Here is a summary of the material covered in this chapter:
Tuned Amplifier Characteristics
There are many types of tuned amplifiers. A tuned amplifier may have a lower cutoff frequency (), an upper cutoff frequency (), or both. An ideal tuned amplifier would have zero () gain up to . Then the gain would instantly jump to until it reaches , when it would instantly drop back to zero. All the frequencies between and are passed by the amplifier. All others are effectively stopped. This is where the terms pass band and stop band come from. The gain of the practical tuned amplifier does not change instantaneously, as shown in Figure 17-1. Note that the value of shown in the figure corresponds to a value of.
FIGURE 17-1 Ideal versus practical pass band characteristics.
There are two basic principles we need to establish:
How close a tuned amplifier comes to having the characteristics of an ideal circuit depends on the quality (Q) of the circuit. The Q of a tuned amplifier is a figure of merit that equals the ratio of its geometric center frequency to its bandwidth. By formula:
The relationship between Q, , and BW is illustrated in Figure 17-2. Note that the lower the Q of an amplifier:
The equation in Figure 17-2 is somewhat misleading as it implies that Q is dependent upon the circuits center frequency and bandwidth. In fact, both Q and are dependent on circuit component values. Once Q and are calculated, BW can be found from
The use of these two equations is demonstrated in Examples 17.1 and 17.2, respectively, of the text.
FIGURE 17-2 Bandwidth versus roll-off rate.
You may remember that is the geometric average of and , found as
If the Q of a tuned amplifier is greater than or equal to 2, then approaches the algebraic average of and , designated as . The value of is found as
This relationship between and (when Q 2) is demonstrated in Example 17.3 of the text. One quick way to determine when Q 2 is to compare and BW. When is at least twice the circuit bandwidth, then Q 2.
Tuned amplifiers can be constructed from either discrete components (FETs and BJTs) or op-amps. Discrete tuned amplifiers normally employ LC (inductive-capacitive) circuits to determine frequency response. Op-amps are normally tuned with RC (resistive-capacitive) circuits. Both these types of circuits are investigated in this chapter.
Tuned op-amp circuits are generally referred to as active filters. There are four basic types of active filters:
Note that the concepts of BW, Q, and are primarily associated with band-pass and notch filters. High-pass and low-pass filters are typically described using only the single cutoff frequency. The frequency-response curves for the four filter types are shown in Figure 17-3.
FIGURE 17-3 Active filter frequency-response curves.
There are several terms that are commonly used when describing active filters. The term pole simply means an RC circuit. A one-pole filter has one RC circuit, a two-pole filter has two RC circuits, and so on. The term order defines how many poles a filter has. A second-order filter has two RC circuits or poles. The reason these terms are important is because the number of poles determines the filters gain roll-off rate. The table below illustrates the relationship between poles and gain roll-off rates for a Butterworth-type filter.
Three common types of active filters are the Butterworth filter, the Chebyshev filter, and the Bessel filter. The Butterworth filter is commonly referred to as a maximally flat or flat-flat filter. These names refer to the fact that is relatively constant across the pass band. Butterworth frequency response is illustrated further in Figure 17.7 of the text.
The Chebyshev filter differs from the Butterworth in two important ways.
Figure 17.8 of the text illustrates the frequency response of a Chebyshev filter and compares it to a Butterworth. Note the ripple in the Chebyshev filter pass band.
The Bessel filter is designed to provide a constant phase shift across its pass band. As a result, it has greater fidelity (ability to accurately reproduce a waveform) than the Butterworth or Chebyshev filters. Its primary disadvantage is its lower initial roll-off rate. Of these three filters, the Butterworth is the most commonly used. A comparison of the Butterworth, Chebyshev, and Bessel response curves is provided in Figure 17.9 of the text.
Low-Pass and High-Pass Filters
Two single-pole low-pass Butterworth filters are illustrated in Figure 17-4. They are usually high-gain circuits or voltage followers (unity gain). Note the RC circuit at the inputs. This RC circuit determines the value of , as follows:
FIGURE 17-4 Single-pole low-pass active filters.
Example 17.4 of the text demonstrates the process for calculating the bandwidth of a single-pole low-pass filter. The response curve for the circuit in this example is shown in Figure 17.12 of the text. Note that the circuit output is down by 3dB at .
Two-pole low-pass filters. As stated earlier, a two-pole low-pass filter has a roll-off rate of 40 dB/decade. A two-pole low-pass active filter is shown in Figure 17-5. The circuit has two RC circuits, each of which introduces a 20 dB/decade rolloff for a total of 40 dB/decade. The value of is calculated using
The voltage gain of the circuit in Figure 17-5 is a function of . The circuit can be converted to a unity-gain configuration by removing the path to ground.
FIGURE 17-5 Common two-pole low-pass filters.
There is one important restriction on two-pole active filters. In order to have a Butterworth (flat) response, the circuit closed-loop gain cannot exceed 1.586 (4 dB). Obviously, a unity gain filter has no problem meeting this requirement. The variable-gain filter (Figure 17-5) is usually designed using the following criteria:
These relationships are illustrated in Example 17.5 of the text.
