FUNDAMENTAL CONCEPTS IN FILTER DESIGN · FUNDAMENTAL CONCEPTS IN FILTER DESIGN 3º Telecomunicación Escuela Superior de Ingenieros, Universidad de Sevilla 2 - 2 2.1. Filters: concept

Curso 2004/05 © Departamento de Electrónica y Electromagnetismo

Lecture 2

FUNDAMENTAL CONCEPTS IN FILTER DESIGN

2.1. Filters: concept and specifications ............... 2-2

2.2. Filters classification........................................ 2-3

2.3. Parameters normalization ............................ 2-11

2.4. Sensitivity ...................................................... 2-11

Problems........................................................ 2-16

Appendices.................................................... 2-23

FUNDAMENTAL CONCEPTS IN FILTER DESIGN 3º Telecomunicación

Escuela Superior de Ingenieros, Universidad de Sevilla 2 - 2

2.1. Filters: concept and specifications

An electric filter is a system, or circuit, that modifies, distorts, deformsor manipulates in any form, the frequency spectrum of a signal —the filter’sinput. This is realized according to a set of requirements, commonly knownas specifications. Filters can be employed to attenuate or amplify thecomponents of the input that are within a certain frequency range. They canbe used also to reject or isolate specific frequency components. A filter canthus be considered to be a transmission system with the ability to let somefrequency components pass and and some others not. In these conditions, thefollowing frequency bands can be defined:

a) Passband (PB): is the frequency range, or ranges, for which inputcomponents are transmitted to the output. Each component of theinput falling within this set of frequencies, is going to show up at theoutput port. Of course, not without some modification of itsamplitude and phase.

b) Stopband (SB) or rejection band: is the frequency range, for whichthe input components are not transmitted to the input. Any inputcomponent at this set of frequencies is rejected, or at least greatlyattenuated.

c) Transition band: frequency range between the passband and thestopband.

The specifications of the filter will consist in:

a) Cutoff and stopband frequencies: are the frequencies where thepassband ends and the stopband begins, respectively.

b) Passband(s) ripple(s) and stopband(s) attenuation(s): are theattenuation limits beyond which the filter does not perform asdesired. There will be a maximum attenuation allowed at eachpassband, the so-called ripple. There will be also a minimumrequired attenuation at each stopband.

c) Other characteristics related with the form of the transfer function:magnitude, phase, group delay, transmission zeroes, etc.

Circuit Analysis and Synthesis Lecture 2

2 - 3 Curso 2004/05 © Departamento de Electrónica y Electromagnetismo

2.2. Filters classification

Lowpass filterThe basic function of a lowpass (LP) filter is to pass low frequencies

from dc to some desired cutoff frequency and to attenuate high frequencies.The specifications for a lowpass filter are shown in Fig. 2.1. The LP filterpassband is defined between 0 and the cutoff frequency . Signals withinthis range will be attenuated as much as dB. The stopband starts andends at infinity. Input components at frequencies above must beattenuated dB at least. The frequency band within and is thetransition band. The specifications of the LP filter are completely describeby the parameters , , , and .

A second-order transfer function implementing a lowpass characteristichas the form:

(2.1)

Attenuation at low frequencies approaches unity, 0dB, as shown in Fig. 2.2.While at higher frequencies it increases with , in other words, with a slope

ωpAp ωs

ωsAs ωp ωs

ωp ωs Ap As

Atte

nuat

ion

dB

ω

passband (PB)

stopband (SB)

transitionband

ωp ωs

Ap

As

Figure 2.1: Lowpass filter specifications.

Gain H s( )VoVi------ b

s2 as b+ +--------------------------

ωo2

s2 ωoQ------s ωo

2+ +

----------------------------------= = = =

s2



of . This function has two poles, as depicted in Fig. 2.2(b).The location of the poles establishes theshape of the filter response in thepassband. For a high , the peak in the passband ripple occursapproximately at . As grows, the peak will be larger and narrower. Acommon application: tone control in Hi-Fi amplifiers.

