Saturday, April 18, 2015

You shall not pass!

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 , 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 0, 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.



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



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