Three-pole low-pass active filters. A three-pole active filter is comprised of a single-pole filter and a two-pole filter. In order for the filter to have a Butterworth response, it must meet the following criteria:
Both filters must be tuned to the same value of . A typical three-pole low-pass filter is shown in Figure 17.16 of the text. It should be noted that active filters can have more than three poles. Regardless of the number of poles, the roll-off rate is equal to the number of poles times 20 dB/decade.
High-Pass Active Filters
High-pass active filters differ from low-pass filters in two respects. Obviously, the component positions are reversed so that the capacitor is in series with the input and limits the low-frequency operation of the circuit. Secondly, the components in the multi-pole circuits are chosen to fulfill the following criteria:
Several high-pass active filters are illustrated in Figure 17-6.
FIGURE 17-6 Typical high-pass active filters.
In order to maintain Butterworth response characteristics, both high- and low-pass multistage filters must meet specific requirements. These gain requirements are summarized in Table 17.1 of the text. Any of the multistage filters in Table 17.1 can be constructed using two-pole and single-pole cascaded stages. You just need to remember two important points:
Figures 17.18 and 17.19 of the text summarize the characteristics of high-pass and low-pass active filters.
Band-Pass and Notch Filters
Band-pass filters are designed to pass all frequencies within their bandwidths. One common way to construct a band-pass filter is to cascade a low-pass filter with a high-pass filter, as shown in Figure 17-7. The low-pass filter determines the value of , and the high-pass filter determines the value of .
FIGURE 17-7 A two-stage band-pass filter.
Once the upper and lower cutoff frequencies are determined, other circuit values can be calculated as follows:
The analysis of a two-stage band-pass filter is demonstrated in Example 17.6 of the text. The obvious disadvantage of this circuit is that it requires two op-amps and a large number of resistors and capacitors. There is an alternative: the multiple-feedback band-pass filter.
The multiple-feedback band-pass filter gets its name from the fact that it has two feedback networks, as shown in Figure 17-8. The operation of this circuit is explained in detail in the text and is illustrated in Figure 17.3.
FIGURE 17-8 A multiple-feedback band-pass filter.
The geometric center frequency of a multiple-feedback band-pass filter can be calculated using:
Example 17.7 demonstrates the use of this equation. Once the value of is determined, we can calculate Q using:
where. The Q and bandwidth of a multiple-feedback band-pass filter are calculated in Example 17.8 of the text. Once these values are known, we can calculate the cutoff frequencies. The approach used to calculate the cutoff frequencies depends on the Q of the circuit. If circuit Q 2, then we can use:
If Q < 2, the cutoff frequencies are calculated using:
Example 17.9 of the text demonstrates that the approximations used (when Q 2) are close enough for most applications. Example 17.10 demonstrates that this is not the case when Q < 2. Finally, Example 17.11 demonstrates how to calculate circuit gain for this circuit using:
A notch filter blocks all frequencies within its bandwidth. As was the case with the band-pass filter, a notch filter can be constructed as a multistage filter or as a multiple-feedback filter. Figure 17-9 shows a block diagram of a multi-stage notch filter and its frequency-response curve. As you can see, the low-pass filter determines and the high-pass filter determines . The gap between the two cutoff frequencies is the bandwidth of the filter. Note that must be lower than for the circuit to have a notch response curve.
FIGURE 17-9 Multi-stage notch filter block diagram and response curve.
The circuit represented in Figure 17-9 is illustrated in Figure 17.26 of the text. As you can see it requires many components. As was the case with the band-pass filter, the multiple-feedback notch filter is simpler to construct. A multiple-feedback notch filter is shown in Figure 17.27.
The multiple-feedback notch filter functions much like its band-pass counterpart, but with a few differences. Briefly, we can calculate the value of using:
It should also be noted that outside of the stop-band.
Several Active Filter Applications
This section of the chapter describes several possible applications for active filters. A crossover network is a circuit that splits an audio frequency signal into high- and low-frequency components. The high frequencies go to a tweeter and the low frequencies go to a woofer. Figure 17-10 shows a block diagram illustrating how a crossover network could be used in an audio system. Figure 17.29 of the text shows the circuit layout.
FIGURE 17-10 Crossover network in an audio system.
A graphic equalizer is a system that is designed to allow you to control the amplitude of different audio-frequency ranges. A simple graphic equalizer can be constructed as shown in Figure 17-11. Although this circuit only uses three band-pass filters, more filters can be employed to increase the sophistication of the system. The gain of each band-pass filter either boosts or attenuates the signal within its frequency range. The final component is the summing amplifier that combines the signals from the various band-pass filters.
FIGURE 17-11 A simple graphic equalizer.
Active Filter Troubleshooting
Troubleshooting active filters is relatively easy if you keep in mind that some of the circuit components are designed to determine the frequency response of the circuit. Tables 17.2 through 17.4 of the text list the common symptoms and causes for the faults in low-pass, high-pass, and band-pass filters. Once you are familiar with these common fault symptoms, you can troubleshoot active filters by doing the following:
Discrete Tuned Amplifiers
Some applications exceed the power handling and/or high-frequency limits of op-amps. In these applications, discrete tuned amplifiers are commonly used. Discrete amplifiers are typically tuned using LC (inductive-capacitive) resonant circuits in place of their collector (or drain) resistors. One such circuit is shown in Figure 17-12.