Highpass filtersA highpass (HP) filter pass all components at frequencies above the

cutoff frequency , while attenuates all frequencies below the stopbandborder at . The passband spans from to , while the stopband rangesfrom 0 to , as shown in Fig. 2.3. The highpass filter specifications arecompletely characterized by the parameters , , and .

A second order transfer fucntion with a highpass characteristic is:

(2.2)

This gain has a pair of coomplex conjugate poles, an a double zero at, as shown in Fig. 2.4(a). La atenuación a alta frecuencia se aproxima

a la unidad mientras que a baja frecuencia aumenta a , comoaparece en la Fig. 2.4(b).

40dB decade⁄

Qωp Q

Figure 2.2: 2nd-order LP function: (a) attenuation (b) poles and zeros.

0dB

Atte

nuat

ion

dB

ω

Slope =40dB/dec.

ωmin ωo≈

σ

jω

(a) (b)

ωpωs ωp ∞ωs

ωs ωp Ap As

VoVi------ s2

s2 as b+ +-------------------------- s2

s2 ωoQ------s ωo

2+ +----------------------------------= =

s 0=40dB década⁄



Bandpass filtersA bandpass (BP) filter only pass components at frequencies within a

particular band. Any component at a frequency outside this band is rejected,as shown in Fig. 2.5. The passband, that goes from to has amaximum attenuation of . The rejection bands, ranging from DC to and from to , have a minimum attenuation of dB.

A second order transfer fuction with a bandpass characteristic is:

(2.3)

Figure 2.3: Highpass filter specifications.

Atte

nuat

ion

dB

ω

passband

stopband

transitionband

ωpωs

Ap

As

Figure 2.4: 2nd-order HP filter: (a) attenuation, (b) poles and zeros.

0dB

Atte

nuat

ion

dB

ω

Slope– 40dB/dec.

ωmin ωo≈

σ

jω

(a) (b)

ωp1ωp2Ap ωs1ωs2

∞ As

VoVi------

ωoQ------s

s2 ωoQ------s ωo

2+ +----------------------------------=



This function has a pair of complex poles in the left-half-plane of the -plane. It has also a zero at the origin and a zero at infinity (Fig. 2.6(a)). Bothat low and high frequencies the attenuation increases like , i. e.

. At the pole frequency, , attenuation is one (Fig. 2.6(b)).

Band reject filters Band reject (BR) filters are employed to a particular band of

frequencies, as shown in Fig. 2.7. The stopband extents from to .The lower passband ranges from 0 to and the upper passband from to . A second order function with band reject characteristic is

Figure 2.5: Bandpass filter specifications.

Atte

nuat

ion

dB

ω

passband

lower

trans

ition

band

ωp2ωp1

Ap

As

rejectionband

upperrejection

band

trans

ition

band

As

ωs1ωs2

s

s20dB decade⁄ ωo

Figure 2.6: 2nd-order BP filter: (a) attenuation, (b) poles and zeros.

0dB

Atte

nuat

ion

dBω

Slope– 20dB/déc.

ωmin ωo=

σ

jω

(a) (b)

Slope20dB/déc.

ωs1ωs2ωp1ωp2∞



(2.4)

This function has complex poles in the left-half-plane of the -plane. It hascomplex zeros in the -axis. If , the frequency of the zerosequals the pole frequency (Fig. 2.8(a)). Attenuation at lower and higher fre-quencies approaches 1. At the zero, , attenuation reaches infinity(Fig. 2.8(b)). This kind of BR filter is referred as “symmetrical notch”.

If , the function in (2.4) represents a LP function with a

Figure 2.7: Band reject filter specifications.

Atte

nuat

ion

dB

ω

rejectionlower

trans

ition

band

ωp2ωp1

Ap

As

passband bandupper

passband

trans

ition

band

As

ωs1ωs2

Ap

VoVi------

s2 ωz2+

s2 ωoQ------s ωo

2+ +----------------------------------=

sjω ωz ωo=

Figure 2.8: 2nd-oder BR filter: (a) attenuation, (b) poles and zeros.

