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Unformatted text preview: 576 a} Chapter 9 e Frequency Response and TimeDependent 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 steadystate 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 Fourierseries 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 steadystate behavior under sinusoidal '
excitation. This information is neatly expressed in the compact, graphical form of a Bode plat (pronounced “Bodee”). 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 frequencydomain 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 highfrequency 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, 6011, 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; 6011 =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 lowfrequency type,
depending on whether its effects are felt above or below the midband range. In an ampliﬁer,
a highfrequency capacitor is deﬁned as one that degrades the gain above the midband range.
Similarly, a lowfrequency 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 highfrequency
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 smallsignal 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 SteadyState 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 frequencydependent
piecewiselinear or smallsignal circuit models. The Bodeplot 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 zeroangle 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
lowfrequency 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 zeroangle reference. In the highfrequency 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—ofIO
increase in a). 578 a Chapter 9 e Frequency Response and TimeDependent Circuit Behavior Figure 3.12 Plot of the
frequency response
of the circuit of
Fig. 9.10: (a) magnitude plot;
(b) phaseangle
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 righthand asymptote slopes downward by a factor of 10 for every
factoroflO 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 phaseangle plot has two jiﬁigtire 9.13 Eliot of the
frequency response
it)!“ the circuit of
:Fig. 9.11: (:1) magnitude plot:
' (b) phaseangle
plot. 579 horizontal asymptotes—one for to << 1 / RC and one for a) >> 1 /RC——located at 0° and —90°,
respectively. The phaseangle 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 SteadyState 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‘ 01' 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 TimeDependent Circuit Behavior The Bodeplot 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 factorof—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
highfrequency 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 lowfrequency 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 SteadyState 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 Bodeplot 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 TimeDependent 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 frequencydomain 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 ‘   Lowfrequency 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 SteadyState 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 Bodeplot 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 TimeDependent 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 frequencyresponse 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 @[email protected] 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 SteadyState 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 signalprocessing 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 ﬂatresponse 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 lowfrequency pole at wa = 10 rad / s and two highfrequency poles—one
atw1=104rad/s and one at £02 = 2 X 104 rad/s. IH I' (dB) Low—frequency limit Highfrequency limit Flat midband region +60
+50 +40 The low and highfrequency 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 TimeDependent 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 lowfrequency and highfrequency
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 highfrequency 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. HighFrequency 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 SteadyState 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—ofpoles approximation at the
highfrequency end of the midband. I According to (9.64), the highfrequency 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. LowFrequency 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 lowfrequency 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 TimeDependent 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 lowfrequency 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 lowfrequency poles (a)a   ' com).
As indicated by Eq. (9.69), the lowfrequency 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 lowfrequency
types. The upper —3dB 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 —3dB 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 superpositionof—poles approximation will
always slightly underestimate the actual value of my and slightly overestimate the actual value
of am. ' EXAMPLE 9.3 Use the superpositionofpoles 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 lowfrequency pole, hence the superpositionofpoles approximation is not
needed to ﬁnd coL.) ...
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 Spring '09
 Horenstein
 Frequency, Decibel, RC RC RC

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