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Amplifier Frequency Response
Chapter Summary

Here is a brief summary of the material covered in this chapter:

Most amplifiers have relatively constant gain across a range, or band, of frequencies. This band of frequencies is referred to as the bandwidth of the circuit. When operated within its bandwidth, the values of , and for an amplifier are calculated as shown earlier in the text. For clarity, these values are referred to as midband gain values, and are designated as , and .

A frequency-response curve is a graphical representation of the relationship between amplifier gain and operating frequency. A generic frequency response curve is shown in Figure 14-1. This particular curve illustrates the relationship between power gain and frequency. As shown:

  • The circuit power gain remains relatively constant across the midband range of frequencies.
  • As operating frequency decreases from the midband area of the curve, a point is reached where the power gain begins to drop off. The frequency at which power gain equals 50% of its midband value is called the lower cutoff frequency ().
  • As operating frequency increases from the midband area of the curve, a point is reached where the power gain begins to drop off again. The frequency at which power gain equals 50% of its midband value is called the upper cutoff frequency ( ).

Note that the bandwidth of the circuit is found as the difference between the cutoff frequencies. By formula,

Bandwidth is calculated as demonstrated in Example 14.1 of the text.

 

 


FIGURE 14-1 A generic frequency-response curve.

The geometric center frequency () of an amplifier is the geometric average of the cutoff frequencies, found as

Power gain is maximum when an amplifier is operated at its geometric center frequency. As frequency varies above (or below) , the power gain decreases slightly. By the time one (or the other) cutoff frequency is reached, power gain has dropped to half its midband value.

The relationship between , , and can also be described using frequency ratios, as follows:

This relationship is illustrated further in Example 14.3 of the text. The relationship allows us to calculate the value of either cutoff frequency when the values of the geometric center frequency and the second cutoff frequency are known. The relationships used are

and

These relationships are illustrated further in Examples 14.4 and 14.5 of the text.

 

Measuring the Cutoff Frequencies

The cutoff frequencies of an amplifier can be measured with an oscilloscope using the following procedure:

  1. Set up the amplifier for the maximum undistorted output signal.
  2. Establish that you are operating in the midband frequency range by varying the frequency of the input signal several kilohertz in both directions. If you are in the midband range, slight variations in operating frequency will not cause any significant changes in the output amplitude of the circuit.
  3. If you are not at midband, adjust until you are.
  4. Adjust the volts/division calibration control on the oscilloscope until the amplifier output waveform fills exactly seven major divisions (peak-to-peak).
  5. To measure the value of , decrease the operating frequency until the amplifier output waveform fills only five major divisions. At this frequency, the amplitude of the amplifier has dropped to of its maximum value. This indicates that we are operating at the lower cutoff frequency.
  6. To measure the value of , increase the operating frequency until the same thing happens on the high-frequency end. The frequency at which this occurs is .

Be sure to use the probe to minimize the effect of the oscilloscope input capacitance on the frequency measurements.

The value of 0.707 is based on the relationship between voltage and power. Power changes with the square of voltage. When voltage gain drops to , power gain drops by a factor of (which is half its midband value).

Gain and Frequency Measurements

Figure 14-1 is a simplified frequency-response curve. A more practical curve is shown in Figure 14-2. In the curve shown:

  • Power gain is represented using a ratio of (at the given frequency) to , expressed in dB.
  • The values of the frequency increments follow a logarithmic progression; i.e., each increment is a whole number multiple of the previous increment.

 


FIGURE 14-2 A more practical frequency-response curve.

A review of dB values and conversions is provided in the Chapter 8 summary on this web site.

The frequency scale shown in Figure 14-2 is a decade scale. A decade is a frequency multiplier of 10. As you can see, the value of each increment in Figure 14-2 is the value of the previous increment. Another commonly used frequency scale is the octave scale. An octave is a frequency multiplier of 2. The value of each increment in an octave scale is the value of the previous increment.

Bode Plots

A Bode plot is a variation on the frequency-response curve shown in Figure 14-2. The blue curve in Figure 14-3 is a Bode plot. For comparison, a frequency-response curve (red, dashed) is included in the figure. The Bode plot shows the change in power gain () to be 0 dB for all frequencies between the cutoff frequencies. Beyond the cutoff frequencies, the amplifier power gain is shown to drop at a rate of 20 dB per decade.

 


FIGURE 14-3 A Bode plot.

 

BJT Amplifier Frequency Response

The frequency-response characteristics of the various circuits in this chapter are a function of the RC circuits they contain. For example, the base circuit of an RC-coupled, voltage-divider biased BJT amplifier can be illustrated as shown in Figure 14-4.

 


FIGURE 14-4

 

The cutoff frequency for the circuit in Figure 14-4 occurs when . For any RC circuit, the frequency at which can be found as

Nearly all of the cutoff frequency calculations introduced in this chapter involve a form of this relationship.