0dB

Atte

nuat

ion

dB

ωωmín ωo=

σ

jω

(a) (b)

s jωz=

ωz ωo>



transmission zero in the stopband (Fig. 2.9(a)I. In this case, the attenuationat higher frequencies is larger than the attenuation at lower frequencies. Thissi known as a "low-pass-notch" filter. If the opposite occurs.Attenuation is larger at lower frequencies than at higher frequencies. Theresulting function is a highpass filter with a transmission cero in thestopband. It is refererred as "high-pass-notch" filter.

ωz ωo<

Figure 2.9: (a) Low-Pass-notch; (b) High-pass notch.

Atte

nuat

ion

dB

ωωo

σ

jω

(a)

(b)

ωz

Atte

nuat

ion

dB

ωωo

σ

jω

ωz



Delay equalizersAll the previous filter types have been defined attending to their

magnitude of gain characteristics. Nothing has been said about their phase.In order to efficiently realize the magnitude specifications all thetransmission zeros has been located at the -axis. The resulting transferfunction has a minimum phase. If the delay has to be modified, a delayequalizer is employed after the filter. A delay equalizer permitscompensating the distortion in the delay introduced by the previous filteringstage. It introduces a modification on the delay that makes the total delay(the sum of the delay introduced by the filter and the delay introduced by theequalizer) as plain as possible. Of course, the delay equalizer must notdistort the attenuation of the filter.

Then, when magnitude and phase are specified, the transfer functioncan not be of minimum phase. This is, must have zeros in the right-half-plane. We can decompose this function into two ccomponents:

(2.5)

where stands form “non-minimum phase”, for “minimum phase”and for “all-pass”. This decomposition is built by adding poles and zerosto at the mirrored position of the zeros. Then is formedby including in the zeros in the right-half-plane and the poles attheir mirrored locations in . includes the rest of the polesand zeros. A 2nd-order delay equalizer is given by:

(2.6)

The complex poles and zeros of this function are symmetrical with respectto the -axis (Fig. 2.10).

In order to explain why this filter is referred as all-pass, consider that in, the roots of are specular to the roots of , that are

in the left-half-plane. Then,

(2.7)

jω

H s( )

HNMP s( ) HMP s( )HAP s( )=

NMP MPAP

HNMP s( ) HAP s( )NAP s( )

DAP s( ) HMP s( )

VoVi------ s2 as– b+

s2 as b+ +--------------------------=

jω

HAP s( ) NAP s( ) DAP s( )

NAP s( ) DAP s–( )±=



an thus:

(2.8)

The magnitude is always . and thus,

(2.9)

where

(2.10)

An allpass network function can realize and arbitray function of thephase without modifying the magnitude characteristic. In order to meetsome particular magnitude and phase specifications, the first thing is findingan appropriate to meet the magnitude requirements. After that anallpass function is designed so as the total phase or the totaldelay meets the phase or delay specifications. When both stages arecascaded phases, and delays, add up:

(2.11)

The total delay increases, but the important thing is that it can be made arbi-trarily plain. The ideal filter has a linear phase or, equivalently, a constantdelay.

Figure 2.10: Poles and zeros in a 2-nd order delay equalizer.

σ

jω

HAP s( )DAP s–( )

DAP s( )---------------------±=

HAP jω( ) 1=

HAP jω( ) ejφAP ω( )

=

φAP ω( ) 2DAPi ω( )

DAPr ω( )----------------------atan–=

H s( )HAP s( ) φT ω( )

τT ω( )

φT ω( ) φ ω( ) φAP ω( )+=

τT ω( ) τ ω( ) τAP ω( )+=



2.3. Parameters normalization

An usual procedure in analog circuit design is normalizing both infrequency and impedance. Normalization does not lead to loss of generality.It is intended to make calculations easier. It helps in hand analysis, avoidingusing large powers of 10. Normalization minimizes rounding error effects.