FIGURE 17-12 Typical BJT tuned amplifier.
The parallel LC (or tank) circuit determines the frequency response of the amplifier. As illustrated in Figure 17.33 of the text, there is a frequency at which . This frequency, called the resonant frequency, is calculated using:
In an ideal resonant circuit, inductor current lags the capacitor current by 180° and the net circuit current is zero. As a result:
Figure 17-13 shows the frequency response of an LC tank circuit. When the input frequency () is lower than, the circuit impedance decreases from its maximum value, and is inductive. When is higher than , the circuit impedance drops again, but is capacitive. When operated at , the impedance of the tank circuit reaches its maximum value. As a result, the gain of the tuned amplifier (Figure 17-12) is also at its maximum value.
FIGURE 17-13 Frequency response of an LC tank circuit.
The equivalent circuit for a discrete tuned amplifier is shown in Figure 17.35 of the text. Example 17.12 demonstrates how to calculate the center frequency for a discrete tuned amplifier.
As stated earlier, the Q of a tuned amplifier equals the ratio of to BW. In a discrete tuned amplifier, it is the Q of the parallel LC circuit that determines the amplifier Q. A more accurate definition of Q is the ratio of energy stored in the circuit to the energy lost per cycle by the circuit, which equals the ratio of reactive power (energy stored per second) to resistive power (energy lost per second). Since inductor Q is much lower than capacitor Q, the overall Q of the tank circuit is determined by the inductor. By formula:
where is the winding resistance of the coil. Example 17.13 of the text demonstrates how this equation is used to determine the Q of a tank circuit.
A resistive load on a tuned amplifier reduces the overall Q of the circuit. As shown in Figure 17.37a of the text, the load is in parallel with the tank circuit. The first step to determine the loaded-Q () is to calculate a parallel equivalent for . The derivation for the following equation can be found in Appendix D of the text:
is in parallel with the load resistance, as shown in Figure 17-37b. Therefore,
And the loaded Q is found as
Example 17.14 of the text demonstrates the procedure used to calculate loaded Q. Once the value of is known, the circuit bandwidth is found using:
as demonstrated in Example 17.15. Once the values of, BW, and are known, we can then calculate the cutoff frequencies using the following equations:
When : and
Discrete Tuned Amplifiers: Practical Considerations and Troubleshooting
It is common to see a significant difference between the calculated and measured center frequencies of a tuned amplifier. Two of the reasons for the difference are as follows:
One method to overcome these problems is to include a variable inductor or capacitor in the circuit. Another method of adjusting the circuit tuning is illustrated in Figure 17.38 of the text. This technique is referred to as electronic or varactor tuning. You may recall that a varactor is a diode that acts as an electronically variable capacitor. It is used to tune the circuit by changing the voltage applied to the varactor (and thus its capacitance).
The most common fault in tuned circuits is frequency drift. In most cases this is a result of the components aging and the circuit can simply be retuned. If this cannot be done, then one or both of the tank components must be replaced. If either the inductor or capacitor were to fail, then the result would be much more than a simple drift problem. Table 17.5 of the text lists the faults and symptoms of either of these components failing.
The characteristics of tuned amplifiers are summarized in Figure 17.39 of the text.
Class C Amplifiers
Class C amplifiers were briefly mentioned in Chapter 11. The transistor in a class C amplifier conducts for less than 180° of the input cycle. A basic class C amplifier is illustrated in Figure 17-14.
FIGURE 17-14 Class C amplifier.
The most important aspect of the dc operation of this amplifier is that it is biased deeply into cutoff, meaning that and. If a negative supply is used to bias the base circuit, the value of usually fulfills the following relationship:
The ac operation of the class C amplifier is based on the characteristics of the parallel-resonant tank circuit. If a single current pulse is applied to the tank circuit, the result is a decaying sinusoidal waveform (as shown in Figure 17.42b of the text). The waveform shown is a result of the charge/discharge cycle of the capacitor and inductor in the tank circuit, and is commonly referred to as the flywheel effect.
If we want to produce a sine wave that does not decay, we must repeatedly apply a current pulse during each full cycle. At the peak of each positive alternation of the input signal, the tank circuit in a class C amplifier gets the current pulse it needs to produce a complete sine wave at the output. This concept is illustrated in Figure 17.43 of the text. Note that , , and are inverted at the output relative to the input. This is due to the fact that a common-emitter amplifier produces a 180° -voltage phase shift. Note that the bandwidth, Q, and characteristics of a class C amplifier are the same as those for any tuned discrete amplifier.
One final point about the class C amplifier. In order for this amplifier to work properly, the tank circuit must be tuned to the same frequency as the input signal, or to some harmonic of that frequency. For instance, you could tune the class C amplifier to the third harmonic of the input and have an output three times higher in frequency. This means that the class C amplifier can be used as a frequency multiplier.