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f CHAPTER BJT and JFET Frequency Response 11 11.1 INTRODUCTION The analysis thus far has been limited to a particular frequency. For the amplifier, it was a frequency that normally permitted ignoring the effects of the capacitive elements, reducing the analysis to one that included only resistive elements and sources of the independent and controlled variety. We will now investigate the frequency effects introduced by the larger capacitive elements of the network at low frequencies and the smaller capacitive elements of the active device at the high frequencies. Since the analysis will extend through a wide frequency range, the logarithmic scale will be defined and used throughout the analysis. In addition, since industry typically uses a decibel scale on its frequency plots, the concept of the decibel is introduced in some detail. The similarities between the frequency response analyses of both BJTs and FETs permit a coverage of each in the same chapter. 11.2 LOGARITHMS There is no escaping the need to become comfortable with the logarithmic function. The plotting of a variable between wide limits, comparing levels without unwieldy numbers, and identifying levels of particular importance in the design, review, and analysis procedures are all positive features of using the logarithmic function. As a first step in clarifying the relationship between the variables of a logarithmic function, consider the following mathematical equations: a  bx, x  logb a (11.1) The variables a, b, and x are the same in each equation. If a is determined by taking the base b to the x power, the same x will result if the log of a is taken to the base b. For instance, if b  10 and x  2, a  bx  (10)2  100 but x  logb a  log10 100  2 In other words, if you were asked to find the power of a number that would result in a particular level such as shown below: 10,000  10x 493 f the level of x could be determined using logarithms. That is, x  log10 10,000  4 For the electrical/electronics industry and in fact for the vast majority of scientific research, the base in the logarithmic equation is limited to 10 and the number e  2.71828. . . . Logarithms taken to the base 10 are referred to as common logarithms, while logarithms taken to the base e are referred to as natural logarithms. In summary: Common logarithm: x  log10 a (11.2) Natural logarithm: y  loge a (11.3) loge a  2.3 log10 a (11.4) The two are related by On today’s scientific calculators, the common logarithm is typically denoted by the key and the natural logarithm by the key. EXAMPLE 11.1 Using the calculator, determine the logarithm of the following numbers to the base indicated. (a) log10 106. (b) loge e3. (c) log10 102. (d) loge e1. Solution (a) 6 (b) 3 (c) 2 (d) 1 The results in Example 11.1 clearly reveal that the logarithm of a number taken to a power is simply the power of the number if the number matches the base of the logarithm. In the next example, the base and the variable x are not related by an integer power of the base. EXAMPLE 11.2 Using the calculator, determine the logarithm of the following numbers. (a) log10 64. (b) loge 64. (c) log10 1600. (d) log10 8000. Solution (a) 1.806 (b) 4.159 (c) 3.204 (d) 3.903 Note in parts (a) and (b) of Example 11.2 that the logarithms log10 a and loge a are indeed related as defined by Eq. (11.4). In addition, note that the logarithm of a number does not increase in the same linear fashion as the number. That is, 8000 is 125 times larger than 64, but the logarithm of 8000 is only about 2.16 times larger 494 Chapter 11 BJT and JFET Frequency Response f than the magnitude of the logarithm of 64, revealing a very nonlinear relationship. In fact, Table 11.1 clearly shows how the logarithm of a number increases only as the exponent of the number. If the antilogarithm of a number is desired, the 10x or ex calculator functions are employed. TABLE 11.1 log10 100 log10 10 log10 100 log10 1,000 log10 10,000 log10 100,000 log10 1,000,000 log10 10,000,000 log10 100,000,000 and so on 0 1 2 3 4 5 6 7 8 EXAMPLE 11.3 Using a calculator, determine the antilogarithm of the following expressions: (a) 1.6  log10 a. (b) 0.04  loge a. Solution (a) a  101.6 Calculator keys: and a  39.81 (b) a  e0.04 Calculator keys: and a  1.0408 Since the remaining analysis of this chapter employs the common logarithm, let us now review a few properties of logarithms using solely the common logarithm. In general, however, the same relationships hold true for logarithms to any base. log10 1  0 (11.5) As clearly revealed by Table 11.1, since 100  1, a log10   log10 a  log10 b b (11.6) which for the special case of a  1 becomes 1 log10   log10 b b (11.7) revealing that for any b greater than 1 the logarithm of a number less than 1 is always negative. log10 ab  log10 a  log10 b (11.8) In each case, the equations employing natural logarithms will have the same format. 