Frequency normalization consists in a change of scale. It isaccomplished by dividing the frequency variable by a normalizationfrequency . This frequency has to be properly selected. Then thenormalized frequency variable will be:

(2.12)

Impedance level normalization is accomplished by dividing everyimpedance by a normalization resistance . Linear resistors, inductors andcapacitors are transformed as follows

(2.13)

yielding the following normalized values:

(2.14)

2.4. Sensitivity

In the synthesis process, the circuit designer must select the best circuitconfiguration available. For this, the deviation of the real components fromtheir nominal behaviour has to be considered. Deviations are due to: a) fabrication process tolerances

b) variations caused by environmental factorsc) chemical changes related with aging

d) the use of low accuracy models

Ωo

snsΩo-------=

Ro

RnRRo------= snLn

sΩo------- ΩoL

Ro------------------------= 1

snCn------------ 1

sΩo-------------- 1

ΩoCRo------------------=

RnRRo------= Ln L

ΩoRo-------= Cn CΩoRo=



For the implementation of a filter, all the coeffiecients of the networkfunction depend on the elements of the circuit. Especially the locationof the poles and zeros. A certain deviation of the filter operation from theideal behaviour can be expected. The magnitude of the error will depend onthe tolerances of the elements and the sensitivity of the circuit to each ofthese elements.

The concept of sensitivity is very useful when comparing differentcircuit configurations. It permits establishing their applicability to matchsome particular specifications. Sensitivity helps to:

a) select the best topology between all the available configurationsb) know if the circuit will continue satisfying the specifications in

the future.For a given circuit element , any characteristic of the circuit will

depend on . If P is a network function, it will depend on the frequency aswell: . In order to find the deviation of caused by an error

of the element , we can perform a Taylor expansion of around :

(2.15)

Let us suppose that an that the curvature of around is not large. Then 2nd derivatives and higher can be neglected:

(2.16)

Considering the relative changes:

(2.17)

and then, the small signal sensitivity of with respect to is:

(2.18)

H s( )

x Px

P P s x,( )= Pdx x xo–= xP s x,( ) xo

P s x,( ) P s xo,( )x∂∂ P s x,( )

xo

xd 12---

x2

2

∂

∂ P s x,( ) xoxd( )2 …+ + +=

dx xo⁄( ) 1« P s x,( ) xo

∆P s xo,( ) P s xo dx+,( ) P s xo,( )x∂∂ P s x,( )

xo

xd≅–=

∆P s xo,( )

P s xo,( )-----------------------

xoP s xo,( )-------------------

x∂∂ P s x,( )

xo

xdxo-----≅

P x

SxP xo

P s xo,( )-------------------

x∂∂ P s x,( )

xo

P P⁄∂x x⁄∂

--------------xo

lnP( )dlnx( )d

----------------xo

= = =



Also, the semirelative sensitivity can be defined as:

(2.19)

It can be useful in some cases, for instance, if is evaluated where ,then , that does not report any information, while does. Also,sometimes absolute variations are more interesting than relative variations,like in the actual location of poles and zeros.

The relative change, variability, of a chanracteristic ,

(2.20)

is times the relative change in the parameter .A good circuit should have small sensitivities. In these conditions, for

an acceptable variability of , this is , elements with a largertolerance, , can be employed, resultiong in a cheaper implementation.Also, in a circuit with small sensitivities, it is less probable that thecomponents move out from the acceptability region during circuit operationbecause of their varying values.

These are the conclusions that can be extracted from this definition:a) The Taylor expansion has been truncated at the second order

term. This implies substituting by its slope at .This means that the sensitivity does not provide useful infor-mation if is not small.

b) If depends on frequency, the sensitivity is a function of fre-quency as well. The frequency ranges has to be considered too.For instance, the sensitivity of a network function to the gain ofan amplifier in dc is useless to evaluate the operation at 10kHz.

c) will depend on more than one parameter. It has been calculatedfor the nominal value of all of them. If any one changes, it has tobe re-calculated. Free parameters are usually employed to reducesensitivities.

d) At the end, the important parameter is the variability and not thesensitivity. A large sensitivity to very stable parameters can beacceptable, while a moderate sensitivity to elements with a largetolerance can render a design useless.

