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Most electrical filters are circuits that select certain bands of frequencies to pass along, or accept, and other bands of frequencies to stop, or reject. The frequency at which the transition between passing and rejecting input signals occurs is called the “cutoff” frequency, often abbreviated as fC. The band of frequencies that is passed is called the “passband” of the filter, also called the “bandwidth.” We will be concerned with filters that operate on signal voltages. There are other types of filters that manipulate the phases of signals that pass through them, but we are not going to cover these in this lab. The simplest filters are constructed of two passive elements -- either a resistor and capacitor (RC), or a resistor and inductor (RL). Each of these two-element filters can be arranged to pass low frequencies and reject high frequencies, or pass high frequencies and reject low frequencies, depending upon which element the output voltage is taken across. Not surprisingly, the filters that pass low frequencies are referred to as “lowpass” filters and the filters that pass high frequencies are referred to as “highpass” filters.
A commonly-encountered arrangement of lowpass and highpass filters can be found in the crossover networks in two-way loudspeaker systems consisting of a low frequency driver called the woofer and a high frequency driver called the tweeter. The lowpass filter passes low frequencies to the woofer, which is designed to reproduce these frequencies, and rejects the high frequencies that the woofer cannot reproduce. Similarly, the highpass filter passes high frequencies to the tweeter, which it is designed to reproduce, and blocks the low frequencies that the tweeter cannot reproduce, or could even damage the tweeter. Ideally, the overall audio output from the loudspeaker system will cover the entire audio range, divided between the woofer and tweeter.
In this experiment we design and build a lowpass RC filter and a highpass RC filter using components available in the ADALP2000 Analog Parts Kit. The cutoff frequency of the lowpass filter is designed to be approximately 100 Hz, and the highpass filter cutoff frequency is designed to be approximately 200 Hz. It is important to note that the lopwass filter passes DC -- the ultimate low frequency signal -- while the highpass filter does not.
After building and testing the two filters we will construct two very simple circuits that detect when signals are approximately in each filter's passband, and drive a LED to indicate this. We can then connect the two filters together and sweep the frequency of the input signal from a very low frequency that is in the lowpass filter's passband but not in the highpass filter's passband up to a high frequency that is not in the lowpass filter's passband but is in the highpass filter's passband. We can then observe the LEDs switch on and off as a function of the input frequency.
To study RC lowpass and highpass filters and design a simple circuit to detect the approximate amplitude of a signal and illuminate a LED when the amplitude is above a predetermined threshold. Upon completion of this lab you should be able to describe the basic operation of RC lowpass and highpass filters, describe the operation of a simple low-accuracy peak detector, explain what a voltage comparator is, and explain how the comparator output can drive a LED.
Series RC circuits can realize the simplest lowpass and highpass filters that operate on voltages, though current-mode operation is also possible. The filter is lowpass when the output voltage is taken across the capacitor and is highpass when the output voltage is taken across the resistor. The filter operates as a two-element voltage divider between the resistance of the resistor, R, and the reactance of the capacitor, 1/2πfC. Since the capacitive reactance varies inversely with frequency, the voltage across the capacitor decreases with frequency, producing the lowpass frequency response. By Kirchhoff's Voltage Law, the voltage across the resistor increases with frequency in a complementary fashion to that of the capacitor, producing the highpass response. The cutoff frequency fC is defined as the frequency at which the capacitive reactance is equal to the resistance. Setting R = 1/2πfCC and solving for fC gives the following result.
At the cutoff frequency, the amplitude of the sinusoidal output voltage in response to a sinusoidal input voltage is equal to the reciprocal of the square-root of two times the input voltage amplitude, or about 70.7% of the input voltage amplitude.
The lowpass filter in the lab is comprised of a 68 Ω resistor and a 22 μF capacitor, and therefore has a cutoff frequency of approximately 106 Hz. Because of the tolerances of the resistor and capacitor, the actual cutoff frequency will deviate from this value. The frequency at which the peak-to-peak output amplitude was measured to drop from about 5 V to (0.707)*5 V ≈ 3.5 V in the lowpass filter part of the lab should have been close to 106 Hz.
