Reactive power \$Q\$ is a measure of the energy \$E\$ that is oscillating between the generator and the load without loss. Specifically:

$$ E = {Q \over 2 \pi f} $$

or

$$ Q = 2\pi fE $$

where \$f\$ is the frequency.

If you follow the convention of using a negative reactance for capacitors, you might end up with a "negative power". But this is just a mathematical convention to distinguish between a 90° or a -90° phase shift; either way the total energy oscillating between the load and generator is the same.

Derivation:

Let's say the load is just an ideal inductor.

The reactance \$X\$ will tell us the ratio of RMS voltage \$V\$ to current \$I\$:

$$ X = {V/I} \tag 1 $$

And reactive power \$Q\$ is just the product of voltage and current since our load is purely reactive:

$$ Q = V I \tag 2 $$

Reactance is related to frequency \$f\$ and inductance \$L\$:

$$ L = {X \over 2 \pi f} \tag 3 $$

Combine (1) and (2) to get:

$$ I = \sqrt{Q/X} \tag 4 $$

Energy \$E\$ stored in an inductor is a function of the inductance and current \$I\$:

$$ E = 1/2\: LI^2 \tag 5 $$

When current is at a peak, so too is the stored energy in the load inductor. We can convert RMS current to a peak instantaneous current like so:

$$ I_\text{peak} = \sqrt{2} \cdot I_\text{RMS} \tag 6 $$

Let's combine these equations, starting with (5):

$$ E = {1\over 2}\: LI_\text{peak}^2 $$

Sub in equations 3 and 6:

$$ E = {1\over 2}\: {X \over 2 \pi f}\: (\sqrt{2}\: I_\text{RMS})^2 $$

Simplify:

$$ E = {1\over 2}\: {X \over 2 \pi f}\: 2I_\text{RMS}^2 $$

Ohm's law or (4):

$$ E = {1\over 2}\: {X \over 2 \pi f}\: 2(\sqrt{Q/X})^2 $$

Simplify:

$$ \require{cancel} E = {1\over 2}\: {X \over 2 \pi f}\: {2Q\over X } \ = {Q \over 2\pi f} $$

Again, that's the peak energy stored in our load inductor, a purely reactive load.

By laws of energy conservation it should be apparent that when the energy stored in the inductor is at a maximum, energy stored in the generator is zero, and energy can't go anywhere else, so this is also at any time the total energy in the system.

Reactive power \$Q\$ is a measure of the energy \$E\$ that is oscillating between the generator and the load without loss. At any instant energy might be moving from load to generator or the other way, and the energy stored in the load will vary over time. But the sum of the energy in both is constant, since by definition there's no loss.

Specifically:

$$ E = {Q \over 2 \pi f} $$

or

$$ Q = 2\pi fE $$

where \$f\$ is the frequency.

If you follow the convention of using a negative reactance for capacitors, you might end up with a "negative power". But this is just a mathematical convention to distinguish between a 90° or a -90° phase shift; either way the total energy oscillating between the load and generator is the same.

Derivation:

Let's say the load is just an ideal inductor.

The reactance \$X\$ will tell us the ratio of RMS voltage \$V\$ to current \$I\$:

$$ X = {V/I} \tag 1 $$

And reactive power \$Q\$ is just the product of voltage and current since our load is purely reactive:

$$ Q = V I \tag 2 $$

Reactance is related to frequency \$f\$ and inductance \$L\$:

$$ L = {X \over 2 \pi f} \tag 3 $$

Combine (1) and (2) to get:

$$ I = \sqrt{Q/X} \tag 4 $$

Energy \$E\$ stored in an inductor is a function of the inductance and current \$I\$:

$$ E = 1/2\: LI^2 \tag 5 $$

When current is at a peak, so too is the stored energy in the load inductor. We can convert RMS current to a peak instantaneous current like so:

$$ I_\text{peak} = \sqrt{2} \cdot I_\text{RMS} \tag 6 $$

Let's combine these equations, starting with (5):

$$ E = {1\over 2}\: LI_\text{peak}^2 $$

Sub in equations 3 and 6:

$$ E = {1\over 2}\: {X \over 2 \pi f}\: (\sqrt{2}\: I_\text{RMS})^2 $$

Simplify:

$$ E = {1\over 2}\: {X \over 2 \pi f}\: 2I_\text{RMS}^2 $$

Ohm's law or (4):

$$ E = {1\over 2}\: {X \over 2 \pi f}\: 2(\sqrt{Q/X})^2 $$

Simplify:

$$ \require{cancel} E = {1\over 2}\: {X \over 2 \pi f}\: {2Q\over X } \ = {Q \over 2\pi f} $$

Again, that's the peak energy stored in our load inductor, a purely reactive load.

