# Impedance/Reactance of capacitors

## Capacitive Reactance
Is a measure of a capacitors opposition to alternating current.

$Xc$  in $\ohm$

$X_{c} = \frac{1}{2 \pi fC}$
$Xc = \textit{Capacity in } \ohm$
f = Frequency in Hertz
C = Capacitance in Farads

![](graphXc.gif)

Higher Frequence $\Rightarrow$ Lower Current Flow
Higher Capacitance $\Rightarrow$ Lower Current Flow

When the Frequency is 0, the capacitor acts as an open circuit
When the Frequency is really high, the capacitor is equal to a simple wire

**Example:**

Calculate the capacitive reactance of a 220nF capacitor at a frequency of 1kHz and 20kHz

$$
\begin{flalign}
&X_{c} = \frac{1}{2 \pi * 1000 * 220 * 10^{-9} } \\
&X_{x} \approx \textbf{723.43} \ohm\\
\\
&X_{c} = \frac{1}{2 \pi * 20000 * 220 * 10^{-9} } \\
&X_{x} \approx \textbf{36.17} \ohm\\
\end{flalign}
$$

Here we can see when the frequency increases the reactive capacitance decreases

**Example 2:**

```circuitjs
$ 1 0.000005 10.20027730826997 50 5 43 5e-11
v 208 256 208 144 0 1 80 5 0 0 0.5
r 208 144 336 144 0 100
c 336 144 336 256 0 0.000029999999999999997 -2.4446139526159825 0.001
w 336 256 208 256 0
```

How would we calculate the $I_{rms}$ of this circuit, we'll basically using Ohms Formular

$$
I_{rms} = \frac{V_{rms}}{R1+X_{c}}
$$
The Problem is, we can't just simply add up R1 and Xc, because Xc is shifted by 90°. We need to add them up as Vectors:

$$
R_{e} = \sqrt{R1^2+X_{c}^2}
$$

Lets fill in the numbers from the circuit above and test it out:

$$
\begin{flalign}
&X_{c} = \frac{1}{2 \pi * 80 * 30 * 10^{-6}} &&\\\
&X_{c} \approx 66.3 \ohm \\
&V_{rms} = 3.5v \\
\\
&I_{rms} = \frac{3.5}{\sqrt{100^2+66.3^2}} \\
&I_{rms} = \frac{3.5}{119.98} \\
&I_{rms} = 0.029171033 A \\
&I_{rms} \approx 29.17mA

\end{flalign}
$$


## Reality
In reality capacitors are not perfect, they are more like:
![](rlc-capacitor.svg)

So the have a $ESR$ and $X_{C}$ and $X_{L} / ESL$

$$
C_{IMP} = ESR + X_{C} + X_{L}
$$

Due to this the frequency to impedance curve of real capacitors look something like this.

![](EMC-9_graf_01.gif)

When we add multiple capacitors we can get a curve looking like this

![](rlc-capacitor-multiple.png)