 

BJT Amplifier Low-Frequency Response

A BJT amplifier is shown in Figure 14-5. Each circuit has its own cutoff frequency, which is calculated using the relationship shown. Note that the lower cutoff frequencies for the base (), collector (), and emitter () circuits are calculated. The highest cutoff frequency calculated is the overall value of for the circuit.

 


FIGURE 14-5 BJT amplifier lower cutoff frequencies.

 

Example 14.8 of the text demonstrates the calculation of . Example 14.9 demonstrates the calculation of . Example 14.10 demonstrates the calculation of .

Gain Roll-Off

The term roll-off rate is used to describe the rate at which an RC circuit causes the voltage gain of an amplifier to decrease once the value of or has been passed. Low-frequency roll-off is calculated using

 

where f is the frequency of operation. The use of this relationship is demonstrated in Example 14.11 of the text.

The equation illustrates the fact that the values of R and C for an RC circuit do not affect the rate at which voltage gain decreases when the circuit is operated beyond its cutoff frequency. The equation can be used to show that every RC circuit has a roll-off rate of 20 dB per decade. A low-frequency roll-off rate of 20 dB per decade is illustrated in Figure 14-6.

 


FIGURE 14-6 A 20 dB per decade roll-off rate.

 

The equation can also be used to show that every RC circuit has a roll-off rate of 6 dB per octave. A low-frequency roll-off rate of 6 dB per octave is illustrated in Figure 14-7.


FIGURE 14-7 A 6 dB per octave roll-off rate.

 

When a circuit (such as the BJT amplifier) contains more than one RC circuit, the effects of the values are additive. This point is illustrated in Figure 14.12 of the text and in the Bode plot shown in Figure 14-8.

 


 

FIGURE 14-8 A Bode plot showing multiple roll-off rates.

 

BJT High-Frequency Response

The high-frequency equivalent of a BJT amplifier is shown in Figure 14-9. Each circuit has its own cutoff frequency, which is calculated using the relationship shown. Note that the upper cutoff frequencies for the base () and collector () circuits are calculated, and the lowest result is the overall value of for the circuit.

 


 

FIGURE 14-9 BJT amplifier upper cutoff frequencies.

 

The value of in Figure 14-9 represents the base-emitter capacitance of the transistor. This value is commonly listed on the spec sheet of a transistor. However, the value on the spec sheet is at a specific value of , which may or may not be accurate for the circuit. A more accurate value of can be found using

where is the current gain-bandwidth product for the transistor (which is listed on the spec sheet).

The equations in Figure 14-9 contain the variables and . These variables represent the Miller capacitance values for the circuit. Miller’s theorem states that a feedback capacitor (such as ) can be represented as separate input and output capacitance values (which simplifies the analysis of the circuit). The values of the Miller capacitances are found using the relationships shown in the figure.

Example 14.14 of the text demonstrates the calculations required to determine the value of for an amplifier. Example 14.15 demonstrates the calculations required to determine the value of for an amplifier.

Theory versus practice. It should be noted that there is usually a large percentage of error between the calculated and measured values of for a BJT amplifier. This high percentage of error is caused by several factors:

  • The BJT internal capacitance values are estimated.
  • The BJT internal capacitances are in the pF range, as is the input capacitance of most oscilloscopes. As a result, the oscilloscope input capacitance can have a major impact on the measured values of and
.

FET Amplifier Low-Frequency Response

A FET amplifier is shown in Figure 14-10. Note that the lower cutoff frequencies for the gate () and drain () circuits are calculated. The highest cutoff frequency calculated is the overall value of for the circuit.

 


 

FIGURE 14-10 FET amplifier lower cutoff frequencies.

 

The value of is calculated as shown in Example 14.17 of the text. The value of is calculated as shown in Example 14.18. The lower cutoff frequency of the source circuit is not considered here for several reasons:

  • The calculation of is extremely complex.
  • The value of is normally much lower than either or . As a result, its value has little effect on the low-frequency operation of the circuit.

FET Amplifier High-Frequency Response

The high-frequency equivalent of a FET amplifier is shown in Figure 14-11. Note that the upper cutoff frequencies for the gate () and drain () circuits are calculated. The lowest cutoff frequency calculated is the overall value of for the circuit. The calculations required to determine the values of and are shown in Example 14.20 of the text.


 

FIGURE 14-11 FET amplifier upper cutoff frequencies.

 

Capacitance specifications. The most commonly used JFET capacitance ratings are listed below.

 

  • is the JFET input (gate) capacitance.
  • is the JFET output (drain) capacitance.
  • is the gate-to-source capacitance.
  • When the above ratings are used, the inter-terminal capacitances of the JFET are found using the following relationships:

     

    Multistage Amplifiers

    The cutoff frequencies for cascaded amplifiers with identical values of and are found using

    and

    where n is the number of cascaded stages. As demonstrated in Examples 14.22 and 14.23 of the text:

    and

    When the cutoff frequencies for the circuit are determined, its bandwidth is found using:

    Once the overall bandwidth has been determined, the Bode plot for the circuit can be plotted as shown in Figure 14.31 of the text.



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