BodePlotNotesEK307--

BodePlotNotesEK307-- - 576 a} Chapter 9 e Frequency...

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Unformatted text preview: 576 a} Chapter 9 e Frequency Response and Time-Dependent Circuit Behavior ' ea er $®i§§im giEfiflYmgafATE $tFiER §E§P®N$E The various capacitances described in Section 9.1, as well as any discrete capacitors specifiers added by the designer, all influence the response of an electronic circuit. Indeed, the body n this chapter deals with methods for dealing with and predicting the effect of capacitance circuit response. Before embarking on a study of these effects, however, we first review scve‘ : key concepts and definitions that pertain to the frequency domain. In the frequency domui i a circuit is assumed to have been excited for some time by a sinusoidal input, such that r natural, transient responses have decayed to zero. Under such sinusoidal steady-state excitatin every voltage and current signal in the circuit acquires the frequency of the input and can 3 represented by a phasor. More importantly, each capacitor in the circuit can be represcmt by a frequency—dependent impedance of value l/jcoC. This feature transforms the differentl equations that normally govern capacitive circuits into simple algebraic equations. Any arbiu‘rn input signal can always be represented as a Fourier-series superposition of sinusoids of diffcre frequencies and amplitudes. Knowledge of the Circuit’s response to the individual sinusoid Fourier components of the input allows the designer to~predict the Circuit’s response to a comple periodic signal. The next three sections are devoted to a review of concepts that are important 1 the frequency domain. The study of actual circuits that contain capacitance begins in Section 9.. $2.3. Eede Pier Representatiee in the Freeeeney Domain The input—output response of a circuit in the frequency domain under sinusoidal steady~stule conditions is called the Circuit’s system function, or sometimes the transfer function.3 The system function contains a wealth of information about the Circuit’s steady-state behavior under sinusoidal ' excitation. This information is neatly expressed in the compact, graphical form of a Bode plat (pronounced “Bo-dee”). When a linear circuit has a frequency—dependent system function, boll! L the magnitude and phase angle of the response are variables of great interest. It is often useful to know their values over very large ranges in frequency spanning several orders of magnitude. Similarly, it is often desirable to assign equal importance to the lower and higher ends of the frequency spectrum. The Bode plot consists of a set of straight lines placed on a graph with the frequency on the horizontal axis and either the output amplitude or phase angle on the vertical axis. The straight lines serve as asymptotes that closely represent the actual circuit response, but are much easier to manipulate and analyze. We shall first develop the Bode plot for the simple circuits of Figs. 9.10 and 9.11. These simple circuits highlight the key role of capacitors in many electronic circuits. We then extend the concept to encompass more complicated circuits having system functions of arbitrary complexity. 3 More accurately, the term transfer function is used to describe the frequency-domain relationship between inpul and outputsignals appearing in different parts of the circuit. The more general term system function include. transfer functions, but can also be used to describe the impedance or admittance of a single port. Section 9.3 e Frequency Response of Circuits Containing Capacitors a 589 Salaries H (jco) begins at low frequencies with a solitary factor of jw and an initial slope of +20 dB/decade. The flat midband region thus begins at the lowest~frequency pole a) = 10 rad / s. The upper —3—dB limit of the midband region can be estimated by superimposing the two remain— ing high-frequency poles: 1 my = 601!le — z 0.67 x 104rad/s (9.73) _ 1/104+1/(2 x104) This estimated value for mg is slightly lower than the true value 60;; = 0.84 X 104 rad / 3 obtained from Eq. (9.58). Find wL, 601-1, and the midband gain of the system function jw/2 0‘0) = 5(1+ja)/2)(1+ jw/105)(1+ 110/106) Answer: wL = Zrad/s; w” w 9.1 x 104rad/s; A, = 5 E 14 dB 9.7 Find cuL, (DH, and the midband gain of the system function of Exercise 9.5. Answer: wL = 2.3 x lO‘frad/s; 601-1 =106rad/s;Ao =1.1 x 105 E 101 dB ‘ . 