QxP x xd

dP=

SxP P 0=

SxP ∞→ Qx

P

P∆PP

------- SxP xd

x-----≅

SxP x

P ∆P P⁄dx x⁄

P s x,( ) x xo=Sx

P

dx x⁄P Sx

P

P



Gain-sensitivity productThe most employed active device in active filters design is the

operational amplifier (OPAMP) (Fig. 2.11(a)). It is a high-gain differentialvoltage amplifier. Its output voltage is defined by:

(2.21)

The open-loop gain, A, is very large. It has also a large variability, .Opamps are employed in feedback configuration with a reduced gain buta smaller variability . Two examples: the non-inverting and invertingamplifiers of Fig. 2.11(b) and Fig. 2.11(c). Consider the non-inverting am-plifier of Fig. 2.11(b):

(2.22)

the closed-loop amplifier gain is:

(2.23)

Similarly, for Fig. 2.11(c) the gain obtained is:

(2.24)

Vo A V+ V––( )=

Figure 2.11: (a) OPAMP symbol, (b) amplifier with non-invertingfeedback, (c) amplifier with inverting feedback.

(b) (c)

Vo

V+

V– AA

R1

R2 K 1–( )R1=

V2

V1

A

R1

R2 K 1–( )R1=

V2V1

(a)

dA A⁄µ

dµ µ⁄

V2 A V1R1

R1 K 1–( )R1+-------------------------------------V2– A V1

1K----V2–

= =

µ1V2V1------ K

1 K A⁄+---------------------

K A«K≅= =

µ2V2V1------ K 1–

1 K A⁄+---------------------

K A«– K 1–( )–≅= =



The sensitivities of the closed-loop gain with respect to the gain in open-loop are:

(2.25)

The sensitivity of with respect to increases with the closed-loop gain. Also the variability of is more reduced that the variability of :

(2.26)

as it can be considered that .for a certain filter characteristic P, that depends on the closed-loop gain

of an amplifier, we have:

(2.27)

and its variability will be:

(2.28)

is proportional to the product of gain and sensitivity, , and to that only depends on the amplifier. Then the variability of a certain

characteristic does not only depend on the sensitivity, but also on the closed-loop gain that is required in the implementation. The gain-sensitivity prod-uct is a better parameter to establish a comparison between different designs.For instance, consider an OPAMP with and . For aparticular design, the sensitivity of a characteristic is . In this de-sign the closed-loop gain is . For a different design with the samefunctionality, the sensitivity is . This time the closed-loop gain re-quired is only . The variabilities obtained are:

SAµ1 K A⁄

1 K A⁄+---------------------

µ1A------= =

SAµ2 K A⁄

1 K A⁄+---------------------

µ2A------–≅= si K 1»

µi Aµi µi A

µidµi-------- SA

µidAA

-------µiA-----dA

A-------≅=

µi A«

SAP Sµ

PSAµ µ

A---Sµ

P= =

dPP

------- µSµP( )dA

A2------- Γµ

PdA

A2-------= =

dP P⁄ ΓµP

dA A2⁄

A 104= dA A⁄ 60%=P Sµ

P1 6=µ 95=

SµP2 38=

µ 4=



(2.29)

what means that the second design is better than the first, though it has alarger sensitivity.

Finally, it is interesting to know that the gain-sensitivity product is thesame in both open and closed-loop configurations:1

(2.30)

From (2.28) it can be extracted that the filter designer must select OPAMPswith the highest , in the frequency range of interest, in order to reduce

.

Solved problems

2.1.- Express the following function, that has non-minimum phase, as theproduct of a minimum phase function and an allpass function:

Solution: has three zeros in the right-half-plane, one at and the

other two at . The first step is adding three poles and threezeros to , at the mirrored positions of , , and :

1. In order to reach that equation, observe that:

dP1P1

--------- 95 6 0.6 10 4–⋅ ⋅ ⋅ 3.4%= =

dP2P2

--------- 4 38 0.6 10 4–⋅ ⋅ ⋅ 0.91%= =

ASAP ASµ

PSAµ µSµ

P= =

ΓAP Γµ

P=

AdA A2⁄

HNMP s( ) s2 2s 6+ +( ) s2 4s– 8+( ) s 3–( )

s2 s 4+ +( ) s2 3s 7+ +( ) s 1+( )-------------------------------------------------------------------------------=

HNMP s( ) s1 3=s2 3, 2 j2±=

HNMP s( ) s1 s2 s3



(2.31)

It can be split into these two parts:

(2.32)

where . The penalty is a higher order in the to-tal system. It is increased in the number of right-half-plane zeros.

