32.3.2 Filters

FIGURE 32.12a is the circuit we’ve just analyzed; the only difference is that the capacitor voltage $v_{C}$ is now identified as the output voltage $v_{out}$ . This is a voltage you might measure or, perhaps, send to an amplifier for use elsewhere in an electronic instrument. You can see from the capacitor voltage graph in Figure 32.11 that the peak output voltage is $V_{ out} \approx ε_{0}$ if $ω ≪ ω_{c}$ , but $V_{ out} \approx 0$ if $ω ≫ ω_{c}$ . In other words,

If the frequency of an input signal is well below the crossover frequency, the input signal is transmitted or passed with little loss to the output.
If the frequency of an input signal is well above the crossover frequency, the input signal is strongly attenuated and the output is very nearly zero.

This circuit is called a low-pass filter.

The figure labeled (“a”) illustrates a circuit diagram of a low pass filter.

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The figure labeled (b) illustrates a circuit diagram of a high pass filter.

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The circuit of FIGURE 32.12b, which instead uses the resistor voltage $v_{R}$ for the output $v_{out}$ , is a high-pass filter. The output is $V_{ out} \approx 0$ if $ω ≪ ω_{c}$ , but $V_{ out} \approx ε_{ 0}$ if $ω ≫ ω_{c}$ . That is, an input signal whose frequency is well above the crossover frequency is transmitted without loss to the output.

Filter circuits are widely used in electronics. For example, a high-pass filter designed to have $f_{c} = 100 Hz$ would pass the audio frequencies associated with speech $(f > 200 Hz)$ while blocking 60 Hz “noise” that can be picked up from power lines. Similarly, the high-frequency hiss from old vinyl records can be attenuated with a low-pass filter, allowing the lower-frequency audio signal to pass.

A simple $R C$ filter suffers from the fact that the crossover region where $V_{R} \approx V_{C}$ is fairly broad. More sophisticated filters have a sharper transition from off $(V_{ out} \approx 0)$ to on $(V_{ out} \approx ε_{ 0})$ , but they’re based on the same principles as the $R C$ filter analyzed here.

EXAMPLE 32.4 Designing a filter

For a science project, you’ve built a radio to listen to AM radio broadcasts at frequencies near 1 MHz. The basic circuit is an antenna, which produces a very small oscillating voltage when it absorbs the energy of an electromagnetic wave, and an amplifier. Unfortunately, your neighbor’s short-wave radio broadcast at 10 MHz interferes with your reception. Having just finished physics, you decide to solve this problem by placing a filter between the antenna and the amplifier. You happen to have a 500 pF capacitor. What frequency should you select as the filter’s crossover frequency? What value of resistance will you need to build this filter?

MODEL You need a low-pass filter to block signals at 10 MHz while passing the lower-frequency AM signal at 1 MHz.

VISUALIZE The circuit will look like the low-pass filter in Figure 32.12a. The oscillating voltage generated by the antenna will be the emf, and $v_{out}$ will be sent to the amplifier.

SOLVE You might think that a crossover frequency near 5 MHz, about halfway between 1 MHz and 10 MHz, would work best. But 5 MHz is a factor of 5 higher than 1 MHz while only a factor of 2 less than 10 MHz. A crossover frequency the same factor above 1 MHz as it is below 10 MHz will give the best results. In practice, choosing $f_{c} = 3 MHz$ would be sufficient. You can then use Equation 32.15 to select the proper resistor value:

\begin{array}{l} R & = \frac{1}{ω_{c} C} = \frac{1}{2 π (3 \times 10^{6} Hz) (500 \times 10^{- 12} F)} \\ = 106 Ω \approx 100 Ω \end{array}

REVIEW Rounding to 100 $Ω$ is appropriate because the crossover frequency was determined to only one significant figure. Such “sloppy design” is adequate when the two frequencies you need to distinguish are well separated.