35 sssss es asses ss as s assesses ' RCEM‘S CQNTA 7‘ ENG The concepts presented in Section 9.2 provide powerful tools for working in the frequency domain. With these tools mastered we can now understand the effects of capacitance (and inductance, where important) on circuit behavior. In the sections that follow, we shall use these tools to analyze and design real electronic circuits. To facilitate the connection between the abstract concepts of Section 9.2 and the real circuits of the rest of the chapter, we first provide several key definitions that help categorize the role of each capacitor in shaping circuit response. Higi’s and LewFreaasney Capaerers The influence of a given capacitance often occurs at a frequency that lies either above or below a circuit’s midband region. Conversely, the midband represents the frequency range over which circuit behavior is unaffected by circuit capacitance. From a frequency—domain point of view, it is often useful to categorize a given capacitor as either a high—frequency or low-frequency type, depending on whether its effects are felt above or below the midband range. In an amplifier, a high-frequency capacitor is defined as one that degrades the gain above the midband range. Similarly, a low-frequency capacitor is defined as one that degrades the gain below the midband range. Because capacitive impedance is inversely proportional to frequency, it follows that a low—frequency capacitor must behave as a short circuit in the midband, while a high-frequency capacitor must behave as an open circuit in the midband. As a general rule, a given capacitor will function as a low—frequency type if it appears in series with a Circuit’s input or output terminal. Conversely, a capacitor will function as a high- frequency type if it shunts an input or output node to small-signal ground. According to this , ‘ §.Z§.® e RC circuit C re capacitor as Q. Li 3. Bit: RC circuit the capacitor iih’tmt element. 'ies element. Section 9.2 @ Sinusoidal Steady-State Amplifier Response a 577 In general, the use of Bode plots is limited to linear circuits. Many nonlinear circuits, how ever, including the amplifier circuits of this chapter, can be represented by frequency-dependent piecewise-linear or small-signal circuit models. The Bode-plot formulation is useful for describ— ing the small—signal frequency response of these circuits as well. A complete Bode plot consists of two separate parts. The first shows the magnitude of the output variable relative to the input variable as a function of frequency. The second part shows the phase angle of the output variable relative to the input variable as a function of frequency. The angle of the input variable is arbitrarily (and for convenience) taken as the zero-angle reference. As an example, consider the Bode plot for the simple circuit of Fig. 9.10, which consists of a series resistor and a shunt, or parallel, capacitor. The system function of this circuit becomes, via voltage division Vout 1 / C 1 V... _ R+1/jcuC _1+ja)RC (9.32) where the capacitor is treated as an element having impedance 1 /ja)C. As an aid in drawing the Bode plot, we note the behavior of the system function at three extremes of frequency. In the low-frequency limit u) << 1 / RC , the imaginary part of the denominator becomes negligible, and the system function (9.32) reduces to Vom/Vin = I so that Vout Vin and 4 Vout = O = 1 (9.33) (9.34.) where the angle of Vin is taken as the zero-angle reference. In the high-frequency limit a) >> 1 / RC , the imaginary term in the denominator of Eq. (9.32) becomes larger than the real term, so that the system function reduces to V°“‘ —> 1 (9.35) Vin ijC with V°“‘ = —1~ and 4V0," = ~90° (9%) Vin CORC In this limit of large a), the magnitude [Vent/Vin] decreases by a factor of 10 for every factor—of-IO increase in a). 