values on the X-axis represents energy that's actually being consumed (e.g., a resistor that converts electrical energy into heat). Obviously, this is not a mathematically rigorous definition; it's more a just a memory aid that people sometimes find useful.

Answer 8

The best answer is actually the comment by Chu.

Remember a reference is only an arbitrary decision by someone, usually to help understanding. So we select references for AC circuits to make it easier to understand.

Current is constant in a series circuit. So current is used as a reference. This is where students start. We choose current as the reference to simplify calculations.

By selecting current (\$I\$) as the reference, \$V_R\$ will be along the x-axis (no y-component). and \$V_C\$ and \$V_L\$ will be along the y-axis (no x-component). This simplifies circuits and calculations to vector addition (no need to break vectors into components) and allows circuits to be solved using right triangles and trigonometry or complex numbers. The alignment allows complex numbers to be used and complex numbers simplifies calculations as circuits become more complex.

$$ \overrightarrow {V_S} = \overrightarrow {V_R}\ +\ \overrightarrow {V_L}\ +\ \overrightarrow {V_C}$$

$$ V_S = \sqrt{(V_R)^2\ +\ (V_L - V_C)^2}\ \measuredangle\ arctan\ \left ({\frac {V_L - V_C} {V_R}} \right ) $$ $$ V_S = V_R\ +\ j (V_L - V_C)$$

For a parallel circuit, voltage is constant, so the source voltage \$V_S\$ is chosen as reference, and vector addition now becomes addition of currents.

$$ \overrightarrow {I_S} = \overrightarrow {I_R}\ +\ \overrightarrow {I_L}\ +\ \overrightarrow {I_C}$$

By changing the reference, circuit can be represented by right triangles and calculations can be simplified. The reference is chosen to make understanding simpler.

For a series/parallel circuit, the source voltage \$V_S\$ is typically chosen as reference, but the vectors typically do not align with the axis. Hopefully the student has some knowledge by that point.

As for the question. \$V_R\$ (or \$I_R\$) along the x-axis represents Real Power, something we get out of the circuit (heat, mechanical power, etc.). This (in my mind) has always corresponded to real numbers.

For a circuit with a capacitor and an inductor, leading capacitive reactive power supplies lagging inductive reactive, decreasing the reactive power the source must supply [vertical leg of right triangle is less than \$V_L\$ (or \$V_C\$)]. Power flows back and forth between capacitor and inductor. Reactive power is required to make the circuit work (create a magnetic field in a motor), but does no useful work. It appears we get something for nothing, which corresponds as imaginary.

So reactances (reactive power) along imaginary and resistance (real power) along real.

Answer 9

The first reason I want to clarify is that, the resistance being the real part of (complex) impedance and the (magnitude of the) reactance being the imaginary part of the (complex) impedance, is a fact, not a convention/rule as some answers may suggest. Before answer the why of your question, I'll try to convince you what I said is true. If you already know all of this, just head to the last four paragraphs of the answer.

I'll assume you're familiar with resistors, inductors and capacitors, sinusoids, and phasors.

—In a resistor of constant resistance, inductor of constant inductance, and capacitor of constant capacitance, it can be mathematically proven or physically demonstrated that instantaneous voltage \$v(t)\$ - instantaneous current \$i(t)\$ relationship is given by:

\$v(t) = R \, i(t) \quad \text{(resistor)} \tag 1\$

\$v(t) = L \, \dfrac{\mathrm di(t)}{\mathrm dt} \quad \text{(inductor)} \tag 2\$

\$i(t) = C \, \dfrac{\mathrm dv(t)}{\mathrm dt} \quad \text{(capacitor)} \tag 3\$

When those devices are driven by sinusoidal current or voltage, the resulting voltage or current is also sinusoidal, of same fundamental frequency, and with zero DC offset.