11.2 Logarithms 495 f EXAMPLE 11.4 Using a calculator, determine the logarithm of the following numbers: (a) log10 0.5. 4000 (b) log10 . 250 (c) log10 (0.6  30). Solution (a) 0.3 (b) log10 4000  log10 250  3.602  2.398  1.204 4000 Check: log10   log10 16  1.204 250 (c) log10 0.6  log10 30  0.2218  1.477  1.255 Check: log10 (0.6  30)  log10 18  1.255 The use of log scales can significantly expand the range of variation of a particular variable on a graph. Most graph paper available is of the semilog or double-log (log-log) variety. The term semi (meaning one-half) indicates that only one of the two scales is a log scale, whereas double-log indicates that both scales are log scales. A semilog scale appears in Fig. 11.1. Note that the vertical scale is a linear scale with equal divisions. The spacing between the lines of the log plot is shown on the graph. Figure 11.1 Semilog graph paper. 496 Chapter 11 BJT and JFET Frequency Response The log of 2 to the base 10 is approximately 0.3. The distance from 1 (log10 1  0) to 2 is therefore 30% of the span. The log of 3 to the base 10 is 0.4771 or almost 48% of the span (very close to one-half the distance between power of 10 increments on the log scale). Since log10 5  0.7, it is marked off at a point 70% of the distance. Note that between any two digits the same compression of the lines appears as you progress from the left to the right. It is important to note the resulting numerical value and the spacing, since plots will typically only have the tic marks indicated in Fig. 11.2 due to a lack of space. You must realize that the longer bars for this figure have the numerical values of 0.3, 3, and 30 associated with them, whereas the next shorter bars have values of 0.5, 5, and 50 and the shortest bars 0.7, 7, and 70. (3) about halfway (0.3) 0.1 0.7 (5) (7) (30) (50) (70) 10 1 f 100 log almost three-fourths (0.5) Figure 11.2 Identifying the numerical values of the tic marks on a log scale. Be aware that plotting a function on a log scale can change the general appearance of the waveform as compared to a plot on a linear scale. A straight-line plot on a linear scale can develop a curve on a log scale, and a nonlinear plot on a linear scale can take on the appearance of a straight line on a log plot. The important point is that the results extracted at each level be correctly labeled by developing a familiarity with the spacing of Figs. 11.1 and 11.2. This is particularly true for some of the log-log plots that appear later in the book. 11.3 DECIBELS The concept of the decibel (dB) and the associated calculations will become increasingly important in the remaining sections of this chapter. The background surrounding the term decibel has its origin in the established fact that power and audio levels are related on a logarithmic basis. That is, an increase in power level, say 4 to 16 W, does not result in an audio level increase by a factor of 16/4  4. It will increase by a factor of 2 as derived from the power of 4 in the following manner: (4)2  16. For a change of 4 to 64 W, the audio level will increase by a factor of 3 since (4)3  64. In logarithmic form, the relationship can be written as log4 64  3. The term bel was derived from the surname of Alexander Graham Bell. For standardization, the bel (B) was defined by the following equation to relate power levels P1 and P2: P2 G  log10  P1 bel (11.9) 11.3 Decibels 497 f It was found, however, that the bel was too large a unit of measurement for practical purposes, so the decibel (dB) was defined such that 10 decibels  1 bel. Therefore, P2 GdB  10 log10  P1 dB (11.10) The terminal rating of electronic communication equipment (amplifiers, microphones, etc.) is commonly rated in decibels. Equation (11.10) indicates clearly, however, that the decibel rating is a measure of the difference in magnitude between two power levels. For a specified terminal (output) power (P2) there must be a reference power level (P1). The reference level is generally accepted to be 1 mW, although on occasion, the 6-mW standard of earlier years is applied. The resistance to be associated with the 1-mW power level is 600 , chosen because it is the characteristic impedance of audio transmission lines. When the 1-mW level is employed as the reference level, the