578 a Chapter 9 e Frequency Response and Time-Dependent Circuit Behavior Figure 3.12 Plot of the frequency response of the circuit of Fig. 9.10: (a) magnitude plot; (b) phase-angle plot. At the boundary between high- and low—frequency extremes, which occurs at the point w = 1/ RC, the magnitude of the real and imaginary terms of the denominator of Eq. (9.32) become equal to each other, so that the magnitude and angle of the system function become Vout 1 1 = —. = —— = 0.707 93% vi. 1+ 1 J5 ( i and 4 Vout = “ 4 (1 + = _450 (938} In Fig. 9.12, the magnitude and phase angle of the circuit of Fig. 9.10 are plotted as functions of frequency on logarithmic scales. The plots include the three limiting region cases described above. Both magnitude and frequency are plotted logarithmically, so that the high and low ends of the axes are given equal graphical weighting. Note that a logarithmic scale has no zero point and a logarithmic graph has no origin; hence the point at which the vertical and horizontal axes cross on the magnitude plot is arbitrary. Breakpoint Magnitude' . co(rad/s) scale RC RC RC RC RC RC scale “t 1.09 - 1109' RC RC RC RC RC RC (1) (rad/s) For u) << 1 / RC , the magnitude plot approaches the horizontal asymptote lVom/Vinl = 1 given by Eq. (9.33). For a) >> 1 / RC , the plot approaches the asymptote given by Eq. (9.36). These asymptotes together constitute the Circuit’s magnitude Bode plot. Above their point of intersection at a) = l / RC , the right-hand asymptote slopes downward by a factor of 10 for every factor-of-lO increase in a). It can be shoWn that at the breakpoint a) = 1 / RC, the actual magnitude curve falls by l/fi from the value at the point of intersection. The phase-angle plot has two jifiigtire 9.13 Eliot of the frequency response it)!“ the circuit of :Fig. 9.11: (:1) magnitude plot: ' (b) phase-angle plot. 579 horizontal asymptotes—one for to << 1 / RC and one for a) >> 1 /RC——located at 0° and —90°, respectively. The phase-angle plot passes through the —45° point at the breakpoint w = 1 / RC . It is often convenient to express the logarithmic magnitude scale of the Bode plot with a unit called the magnitude decibel, defined by Section 9.2 a Sinusoidal Steady-State Amplifier Response a (9.39) The decibel is a logarithmic unit; hence a dB scale used in a logarithmic plot appears linear, as in Fig. 9.12(a). We next consider the circuit of Fig. 9.1 1, which consists of a series capacitor and a shunt resistor. The system function of this circuit is given, again using voltage division, by Vout _ R _ ijC Vin “ R+1/ij _ 1+ij€ (9.40) IVout/VinI l 0dB=l' —20 dB =0.l' I I I I I I I I | | —40 dB = 0.01” Slope = +20 B/dec in 0) —60 dB = 0.001,. __.______.____ D. —80 dB = .0001 1220' RC RC RC RC RC RC (0 (rad/s) 0._01‘ 0-1' 1' a 192' RC RC RC RC RC RC 1&0' C0 (rad/s) 580 e EXERCHSE 9.1 9.2 9.2.2 Cemeiexity Chapter 9 e Frequency Response and Time-Dependent Circuit Behavior The Bode-plot asymptotes can be found from the three limits at to << 1 / RC , a) >> 1 / RC , and a) = 1 / RC : Vom jcoR C At w<<1/RC: —>| 1 l=wRC andeOm—>+90° (eat) In At w>>1/RC: 91% =1 andngom—>O° (9.43:2; At a): l/RC: andeout=90°—45°=45° (Mt) In The low—frequency limit (9.41) has a factor of a) in the numerator. For a) << 1/ RC , the magnitude plot thus approaches an asymptote with an upward slope of 20 dB per factor-of—lO change in a), as shown in Fig. 9.13. Similarly, for a) >> 1 / R C , the magnitude plot asymptotically approaches a constant value of unity. It can be shown for this system function that the low— and high-frequency asymptotes cross at the breakpoint a) =: 1 / RC, where the actual magnitude plot passes 1 /«/2 below the breakpoint'crossing. The factor of 1/«/2 = 0.707 can also be expressed in decibels as dB = 20 loglo 0.707 % ~3 dB At low frequencies, the Bode plot of Fig. 9.13 has an upward slope of +20 dB per decade in a). This slope results because the low-frequency limit (9.41) has a factor of a) in the numerator. Suppose, for example, that at some low frequency (1)1 << 1 / RC, the magnitude has a decibel value of dBl = 20 logic lVout/Vinl = 20 10g10 w1RC (94%} where [Vom /Vin] is expressed using the limiting case (9.41). At some higher frequency 602 = 10601 that still satisfies the limit 602 << 1 /RC, the decibel value becomes (“32 = 20 102310 10601RC = 2010g10 00ch + 20 lOg10 10 = (131 + 20 (9.933) This value is 20 decibels more than the decibel value at col. Draw the magnitude and angle Bode plots for the circuits of Figs. 9.10 and 9.11 if the capacitor is replaced by an inductor of value L. Show that the slopes of the nonhorizontal portions of the magnitude plots of Figs. 9.12 and 9.13 have values equal to :i:6 dB per octave, where an octave is a factor—of—Z change in frequency. ede—Pet Reeteeetttetten et System Feeetiees et Arbtttety In later sections of this chapter, we will examine circuits with system functions that are far more complex than those of Eqs. (9.32) and (9.40). Fortunately, the task of constructing the Bode plot of any circuit, no matter how complex, is greatly simplified if its system function can be expressed in the general form MG ‘1‘ + jw/w4) . . . <1 +jw/wi)(1 + jw/w3)(1 + jw/ws) - -- H (jco) = A The numbered frequencies col - - - a)” are the breakpoints of the system function, and A is a constant. The solitary factor of flu in the numerator is not present for all circuits. If the binomial containing a given breakpoint frequency a)” appears in the numerator, then a)” is called a zero of the system function. If the binomial appears in the denominator, then can is called a pole. Section 9.2 s Sinusoidal Steady-State Amplifier Response a 581 Regardless of its type, a binomial term containing (0,, will affect the circuit’s magnitude and phase response as the driving frequency approaches and passes through the value a)". Suppose that the frequency a) of the input signal driving the circuit initially lies well below a)”. In such a case, the binomial term containing can will alter neither the magnitude nor the phase of the system function, but will simply multiply the system function by unity. This statement can be verified by observing the characteristics of a single binomial term for frequencies well below a)": {1+ 3 e 1 for 0) << w, (9.48) a)” and 4 (1+ z 0° (9.49) Conversely, if the driving frequency a) lies well above a given breakpoint frequency a)", the binomial term associated with can will contribute a factor of w/wn to the magnitude of the system function and an angle factor of 90°. This statement can be verified by noting that ja) a) ll + — 3 ~—— for a) >> can (9.50) can can and ‘ 4 (1+ E) e 4 L“? = 90° (9.51) can can If the binomial appears in the numerator as a zero, the factor of w/wn will appear in the numerator, and the angle contribution of 90° will be added to the overall angle. If the binomial appears in the denominator as a pole, the contributed factor of w/wn will appear in the denominator, and the angle contribution of 90° will be subtracted from the overall angle. The transition between the extremes 0:) << 60,, and a) >> con occurs at a) = a)”. At this frequency, the binomial of con contributes a factor of J? to the magnitude of the system function and an angle of 45°. The validity of this statement can be shown by noting that at a) = con, ll+i3=|1+j1=¢12+12=¢§ (9.52) 1'60 . o and 4 (1+3) = 4(l+])=45 (9.53» It When the numerator of the system function contains a single non—binomial factor of ja), a factor of a) will be contributed to the magnitude and a constant factor of 90° will be contributed to the phase angle at all values of the driving frequency a). Given these guidelines, the Bode-plot asymptotes that describe the magnitude and phase response of a system function of the form (9.47) are easily constructed. We briefly review the procedure here. The processbegins by considering frequencies well below the lowest break— point of the system function. At such frequencies, the response will be flat (zero slope) with magnitude A and phase angle zero. (If the numerator contains a solitary factor of ja), the re- sponse at low frequencies will instead have a magnitude of Aw, a phase angle of 90°, and an initial slope of +20 dB/decade.) The system function is next evaluated as the frequency is in- creased. As the frequency passes through a given breakpoint con, its binomial term will begin to contribute a factor of cu/wn to the magnitude of the system function. If the binomial appears in the numerator, the slope of the asymptote describing the magnitude response will increase by +20 dB/decade. If the binomial term appears in the denominator, the slope of the asymptote will decrease by —20 dB/decade. 