—Given a sinusoidal signal \$x(t) = X_\text{m} \cos{(\omega t + \phi)}\$, its phasor is defined as \$\tilde X = X_\text{m} e^{j \phi}\$. Notice that the defining equation that relates a sinusoidal signal to its phasor is \$x(t) = \Re [\tilde X e^{j \omega t}]\$. With this fact, it can be mathematically proven (as I did in this Quora answer) that the phasor voltage \$\tilde V\$ - phasor current \$\tilde I\$ relationship of the three previous devices, provided they're operating with sinusoidal voltages and currents of same frequency, are:

\$\tilde{V} = R \tilde{I} \quad \text{(resistor)} \tag 4\$

\$\tilde{V} = j \omega L \tilde{I} \quad \text{(inductor)} \tag 5\$

\$\tilde{I} = j \omega C \tilde{V} \quad \text{(capacitor)} \tag 6\$

The (complex) impedance \${\hat Z}\$ of a two-terminal passive device/circuit is defined as the ratio of the phasor voltage \$\tilde{V}\$ across the device to the phasor current \$\tilde{I}\$ through the device:

\${\hat Z} := \dfrac{\tilde{V}}{\tilde{I}} \tag 7\$

Thus, from equations (4) to (6) we easily obtain the (complex) impedance of a resistor, inductor and capacitor, as follows (I explained this in more details in this Quora answer):

\${\hat Z} = R \quad \text{(resistor)} \tag 8\$

\${\hat Z} = j \omega L \quad \text{(inductor)} \tag 9\$

\${\hat Z} = \dfrac{1}{j \omega C} = -j \dfrac{1}{\omega C} \quad \text{(capacitor)} \tag {10}\$

—It can be mathematically proven (as I did in this Quora answer; do not read the last section where I talk about the sign of reactance) that the maximum voltage \$V_\text{m}\$ - maximum current \$I_\text{m}\$ relationship of the inductor and capacitor, provided they're operating with sinusoidal voltages and currents of same frequency, are:

\$V_\text{m} = \omega L I_\text{m} \quad \text{(inductor)} \tag {11}\$

\$I_\text{m} = \omega C V_\text{m} \quad \text{(capacitor)} \tag {12}\$

The magnitude of the reactance \$X\$ of an inductor or capacitor operating in sinusoidal steady-state is defined as the ratio of the amplitude/maximum value/peak value \$V_\text{m}\$ of the instantaneous voltage across the device to the amplitude/maximum value/peak value \$I_\text{m}\$ of the instantaneous current through the device:

\$|X| := \dfrac{V_\text{m}}{I_\text{m}} \tag {13}\$

Thus, from equations (11) and (12) we easily obtain the magnitude of the reactance of an inductor and capacitor, as follows:

\$|X| = \omega L \quad \text{(inductor)} \tag {14}\$

\$|X| = \dfrac{1}{\omega C} \quad \text{(capacitor)} \tag {15}\$

—Substituting equations (14) and (15) into (9) and (10), respectively, we get:

\${\hat Z} = j |X| \quad \text{(inductor)} \tag 9\$

\${\hat Z} = -j |X| \quad \text{(capacitor)} \tag {10}\$

Aha! This shows/proves that the reactance is the imaginary part of impedance. This is not a convention. Throughout the derivation we didn't make an arbitrary rule, except for defining (complex) impedance as the ratio of phasor voltage to phasor current, but that's to make it analogous to resistance and thus make the equation look like Ohm's law.

---

Now that I've hopefully convinced you that \$X\$ being the imaginary part of \$\hat Z\$ is not a convention/rule but a proof/theorem, let's address the why of your question.

In brief, it's because of the following.

It can be mathematically proven and physically demonstrated that, in sinusoidal steady-state, in a resistor the instantaneous voltage is in phase with the instantaneous current, in an inductor the instantaneous voltage leads by 90° the instantaneous current, and in a capacitor the instantaneous current leads the instantaneous voltage by 90°.

But, multiplying a complex number by the imaginary unit \$j\$ makes the complex number to be rotated 90° counterclockwise, making the new complex number lead the original complex number by 90° in the complex plane (as I proved in this and this Quora answer).

Thus, it follows that in the phasor relationship for a resistor there mustn't be a \$j\$ present (because the phasor voltage must be in phase with the phasor current), in an inductor there must be a \$j\$ multiplying the phasor current (along with \$\omega\$ and \$L\$) to produce the phasor voltage (because the phasor voltage must lead by 90° the phasor current), and in a capacitor there must be a \$j\$ multiplying the phasor voltage (along with \$\omega\$ and \$C\$) to produce the phasor current (because the phasor current must lead by 90° the phasor voltage).

Therefore, when we compute the (complex) impedance, which is defined as the ratio of phasor voltage to phasor current, we still get no \$j\$ in the resistor, a \$+j\$ in the inductor, and a \$-j\$ in the capacitor.