decibel symbol frequently appears as dBm. In equation form,  P2 GdBm  10 log10  1 mW 600  dBm (11.11) There exists a second equation for decibels that is applied frequently. It can be best described through the system of Fig. 11.3. For Vi equal to some value V1, P1  V 12/Ri, where Ri , is the input resistance of the system of Fig. 11.3. If Vi should be increased (or decreased) to some other level, V2, then P2  V22/Ri. If we substitute into Eq. (11.10) to determine the resulting difference in decibels between the power levels, P2 V 22/Ri V2 GdB  10 log10   10 log10 2  10 log10  P1 V1 V 1/Ri   and V2 GdB  20 log10  V1 dB 2 (11.12) Figure 11.3 Configuration employed in the discussion of Eq. (11.12). Frequently, the effect of different impedances (R1  R2) is ignored and Eq. (11.12) applied simply to establish a basis of comparison between levels—voltage or current. For situations of this type, the decibel gain should more correctly be referred to as the voltage or current gain in decibels to differentiate it from the common usage of decibel as applied to power levels. One of the advantages of the logarithmic relationship is the manner in which it can be applied to cascaded stages. For example, the magnitude of the overall voltage gain of a cascaded system is given by (11.13) Av   Av Av Av …Av  T 498 Chapter 11 1 2 BJT and JFET Frequency Response 3 n f Applying the proper logarithmic relationship results in TABLE 11.2 Gv  20 log10 AvT  20 log10 Av1  20 log10 Av2  20 log10 Av3  Voltage Gain,  20 log10 Avn (dB) (11.14) In words, the equation states that the decibel gain of a cascaded system is simply the sum of the decibel gains of each stage, that is, Gv  Gv1  Gv2  Gv3   Gvn dB (11.15) In an effort to develop some association between dB levels and voltage gains, Table 11.2 was developed. First note that a gain of 2 results in a dB level of 6 dB while a drop to 12 results in a 6-dB level. A change in Vo /Vi from 1 to 10, 10 to 100, or 100 to 1000 results in the same 20-dB change in level. When Vo  Vi, Vo /Vi  1 and the dB level is 0. At a very high gain of 1000, the dB level is 60, while at the much higher gain of 10,000, the dB level is 80 dB, an increase of only 20 dB—a result of the logarithmic relationship. Table 11.2 clearly reveals that voltage gains of 50 dB or higher should immediately be recognized as being quite high. Vo /Vi dB Level 0.5 0.707 1 2 10 40 100 1000 10,000 etc. 6 3 0 6 20 32 40 60 80 EXAMPLE 11.5 Find the magnitude gain corresponding to a decibel gain of 100. Solution By Eq. (11.10), P2 P2 GdB  10 log10   100 dB → log10   10 P1 P1 so that P2   1010  10,000,000,000 P1 This example clearly demonstrates the range of decibel values to be expected from practical devices. Certainly, a future calculation giving a decibel result in the neighborhood of 100 should be questioned immediately. The input power to a device is 10,000 W at a voltage of 1000 V. The output power is 500 W, while the output impedance is 20 . (a) Find the power gain in decibels. (b) Find the voltage gain in decibels. (c) Explain why parts (a) and (b) agree or disagree. EXAMPLE 11.6 Solution Po 500 W 1 (a) GdB  10 log10   10 log10   10 log10   10 log10 20 Pi 10 kW 20  10(1.301)  13.01 dB Vo P R  (5 00  W2 )(0) (b) Gv  20 log10   20 log10   20 log10  Vi 1000 1000 V 100 1  20 log10   20 log10   20 log10 10  20 dB 1000 10 Vi2 (1 kV)2 106 (c) Ri      4  100   Ro  20  Pi 10 kW 10 11.3 Decibels 499 f EXAMPLE 11.7 An amplifier rated at 40-W output is connected to a 10- speaker. (a) Calculate the input power required for full power output if the power gain is 25 dB. (b) Calculate the input voltage for rated output if the amplifier voltage gain is 40 dB. Solution (a) Eq. (11.10): (b) Gv  20 log10 40 W 25  10 log10  Pi 40 W 40 W Pi     antilog (2.5) 3.16  102 40 W    126.5 mW 316 Vo Vo  40  20 log10  Vi Vi Vo   antilog 2  100 Vi Vo  P R   (4 0)( W10) V  20 V Vo 20 V Vi      0.2 V  200 mV 100 100 11.4 GENERAL FREQUENCY CONSIDERATIONS The frequency of the applied signal can have a pronounced effect on the response of a single-stage or multistage network. The analysis thus far has been for the midfrequency spectrum. At low frequencies, we shall find that the coupling and bypass capacitors can no longer be replaced by the short-circuit approximation because of the increase in reactance of these elements. The frequency-dependent parameters of the small-signal equivalent circuits and the stray capacitive elements associated with the active device and the network will limit the high-frequency response of the system. An increase in the number of stages of a cascaded system will also