582 e Chapter 9 e Frequency Response and Time-Dependent Circuit Behavior The angle portion of the Bode plot can be constructed in a similar fashion. When t binomial term appears in the numerator, the angle of the system function will undergo a pha shift of +90° as the frequency passes through the breakpoint. If the binomial term appears in t denominator, the phase shift will be ~90°. The phase shift contributed at the breakpoint will i equal to +45° or ~45°, respectively. If a solitary factor of ja) appears in the numerator of ti system function, the Bode plot will begin with an upward slope of +20 dB/decade and a pha angle of +90° at low frequencies. In the following examples, the techniques for constructing a Bode plot are illustrated fi two cases. The first involves a system function whose response is flat at low frequencies. Tl second involves a system function with a factor of ja) in the numerator. WW». EXAMPLE 9.1 Figure 9.14 Magnitude plot of the system function of Eq. (9.54). Draw the magnitude and angle Bode plots of a circuit that has a frequency-domain system functic given by V0... _ 100 Vin _ (1 +J'w/102)(1+J'w/106) H (10)) = (9.5 Seiutien The system function (9.54) has one pole at a) = 102 rad / s and one at 106 rad / 5. At frequencies we below the lowest pole at w = 102 rad / s, the magnitude of the system function is flat and approache the limit IHI = 100 _=. +40 dB, as shown in Fig. 9.14. Above the pole at a) = 102 rad/s, th asymptote describing the magnitude response acquires a slope of ~20 dB/decade. The actua magnitude curve lies —3 dB below the asymptote intersection at point A. Above the second poll at a) = 106 rad /s, the asymptote acquires an additional slope of ~20 dB/decade, for a total slop< of ~40 dB/decade. With no other poles or zeros in the system function, this new slope continue; indefinitely for all higher frequencies. The actual magnitude curve again lies —3 dB below thl asymptote intersection at point B. IVom/Vinl dB ‘ ----- -- Low-frequency asymptote (0 dB/dec) 20' —20 dB/dec asymptote 100 104 1000 108 co (rad/s) The angle plot of Eq. (9.54) is shown in Fig. 9.15. Well below a) = 102 rad/s, the angle of the system function approaches zero. As the first pole at a) = 102 rad /s is passed, the angle undergoes a net phase shift of ~90°, with its value precisely at a) = 102 rad/s equal to ——45"‘,__¢ As the second pole at a) = 106 rad /s is passed, it contributes an additional phase shift of —9()“;i for a total phase shift of —180° well above a) = 106 rad/s. The total phase shift precisely it a) = 106 rad/s is ~135°, with ~90° contributed from the pole at a) = 102 rad/s and “15$ contributed by the pole at a) = 106 rad / s iléE 9.2 Section 9.2 e Sinusoidal Steady-State Amplifier Response e 583 4 Vent/Vin 00‘ V —90° phase shift' contributed by the pole" at a) = 102 rad/s —45°' —90°' —,90° phase' shift contributed" by the pole' at 106 rad/s —l35°' RR“ 7. MM —18_0° 10° 101 102 103 104 105 106- 10 18 a) (rad/s) Construct the Bode plot of a circuit whose input—output system function is given by , V ja)(1 + 'w 10) H(_]a)) = ‘om = 50—-_—4-‘-]-——/,—7~ (9.55) Vin (1+ Jw/10)(1+ Jw/lO) This system function has a solitary factor of jaw in the numerator. |Vent/VinI (dB) 240' 40 dB/dec 200' 20 dB/dec 160' 120' 80' 40' 20 dB/dec 10 100 103 104 105 106 107 108 109 l_*_l Breakpoints 1 a) (rad/s) 20 Seution The magnitude Bode plot of Wont/Vinl for the system function (9.55) is shown in Fig.9.16. The point of intersection of the two axes is arbitrary. The system function contains a solitary factor of flu in the numerator, hence the plot begins with a positive slope of +20 dB/decade for frequencies below the lowest breakpoint a) = 10rad/s. At the frequency a) = 10, the zero in the numerator takes effect and the slope of the Bode-plot asymptote acquires an additional factor of +20 dB/decade to become +40 dB/decade. At the frequency a) = 104 rad / s, the first pole in the denominator is encountered, and the asymptote slope is reduced by ~20 dB/decade to again become +20 dB/decade. Finally, at the second pole frequency a) = 107 rad /s, the asymptote acquires another factor of —20 dB/decade and becomes horizontal for all frequencies greater than 584 a Chapter 9 a Frequency Response and Time-Dependent Circuit Behavior (0 = 107 rad / s, which is the highest breakpoint of the system function. At each of the breakpoints in the system function (9.55), the actual frequency-response curve falls +3 dB or —3 dB above or below the intersection points of the asymptotes. Well above the highest breakpoint frequency a) := 107, the magnitude of the system function can be approximated by lVoutl N w(w/10) _. 