- Frequency measurements (bandwidth, cutoff frequency, octaves, decades, etc.)

- A review of decibel gain measurements

- Frequency response curves and Bode plots

- BJT amplifier frequency response

- Gain roll-off rates

- FET amplifier frequency response

**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**( ).

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

*Measuring the Cutoff Frequencies*- Set up the amplifier for the maximum undistorted output signal.

- 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.

- If you are not at midband, adjust until you are.

- Adjust the
*volts/division*calibration control on the oscilloscope until the amplifier output waveform fills exactly*seven*major divisions (peak-to-peak).

- 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.

- 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 .

*square*of voltage. When voltage gain drops to , power gain drops by a factor of (which is half its midband value).

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

*Gain and Frequency Measurements*- 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.**

**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.

**A**

*Bode Plots***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.**

**The frequency-response characteristics of the various circuits in this chapter are a function of the**

*BJT Amplifier Frequency Response**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.

**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**

*BJT Amplifier Low-Frequency Response**highest*cutoff frequency calculated is the overall value of for the circuit.

**FIGURE 14-5 BJT amplifier lower cutoff frequencies.**

**The term**

*Gain Roll-Off***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.

*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.

**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.

**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**

*BJT High-Frequency Response**lowest*result is the overall value of for the circuit.

**FIGURE 14-9 BJT amplifier upper cutoff frequencies.**

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

**A FET amplifier is shown in Figure 14-10. Note that the lower cutoff frequencies for the gate () and drain () circuits are calculated. The**

*FET Amplifier Low-Frequency Response**highest*cutoff frequency calculated is the overall value of for the circuit.

**FIGURE 14-10 FET amplifier lower cutoff frequencies.**

- 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.

**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**

*FET Amplifier High-Frequency Response**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.

**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:

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