**Capacitive impedance** is the ratio of *phasor voltage* to *phasor current* in a *capacitor*. It also means the opposition of a capacitor to an alternating current. Impedance of a capacitor is dependent on the frequency of the voltage or current across it.

**What is Impedance?**

When electrical current flows through a material, it meets resistance primarily due to the nature of the material. *Conductors* will resist less and *insulators* will resist more. Voltage pushes the current; the stronger the voltage, the higher the current. And since the higher the resistance, the lesser the current, we have this math relationship:

This is popularly known as * Ohm’s Law*. Here, voltage and current are

*direct*and have

*magnitude*only.

However, *alternating *current and voltages both have magnitude and *phase* since they are sine/cosine quantities. The equation for household AC mains, for example, is :

**Phasors **are invented to avoid exhausting calculations of sinusoidal equations. In circuit theory, sinusoids are functions of time and are therefore *time-domain *quantities.

Phasors are **complex numbers** and can be written in rectangular or polar form.

Here, *X* is the *real part* and *Y* is the *imaginary part*. * j* is the imaginary number equal to √-1.

**j**instead of i in circuits to avoid confusion with the symbol for current.

In polar form, *A* is the magnitude and *ϴ* is the phase.

Conversion between two phasor forms are as follows:

*Rectangular to Polar:*

*Polar to Rectangular:*

A cosine equation easily converts to polar phasor:

And so instead of writing AC voltage like this:

We can just have it like this:

Note that phasor quantities are often in bold, capital letters.

Also instead of dealing with this:

We can just use phasor multiplication which is much simpler:

But if voltage and current are now phasors, does Ohm’s Law’s equation hold true? The answer is **no** since you now have both magnitude and phase:

So, what do we now call the ratio of phasor voltage and current? Not resistance but **impedance (Z).**

**Impedance and Reactance**

If an impedance has no phase difference:

In rectangular form,

As you can see, impedance becomes equal to resistance. In fact, in a *resistor*, there is no phase difference between the current and the voltage.

What if the phase is non-zero? Remember that the imaginary component sinϴ will only be zero if ϴ is zero (or 180 but it’s supplementary to 0). Thus, there will always be an imaginary component in impedance if the phase is not zero.

This *imaginary component* of impedance is called **reactance**. Moreover, the *real part* of impedance is the resistance.

**Capacitive Impedance**

Now with a capacitor, the voltage and current have a relationship of:

The equation is due to the ability of the capacitor to hold electric charge. This means that the current in a capacitor is proportional to the change of voltage in a *given point in time*. In short, a steady voltage produces no current across a capacitor.

Now besides sine/cosine equations, you can also convert differential equations to phasor too.

Here, ω is the radian frequency of the voltage or current.

So, the current-voltage differential equation for capacitors now becomes:

Solving for the ratio of phasor voltage and current:

Hence, the impedance of the capacitor is:

As you can see, this value changes when the radian frequency changes. The higher the value of ω, the lower the Z and vice versa.

Finally, since the impedance of a capacitor is an imaginary number, it is also called **capacitive reactance**.

You can also write the above equation like this:

Converting to polar form,

This means that in a capacitor, the voltage is ahead of current in phase by 90°.