50(104)(107) __ 11 fl wm’V afififiaimfi-"*—fir—-~SUO>=2MdB @5@ Note that the factors of a)2 cancel out in the numerator and denominator in Eq. (9.56), leaving a term that is constant with frequency. The angle portion of the Bode plot of Eq. (9.55) is shown in Fig. 9.17. In this case, the solitary factor of ja) in the numerator contributes an initial angle of +90° to the plot. Above the zero at a) = 10 rad/s, an additional angle of +90° is contributed, making the total angle +180°. Above the next breakpoint at a) = 104, which is a pole, the angle is reduced by —90° to +90°. Above the highest pole at a) = 107, the total system function angle is again reduced by —90° to zero, which is a result consistent with the horizontal slope of the magnitude plot at high frequencies. Note that precisely at the location of each of the breakpoints, the system function angle is shifted by half the overall 90° angle shift contributed by the breakpoint. Figure 4 Vout/Vin Angle plot of the system function of +1800 _______ _._ t29i. -. ___.__________I::. l I i | l l | l 0° ————-—}———————————:— I | l | I I —90° 1 l i l 1 10 100 103 104 105 106 107 108 109 w(rad/s) l—Breaifpoims——l EXERCBSE 9.3 Draw theimagnitude and angle Bode plots of the circuits of Figs. 9.10 and 9.11 if R = 5 k9 and C = 10 ,u.F. 9.4 A circuit has a system function with poles at a) = 500 rad /s and 3 x 105 rad /s. At a) = O, the system function has a value of 50. Draw its magnitude and angle Bode plots. 9.5 Draw the magnitude and angle Bode plots of the system function jw(l +jw/50) [1 + jaw/(3 >< 102)][1+ja)/(2 x103)][1+ja)/106] H(ja)) = 9.5 W Section 9.2 9 Sinusoidal Steady-State Amplifier Response 9 585 Higi’h town, and MidbandsFrequeney Limits .1 8 le plot of n function )wing two )aced rad / s and :104 rad/s. Many signal-processing applications require a circuit or system to have a flat, constant response over a specified range of frequencies called the midband. If the frequency components of the input signal are confined to this range, the output will replicate the form of the input and have the same spectral content. For such circuits, the locations of the specific poles and zeros of the system function are of less interest than the frequency range over which the response may be considered flat. The flat-response region is usually the portion of the Bode plot with maximum magnitude. Its limits are therefore defined as those frequencies wL and why at which the magnitude of the system function first falls by a factor of l/fl, or —3 dB, from the horizontal. A magnitude reduction of 1 /«/5 corresponds to halving of the power delivered to a resistive load. The limits of the midband region may not always coincide with individual poles. Multiple poles may contribute simultaneously to the degradation of the Circuit’s output amplitude. This concept is illustrated in Fig. 9.18, which depicts the magnitude Bode plot of the system function: jw/lO (1+jw/10)(1+ja)/104)[1+jw/(2 X 104)] H(ja)) = 1000 (9.57) Equation (9.57) has a low-frequency pole at wa = 10 rad / s and two high-frequency poles—one atw1=104rad/s and one at £02 = 2 X 104 rad/s. IH I' (dB) Low—frequency limit High-frequency limit Flat midband region +60 +50 +40 The low- and high-frequency limits (0], ande are used to designate the ends of the flat midband region, which has a magnitude of [H l = 1000 2 +60 dB. One might assume from the discussion of Section 9.2.1 that (01 = 104 rad /s, the first pole to be encountered above the midband, represents am. The system function (9.57) has another nearby pole at w; = 2 X 104 rad / 5, however, which also contributes to the reduction of the Bode plot magnitude at an. The exact value of (0;; can be computed by solving for the frequency (0;; at which |H l falls by 1 /«/§ from its midband value of 1000: IHI _ IOOwH _ 1000 (9 58) mm” _ [1+(wH/10)2]1/2[1+(wH/104)2]1/2[1+(C0H/2 x 1002]”2 _ «0 ° This equation can be solved for am to yield (0,, m 0.84 x 104rad/s (9.59) This frequency is lower than the breakpoint (01 = 104 because the nearby pole at wz = 2 X . AA 1 I L , _‘_ _._- 4-- n..