Sometimes in electronics you only want certain frequencies of a
signal to pass -- an example would be an audio system. Usually by design,
the lower frequency signals are passed to the subwoofer (giving it the deep
bass), while the higher frequencies are passed to the tweeter (higher dynamic
pitch). This post will touch base on two different passive filters:
low-pass, and high-pass.

Before you start,
let's review what a capacitor and inductor looks like at various frequencies.

The impedance of a
capacitor is defined as:

Sometimes displayed as:

Right off the bat you should notice that a capacitor looks like a
variable resistor across frequency that is inversely proportional to its
capacitance and frequency value. Two
important frequencies to note are f = 0 Hz and f = ∞ Hz. At f = 0 Hz the impedance is infinite, which
is the same as an open circuit. When f =
∞ Hz, the impedance value is 0 Ω, which is a short.

The impedance of
an inductor is defined as:

Sometimes displayed as:

For the inductor case, at f = 0Hz, the impedance is 0Ω, which is a short. When the frequency is infinite, f = ∞ Hz, the
impedance is infinite, which is the same as an open circuit.

__Lowpass Filter__
As the name
suggests, this passive filter only allows low frequencies through. A single pole low-pass filter is illustrated
in Figure 1.

Figure 1. Low-Pass Filter

Before you derive the transfer function think about this circuit
from another point of view. You can develop an intuitive understanding of
the circuit when you analyze it around two frequency values, 0Hz and ∞ Hz.

At 0Hz the capacitor acts like an open-circuit, therefore Vout =
Vin. At ∞ Hz the capacitor resembles a
short, and Vout = GND = 0V.

The transfer function of
this circuit is shown in Figure 2.

Figure 2. Transfer function of low-pass
filter

The next step is to plot the magnitude and phase of this transfer
function.

Magnitude:

Phase:

__Highpass Filter__
This filter only
allows higher frequencies through. A single pole high-pass filter is displayed
in Figure 3.

At 0Hz the capacitor acts like a short, therefore Vout = GND =
0V. At ∞ Hz the capacitor resembles an
open-circuit, and Vout = Vin.

Figure 3. High-Pass Filter

The transfer function for Figure 3 is shown below.

Figure 4. Transfer function of high-pass
filter

Magnitude:

Phase:

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