The highpass filter in the lab is comprised of a 68 Ω resistor and a 10 μF capacitor, and therefore has a cutoff frequency of approximately 234 Hz. As with the lowpass filter, component tolerances will introduce a small cutoff frequency error. The frequency at which the peak-to-peak output amplitude was measured to drop from about 5 V to (0.707)*5 V ≈ 3.5 V in the highpass filter part of the lab should have been close to 234 Hz.
It's important to note that the resistor in the highpass filter is referenced to +2.5 V instead of ground. The highpass filter cannot pass DC, and it can therefore be viewed as a circuit the removes the DC level from its input signal. The DC level on the other side of the resistor sets the DC level on the output side of the highpasss filter as long as the output does not have a significant DC load. This is true because DC current cannot flow back through the capacitor and there is no significant load current, so the DC voltage drop across the resistor in the highpass filter is effectively zero. Placing the output DC level at 2.5 V places the output swing in the center of the M1K input range. If the resistor were referenced to ground, the highpass filter output would swing above and below ground, and the negative excursions would not be visible on the M1K.
After making basic characterizations of the filters, simple, rather inaccurate, peak detectors consisting of series diodes and shunt capacitors are added to the filter outputs in order to give a rough indication of the filter output level. The peak detector allows the capacitor to charge when the voltage out of the filter is greater than the capacitor voltage plus the forward diode drop. The diode is a nonlinear element with a varying forward voltage drop, so the capacitor will not be able to charge to the actual peak voltage out of the filter. Once the signal out of the filter passes its peak, the diode becomes reverse-biased and conducts very little current, allowing to the capacitor to hold a rough estimate of the peak level that includes the error due to the diode drop. The capacitor voltage does not stay perfectly constant while the diode is reverse biased because the diode leaks current when reverse biased, and current leaks into the M1K when the when the capacitor voltage is measured. The current leakages cause the capacitor voltage to sag during reverse-biased parts of the cycle. The sag is more noticeable for low frequency inputs when the reverse-biased times are long. This brings up an important dilemma that is encountered in all peak detectors. A small capacitor is desirable for fast charging times required in detecting the peaks of high frequency signals, but a large capacitor is required to achieve long hold times for low frequency signals. Ultimately, a compromise must be reached in selecting the capacitor value in a given application. Much more accurate peak detectors can be built using negative feedback circuits that remove the diode drop error from the measurement.
Since these peak detectors include errors, it is best to measure their outputs for various levels output from the filters. In the lab, frequencies roughly a quarter-decade apart are used to characterize the peak detectors. One of these frequencies is chosen to be approximately equal to the cutoff frequency of the respective filter -- 100 Hz for the lowpass filter and 200 Hz for the highpass filter. These peak detector output levels are used to set the threshold levels of the comparators that detect when the signals are in the filter passbands and drive the corresponding LEDs.
A comparator is a high-gain amplifier with a differential input and a binary output that is used to detect when input signals are above or below a predetermined threshold voltage. A comparator is similar to an operational amplifier that is set up in an open-loop configuration, though operational amplifiers should not be used as comparators for a number of reasons. There are many comparators on the market that are specifically designed to be operated open-loop and provide a specific type of digital logic output. Comparators can also be operated in a closed-loop positive-feedback configuration to produce hysteresis. Hysteresis is a feature in which the operation of the comparator depends on its history. It provides two thresholds -- a higher one for increasing inputs and a lower one for decreasing inputs. One of the major benefits of hysteresis is that it prevents the comparator output from “chattering” when the input signal is changing very slowly about the threshold. Hysteresis would be a nice feature in this application if the input frequencies were hovering near the filter cutoff frequencies, but it was not included for simplicity's sake. It is introduced and explained in the “Simple Proximity Detector” lab.
The comparators are set up with threshold voltages on their inverting inputs. The output logic goes to a “high” level when the voltage on the non-inverting input exceeds the threshold voltage by a small amount. We want to drive the LED when the threshold is exceeded, indicating that the input signal is in the filter passband, and it is best to drive the LED when the comparator output is in the “low” logic state. The AD8561 has true and complimentary outputs. The true output goes high when the input voltage exceeds the threshold voltage is exceeded and low when the input voltage is below the threshold voltage. The complementary output beaves in the opposite fashion. Since the complementary output goes “low” when the input voltage exceeds the threshold voltage, it is used to drive the LED. If the comparator only had a true output, the same result could be achieved by swapping the input signals, i.e., placing the threshold voltage on the non-inverting input and the input signal on the inverting input.