- “mime. rnnnfincn of Frpniipnnipc near my, 586 3 Chapter 9 a Frequency Response and Time-Dependent Circuit Behavior 9.20% Superposton sf Poles For a system function like Eq. (9.57), which exhibits a clearly defined midband region, the 10- cations of w” and an, can always be found exactly by solving an equation of the form (9.58). Such calculations, however, become tedious for system functions with many closely spaced poles. In such cases, a simplifying technique called the superposition—of—poles approximation provides reasonable estimates of wL and am while eliminating much of the tedious algebra. The superposition—of—poles approXimation can be applied to any system function that can be put in the form of a midband—gain multiplied by separate low-frequency and high-frequency system functions. Such a system function will have the overall form H(jw)=Ao - HL ' HH = [ (flu/(0a) (flu/60b) (jw/wm) ] 0 (1 + jw/walfl + 160/6012) (1 + jw/wm) >< (9.6%) (1+ Jw/w1)(1+ Jw/wz) ‘ - ' (1 + Jw/wn) l 1 HL th HH Here A, is a constant equal to the magnitude of the system function in the flat midband region, and HL and H H constitute the low- and high-frequency contributions, respectively, to H ( jaw). The breakpoints coa - ‘ . cum of HL jointly define the low—frequency limit of the midband. The breakpoints col - - 'can of _HH define the high—frequency limit of the midband. Note that Eq. (9.57) is of the form (9.60), with A0 = 1000, w, = 10, col = 104, and 602 = 2 x 104. High-Frequency Limit At the high—frequency end of the midband, the poles of H L have no effect on the response. At w = {0}], for example, each of the binomials in the denominator of HL approaches the value j wH /wm, canceling the corresponding factor ij /a)m in the numerator of HL, so that 1H LI ——> 1. At frequencies near my, H (jw) therefore can be approximately expressed by H(ja))%A HH——-—————fi’——~—_ (961} " (1+jw/w1)(1+jw/w2)---(1+jw/wn) The denominator of Eq. (9.61) consists of a product of binomials that can be multiplied out and put in the form , 1 1 1 _ 2 1 1 1 1 1+1w —+——+---+——- +(Jw) r——+ + +--»+ (01 (1)2 a)" 60le (0160,, 60260" ijn (9 $2) . 1 I n + (J'w)3 (terms of the form > + .. . + . (1w) ijkwn colon. . . a)” The second term in Eq. (9.62) contains the factor jcu/wn from each binomial; the third term contains all possible combinations of w2 /wj wk; the fourth term contains all possible combinations of order co3, and'so on. The final term is equal to Ow)" /(w1w2 - ~ - can). By definition, all of the poles (01 through can of Eq. (9.61) are higher than the midband endpoint my. Thus, at frequencies near my, terms of order a)2 or higher in Eq. (9.62) may be ignored, because these terms will be much smaller than terms of order a). This approximation is weakest when two poles coincide exactly near am, but can be shown to yield moderately good Section 9.2 @ Sinusoidal Steady-State Amplifier Response e 587 results even in such a case (see Problem 9.36). Neglecting terms of order (02 or higher in the denominator of equation (9.62) allows the approximate high—frequency system function (9.61) to be further approximated by A0 1+jw(1 +wi2+---+a,17) Eu? H(jw) m (9.63) The denominator of Eq. (9.63) contains a single binomial term that causes [H I to fall by —3 dB when the imaginary part of the denominator equals the real part. The high—frequency —3 dB point am of the system function (9.60), which constitutes the upper limit of the midband region, will thus be given approximately by 1 1 1 “1 a)”: _+___|_...