### Berkeley Electronic Interfaces Course – Capacitor revisited

We have previously talked about using a capacitor in our amplifier module. However we didn’t mention about the frequency behavior of this circuit. As we know, the voltage output of some RC circuit is frequency dependent. To make things easier to analysis, we need to talk about phasor – or phase vector first.

# What is Phasor?

For now we just need to remember that phasor is introduced to simplify calculation. Phasor is a complex number representing a sinusoidal function whose amplitude (A), angular frequency (ω), and initial phase (θ) are time-invariant. Basically speaking it is brought to us by this way, as we all know:

$v(t) = R \cdot i(t)$

where v and i are the voltage and current of some current with resist R and they are time dependent variables. And now let’s for now just switch R as Z, which is called impedance, later you will see why it is convenient to do that here. For now we simply assume Z is something like R. (Or for resisters only, Z=R).

$v(t) = Z \cdot i(t)$

And, as we know those time depended v and i can simply be presented as a cos(ωt + θ) function, (or actually, you can use a group of an infinite number of cos function to linearly add up to form any time dependent wave function you may have), together with Euler’s equation, we got something look like this:

$\Re \lbrace \mathbb{V} e^{j \omega t} \rbrace = \Re \lbrace Z \cdot \mathbb{I} e^{j \omega t} \rbrace$

where phasor $\mathbb{V} = V e^{j \theta}$ means the phasor is a combination of amplitude and initial phase. Big $\Re$ means for taking the real number value of the complex value inside the brackets. Removing it together with the time dependent part we get:

$\mathbb{V} = Z \cdot \mathbb{I}$

$Z=R+jX$

So we can see in phasor world we can do simple calculations with phasors as if we used to do calculation with voltage, current and resistors. Noticing the R is actually resistance and X is reactance, which describe the energy storage characteristic in the system.

So how we can conclude now for resistor, capacitor and inductor, Z is represented as:

$Z_R = R$

$Z_C = {{1} \over {j \omega C}}$

$Z_L = j \omega L$

So now we have a group of tools to mathematically describe things easily, since the following rules are apply:

Components in serial $Z_{eq} = Z_1 + Z_2 + Z_3 ...$

Components in parallel ${1 \over Z_{eq}} = {1 \over Z_{1}} + {1 \over Z_{2}} + {1 \over Z_{3}} ...$

## Example – RC circuit – Low Pass Filter

Now we see an example how to utilize the above math component. Consider the voltage Vc, it can be calculated as if the capacitor is like a resistor in the same position. (Note, later on big letters are all presented as complex numbers, or phasors.)

$V_C = {Z_C \over {Z_C+Z_R}} V_S$

$V_C = {1 \over {1 + j \omega RC}} V_S$

or

$V_C = H \cdot V_S;$ $H={1 \over {1 + j \omega RC}}$

Notice that |H| is the magnitude and it changes with frequency. And when the frequency is low as zero, there is output on Vc, when frequency goes higher and higher, the output on Vc goes lower and lower. This is called a low pass filter. And we defined the so called cutoff frequency of the filter as when the magnitude of power reduce by half, or the magnitude of H reduce by around 0.707. And one can do some calculation to prove that when let

$|H| = {1 \over \sqrt{2}} |H(0)|$

one gets the cutoff frequency as:

$\omega = {1 \over {RC}}$

## Example – RC circuit – High Pass Filter

Similarly, for the high pass filter, we have:

$H = {j \omega RC \over j \omega RC + 1}$

And the cutoff frequency as:

$|H| = {1 \over \sqrt{2}} |H(\infty)|$

$\omega = {1 \over {RC}}$

# Experimenting with real circuit

So now let’s try built some RC circuit on our own and try to measure the H magnitude changes with frequency, also to see the whether the cutoff frequency are correct with our equation above.

We have two setup cases here, one with a different resistor but with the same capacitor, parameters as below:

 Case R C Cutoff freq 1 2.7k Ohm 1uF 58.95 Hz 2 300 Ohm 1uF 530.52 Hz

2.7k Ohm, 1 uF, fc=59 Hz

2.7k Ohm, 1 uF, fc=59 Hz

300 Ohm, 1 uF, fc=59 Hz

300 Ohm, 1 uF, fc=59 Hz

Wala, the measured results fit so well with the theoretical calculation. I am happy 🙂