

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 171. Note that the value of shown in the figure corresponds to a value of. FIGURE 171 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 172. Note that the lower the Q of an amplifier:
The equation in Figure 172 is somewhat misleading as it implies that Q is dependent upon the circuit’s 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 172 Bandwidth versus rolloff 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 opamps. Discrete tuned amplifiers normally employ LC (inductivecapacitive) circuits to determine frequency response. Opamps are normally tuned with RC (resistivecapacitive) circuits. Both these types of circuits are investigated in this chapter. Active Filters Tuned opamp 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 bandpass and notch filters. Highpass and lowpass filters are typically described using only the single cutoff frequency. The frequencyresponse curves for the four filter types are shown in Figure 173. FIGURE 173 Active filter frequencyresponse curves. There are several terms that are commonly used when describing active filters. The term pole simply means an RC circuit. A onepole filter has one RC circuit, a twopole filter has two RC circuits, and so on. The term order defines how many poles a filter has. A secondorder filter has two RC circuits or poles. The reason these terms are important is because the number of poles determines the filter’s gain rolloff rate. The table below illustrates the relationship between poles and gain rolloff rates for a Butterworthtype 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 flatflat 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 rolloff 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. LowPass and HighPass Filters Two singlepole lowpass Butterworth filters are illustrated in Figure 174. They are usually highgain circuits or voltage followers (unity gain). Note the RC circuit at the inputs. This RC circuit determines the value of , as follows:
FIGURE 174 Singlepole lowpass active filters. Example 17.4 of the text demonstrates the process for calculating the bandwidth of a singlepole lowpass 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 . Twopole lowpass filters. As stated earlier, a twopole lowpass filter has a rolloff rate of 40 dB/decade. A twopole lowpass active filter is shown in Figure 175. 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 175 is a function of . The circuit can be converted to a unitygain configuration by removing the path to ground. FIGURE 175 Common twopole lowpass filters.
There is one important restriction on twopole active filters. In order to have a Butterworth (flat) response, the circuit closedloop gain cannot exceed 1.586 (4 dB). Obviously, a unity gain filter has no problem meeting this requirement. The variablegain filter (Figure 175) is usually designed using the following criteria: These relationships are illustrated in Example 17.5 of the text. Threepole lowpass active filters. A threepole active filter is comprised of a singlepole filter and a twopole 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 threepole lowpass 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 rolloff rate is equal to the number of poles times 20 dB/decade.
HighPass Active Filters Highpass active filters differ from lowpass filters in two respects. Obviously, the component positions are reversed so that the capacitor is in series with the input and limits the lowfrequency operation of the circuit. Secondly, the components in the multipole circuits are chosen to fulfill the following criteria: Several highpass active filters are illustrated in Figure 176. FIGURE 176 Typical highpass active filters.
In order to maintain Butterworth response characteristics, both high and lowpass 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 twopole and singlepole cascaded stages. You just need to remember two important points:
Figures 17.18 and 17.19 of the text summarize the characteristics of highpass and lowpass active filters. BandPass and Notch Filters Bandpass filters are designed to pass all frequencies within their bandwidths. One common way to construct a bandpass filter is to cascade a lowpass filter with a highpass filter, as shown in Figure 177. The lowpass filter determines the value of , and the highpass filter determines the value of . FIGURE 177 A twostage bandpass filter. Once the upper and lower cutoff frequencies are determined, other circuit values can be calculated as follows: The analysis of a twostage bandpass filter is demonstrated in Example 17.6 of the text. The obvious disadvantage of this circuit is that it requires two opamps and a large number of resistors and capacitors. There is an alternative: the multiplefeedback bandpass filter. The multiplefeedback bandpass filter gets its name from the fact that it has two feedback networks, as shown in Figure 178. The operation of this circuit is explained in detail in the text and is illustrated in Figure 17.3. FIGURE 178 A multiplefeedback bandpass filter. The geometric center frequency of a multiplefeedback bandpass 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 multiplefeedback bandpass 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: and If Q < 2, the cutoff frequencies are calculated using: and 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: Notch Filters A notch filter blocks all frequencies within its bandwidth. As was the case with the bandpass filter, a notch filter can be constructed as a multistage filter or as a multiplefeedback filter. Figure 179 shows a block diagram of a multistage notch filter and its frequencyresponse curve. As you can see, the lowpass filter determines and the highpass 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 179 Multistage notch filter block diagram and response curve. The circuit represented in Figure 179 is illustrated in Figure 17.26 of the text. As you can see it requires many components. As was the case with the bandpass filter, the multiplefeedback notch filter is simpler to construct. A multiplefeedback notch filter is shown in Figure 17.27. The multiplefeedback notch filter functions much like its bandpass counterpart, but with a few differences. Briefly, we can calculate the value of using: It should also be noted that outside of the stopband.
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 lowfrequency components. The high frequencies go to a tweeter and the low frequencies go to a woofer. Figure 1710 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 1710 Crossover network in an audio system.
A graphic equalizer is a system that is designed to allow you to control the amplitude of different audiofrequency ranges. A simple graphic equalizer can be constructed as shown in Figure 1711. Although this circuit only uses three bandpass filters, more filters can be employed to increase the sophistication of the system. The gain of each bandpass 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 bandpass filters. FIGURE 1711 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 lowpass, highpass, and bandpass 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 highfrequency limits of opamps. In these applications, discrete tuned amplifiers are commonly used. Discrete amplifiers are typically tuned using LC (inductivecapacitive) resonant circuits in place of their collector (or drain) resistors. One such circuit is shown in Figure 1712. FIGURE 1712 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 1713 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 1712) is also at its maximum value. FIGURE 1713 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 loadedQ () 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 1737b. 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 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 1714. FIGURE 1714 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 parallelresonant 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 commonemitter 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.