+_~ (9,64) w] 602 can As Eq. (9.64) suggests, am can be expressed in “parallel combination” notation as col sz - - - “run and can be thought of as the “parallel” superposition of all the individual high—frequency poles col through a)". Equation (9.64) is known as the superposition—of-poles approximation at the high-frequency end of the midband. I According to (9.64), the high-frequency poles with the lowest frequency will make the most contribution to am. If one pole is significantly lower in frequency, it will dominate coy. Similarly, poles located near each other will make nearly equal contributions to my. Any poles located well above my will make little contribution to the value of my. Low-Frequency Limit A similar approach applies at the low—frequency end of the midband. Near the low—frequency end of H ( jw), the poles of the high~frequency function H H have little effect on the response. At such frequencies, each of the binomial terms in H H approaches unity. At frequencies near the low-frequency limit wL, the system function H (jw) given by Eq. (9.60) thus can be approximately expressed by H(jw)%A0HL=AOMMU.M (9°65) (1 + jw/wa) (1 + J'w/wb) (1+ jw/wm) If each of the factors jw/wa through jw/wm is divided into numerator and denominator, Eq. (9.65) becomes 1 H'a) NIAH=A—————-————————————————— 9,66 0 ) " L " (ma/jaw + mob/1w +1)---<wm/jw +1) ( l The denominator of Eq. (9.66) can be expressed in polynomial form as 1 1+.—‘(wa +wb + +wm) + . 2(wawb‘i'wawm +wbwm + "'+wja)m)+ 160 (1a)) 1 (9.67) ---+ [terms ofthe form . 3 (ijkwm):l + - - - + , (warm, - - -wm) (1w) (1%)“ By definition, all the poles cod - - -wm are lower in frequency than the actual low—frequency midband endpoint wL. Hence, at frequencies near a) = wL, the terms of order l/a)2 or higher may be ignored. These terms are presumed to be much smaller than terms of order 1 /a). This 366 e Chapter 9 e Frequency Response and Time-Dependent Circuit Behavior approximation is weakest when two poles coincide exactly near wL. It can be shown, however, that the approximation yields good results even in such a case (see Problem 936). Neglecting terms of order (1/co)2 or higher allows the approximate low-frequency system function (9.65) to be further approximated by 1 H 'a) WA ———-—«-———————.. (J) “1+(1/jw><wa+wb+-~+wm) Multiplying numerator and denominator by jw and dividing both by (and + w}, + - - -+ com) results in . H (jw) = ——-————————Jw./(w“ + w” + ' ' ' + ‘0’") (9.68) 1+[Jw/(wa+wb+'”+wm)] The denominator of this expression contains a single complex binomial that describes the lower ~3—dB endpoint of the system function (9.60). According to Eq. (9.68), the value of this low—frequency limit will be given approximately by wmea+wb+"'+wm (969) As this expression suggests, the low—frequency —3—dB limit of the midband region may be expressed as an additive “series” superposition of all the low-frequency poles (a)a - - ' com). As indicated by Eq. (9.69), the low-frequency poles with the highest value will make the most contribution to coL. If one pole is significantly higher in frequency, it will dominate. Poles located near each other will contribute in nearly equal amounts to wL. Similarly, any poles located well below 0);, will make little contribution to the value of am. Summary of Method In summary, when a system function has a clearly defined midband region, the superposition- of—poles approximation may be applied by classifying all poles as either high- or low-frequency types. The upper —3-dB point 6011 of the flat midband region can be estimated by a parallel superposition of poles: 1 PVme a, 9°70 1/w1+1/w2+..v.+1/wn liiwz n ( ) (OH The lower -—3-dB point wL of the flat midband region can be estimated by a series superposition of poles: wwaa+wb+~~+wm ‘ (9.71) If multiple poles exist at either end of the midband, the superposition-of—poles approximation will always slightly underestimate the actual value of my and slightly overestimate the actual value of am. ' EXAMPLE 9.3 Use the superposition-of-poles approximation to estimate the upper —3—dB endpoint mg of the system function: jw/lO H Um) = WWW (9.72) Compare the result to the true value (9.59) obtained from Eq. (9.58). (This system function contains only one low-frequency pole, hence the superposition-of-poles approximation is not needed to find coL.) ...
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BodePlotNotesEK307-- - 576 a} Chapter 9 e Frequency...

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