By laws of energy conservation it should be apparent that when the energy stored in the inductor is at a maximum, energy stored in the generator is zero, and energy can't go anywhere else, so this is also at any time the total energy in the system.

Reactive power \$Q\$ is a measure of the quantity of energy \$E\$ that is oscillating between the generator and the load without loss. Specifically:

$$ E = {Q \over 2 \pi f} $$

or

$$ Q = 2\pi fE $$

where \$f\$ is the frequency.

If you follow the convention of using a negative reactance for capacitors, you might end up with a "negative power". But this is just a mathematical convention to distinguish between a 90° or a -90° phase shift; either way the total energy oscillating between the load and generator is the same.

Derivation:

Let's say the load is just an ideal inductor.

The reactance \$X\$ will tell us the ratio of RMS voltage \$V\$ to current \$I\$:

$$ X = {V/I} \tag 1 $$

And reactive power \$Q\$ is just the product of voltage and current since our load is purely reactive:

$$ Q = V I \tag 2 $$

We can calculate the reactance from frequency \$f\$ and inductance \$L\$:

$$ X = 2 \pi f L \tag 3 $$

Energy \$E\$ stored in an inductor is a function of the inductance and current \$I\$::

$$ E = 1/2\: LI^2 \tag 4 $$

When current is at a peak, so too is the stored energy in the load inductor. We can convert RMS current to a peak instantaneous current like so:

$$ I_\text{peak} = \sqrt{2} \cdot I_\text{RMS} \tag 5 $$

Combining all these equations we get:

$$ E = {1\over 2}\: LI_\text{peak}^2 $$

Sub in equations 3 and 5:

$$ E = {1\over 2}\: {X \over 2 \pi f}\: (\sqrt{2}\: I_\text{RMS})^2 $$

Simplify:

$$ E = {1\over 2}\: {X \over 2 \pi f}\: 2I_\text{RMS}^2 $$

Ohm's law:

$$ E = {1\over 2}\: {X \over 2 \pi f}\: 2(\sqrt{Q/X})^2 $$

Simplify:

$$ \require{cancel} E = {1\over 2}\: {X \over 2 \pi f}\: {2Q\over X } \ = {Q \over 2\pi f} $$

Again, that's the peak energy stored in our load inductor, a purely reactive load.

By laws of energy conservation it should be apparent that when the energy stored in the inductor is at a maximum, energy stored in the generator is zero, and energy can't go anywhere else, so this is also at any time the total energy in the system.

Reactive power \$Q\$ is a measure of the energy \$E\$ that is oscillating between the generator and the load without loss. Specifically:

$$ E = {Q \over 2 \pi f} $$

or

$$ Q = 2\pi fE $$

where \$f\$ is the frequency.

If you follow the convention of using a negative reactance for capacitors, you might end up with a "negative power". But this is just a mathematical convention to distinguish between a 90° or a -90° phase shift; either way the total energy oscillating between the load and generator is the same.

Derivation:

Let's say the load is just an ideal inductor.

The reactance \$X\$ will tell us the ratio of RMS voltage \$V\$ to current \$I\$:

$$ X = {V/I} \tag 1 $$

And reactive power \$Q\$ is just the product of voltage and current since our load is purely reactive:

$$ Q = V I \tag 2 $$

Reactance is related to frequency \$f\$ and inductance \$L\$:

$$ L = {X \over 2 \pi f} \tag 3 $$

Combine (1) and (2) to get:

$$ I = \sqrt{Q/X} \tag 4 $$

Energy \$E\$ stored in an inductor is a function of the inductance and current \$I\$:

$$ E = 1/2\: LI^2 \tag 5 $$

When current is at a peak, so too is the stored energy in the load inductor. We can convert RMS current to a peak instantaneous current like so:

$$ I_\text{peak} = \sqrt{2} \cdot I_\text{RMS} \tag 6 $$

Let's combine these equations, starting with (5):

$$ E = {1\over 2}\: LI_\text{peak}^2 $$

Sub in equations 3 and 6:

$$ E = {1\over 2}\: {X \over 2 \pi f}\: (\sqrt{2}\: I_\text{RMS})^2 $$

Simplify:

$$ E = {1\over 2}\: {X \over 2 \pi f}\: 2I_\text{RMS}^2 $$

Ohm's law or (4):

$$ E = {1\over 2}\: {X \over 2 \pi f}\: 2(\sqrt{Q/X})^2 $$

Simplify:

$$ \require{cancel} E = {1\over 2}\: {X \over 2 \pi f}\: {2Q\over X } \ = {Q \over 2\pi f} $$

Again, that's the peak energy stored in our load inductor, a purely reactive load.

By laws of energy conservation it should be apparent that when the energy stored in the inductor is at a maximum, energy stored in the generator is zero, and energy can't go anywhere else, so this is also at any time the total energy in the system.