limit both the high- and low-frequency responses. The magnitudes of the gain response curves of an RC-coupled, direct-coupled, and transformer-coupled amplifier system are provided in Fig. 11.4. Note that the horizontal scale is a logarithmic scale to permit a plot extending from the low- to the high-frequency regions. For each plot, a low-, high-, and mid-frequency region has been defined. In addition, the primary reasons for the drop in gain at low and high frequencies have also been indicated within the parentheses. For the RC-coupled amplifier, the drop at low frequencies is due to the increasing reactance of CC, Cs, or CE, while its upper frequency limit is determined by either the parasitic capacitive elements of the network and frequency dependence of the gain of the active device. An explanation of the drop in gain for the transformer-coupled system requires a basic understanding of “transformer action” and the transformer equivalent circuit. For the moment, let us say that it is simply due to the “shorting effect” (across the input terminals of the transformer) of the magnetizing inductive reactance at low frequencies (XL  2 fL). The gain must obviously be zero at f  0 since at this point there is no longer a changing flux established through the core to induce a secondary or output voltage. As indicated in Fig. 11.4, the high-frequency response is controlled primarily by the stray capacitance between the turns of the primary and secondary wind500 Chapter 11 BJT and JFET Frequency Response f Figure 11.4 Gain versus frequency: (a) RC-coupled amplifiers; (b) transformercoupled amplifiers; (c) direct-coupled amplifiers. ings. For the direct-coupled amplifier, there are no coupling or bypass capacitors to cause a drop in gain at low frequencies. As the figure indicates, it is a flat response to the upper cutoff frequency, which is determined by either the parasitic capacitances of the circuit or the frequency dependence of the gain of the active device. For each system of Fig. 11.4, there is a band of frequencies in which the magnitude of the gain is either equal or relatively close to the midband value. To fix the frequency boundaries of relatively high gain, 0.707Avmid was chosen to be the gain at the cutoff levels. The corresponding frequencies f1 and f2 are generally called the corner, cutoff, band, break, or half-power frequencies. The multiplier 0.707 was chosen because at this level the output power is half the midband power output, that is, at midfrequencies, AvmidVi2 V2o Pomid     Ro Ro and at the half-power frequencies, 0.707AvmidVi2 AvmidVi2 PoHPF    0.5  Ro Ro 11.4 General Frequency Considerations 501 f PoHPF  0.5Pomid and (11.16) The bandwidth (or passband) of each system is determined by f1 and f2, that is, bandwidth (BW)  f2  f1 (11.17) For applications of a communications nature (audio, video), a decibel plot of the voltage gain versus frequency is more useful than that appearing in Fig. 11.4. Before obtaining the logarithmic plot, however, the curve is generally normalized as shown in Fig. 11.5. In this figure, the gain at each frequency is divided by the midband value. Obviously, the midband value is then 1 as indicated. At the half-power frequencies, the resulting level is 0.707  1/2. A decibel plot can now be obtained by applying Eq. (11.12) in the following manner: Av Av dB  20 log10  Avmid Avmid (11.18) Aυ A υmid 1 0.707 f1 100 10 1000 10,000 100,000 f2 1 MHz 10 MHz f (log scale) Figure 11.5 Normalized gain versus frequency plot. At midband frequencies, 20 log10 1  0, and at the cutoff frequencies, 20 log10 1/2  3 dB. Both values are clearly indicated in the resulting decibel plot of Fig. 11.6. The smaller the fraction ratio, the more negative the decibel level. Aυ Aυ 10 0 dB mid (dB) f1 100 1000 10,000 100,000 f2 1 MHz 10 MHz f (log scale) − 3 dB − 6 dB − 9 dB − 12 dB Figure 11.6 Decibel plot of the normalized gain versus frequency plot of Fig. 11.5. For the greater part of the discussion to follow, a decibel plot will be made only for the low- and high-frequency regions. Keep Fig. 11.6 in mind, therefore, to permit a visualization of the broad system response. It should be understood that most amplifiers introduce a 180° phase shift between input and output signals. This fact must now be expanded to indicate that this is the case only in the midband region. At low frequencies, there is a phase shift such that Vo lags Vi by an increased angle. At high frequencies, the phase shift will drop below 180°. Figure 11.7 is a standard phase plot for an RC-coupled amplifier. 502 Chapter 11 BJT and JFET Frequency Response