2.2.- The circuit in Fig. 2.12 implements a 2nd-order bandpass functionwith a pole frequency and a quality factor . findthe corresponding values for the elements. Obtain the sensitivities and , where represents any passive element. is the positive gainof a voltage amplifier. Ignore its limited bandwith.

HNMP s( ) s2 2s 6+ +( ) s2 4s– 8+( ) s 3–( ) s 3+( ) s2 4s 8+ +( )

s2 s 4+ +( ) s2 3s 7+ +( ) s 1+( ) s 3+( ) s2 4s 8+ +( )-------------------------------------------------------------------------------------------------------------------------------=

HAP s( )NAP s( )

DAP s( )------------------ s 3–( ) s2 4s– 8+( )

s 3+( ) s2 4s 8+ +( )------------------------------------------------= =

HMP s( )NMP s( )DMP s( )------------------- s 3+( ) s2 4s 8+ +( ) s2 2s 6+ +( )

s 1+( ) s2 s 4+ +( ) s2 3s 7+ +( )-------------------------------------------------------------------------------= =

HNMP s( ) HMP s( )HAP s( )=

fo 3kHz= Q 20=Sxωo

SKQ x K

Figure 2.12: Second order bandpass filter.

K

V2V1

V' V''

R1 C1

C2

R2



Solution:Let us suppose that .Writing the node equations:

(2.33)

Comparing this equation with the common expression for a second ordernetwork function, the pole frequency and the quality factor can be easilyidentified:

(2.34)

where

(2.35)

It can be noticed that is not depending on the amplifier gain , then

(2.36)

which is very convenient. On the other side:

(2.37)

This last one is because of the initial assumption: . If then

(2.38)

C1 C2 C= =

H s( )V2V1------ K

K 1–-------------

s R1C⁄

s2 s 2R2C---------- 1

R1C---------- 1

K 1–-------------–

1

R1R2C2--------------------+ +

---------------------------------------------------------------------------------------–= =

ωo1

C2R1R2

------------------------= Q r

2 r2 K 1–( )⁄–-----------------------------------=

rR2R1------=

ωo K

SKωo 0=

SRi

ωo Riωo------

Ri∂

∂ωo 12---–= = SC

ωo Cωo------

C∂∂ωo 1–= =

1– C1 C2 C= =C1 C2≠

ωo1

C1C2R1R2-------------------------------=



and, consequently (2.39)

It is interesting to notice that analyzing the dimensions of it hasdimensions of . It has to be inversely proportional to the squareroot of a product of time constants. Then, is the minimum value that thesensitivity of with respect to any passive element of the filter can have.One of the design goals has to be yielding the minimum sensitivity withrespect to the passive elements. Another is to avoid any dependence on theactive elements.

Let us consider now the quality factor. There are two parameters, and that can help achieving . They can be arbitrarily assigned, as long

as , because must be positive. If we choose forconvenience (equal resistors) the following results:

(2.40)

and if then . This makes the sensitivity of with respect to to be:

(2.41)

then a change in the amplifier gain of a triggers a change in of a. As a tolerance is rarely achieved in active devices, this circuit

is useless. This is due to the abrupt slope of in the proximities of the poleat . Observe that a change of a in produces oscillations,because becomes negative for .

Moreover, recalling what was said in paragraph (a), a change isnot a small change. In this case (2.41) predicts a variation of in while applying (2.40) yields errors as large as and .

Before rejecting this configuration, let us remember paragraph (c). Itsays that sensitivity is a function of all the parameters, and that freeparamenters can be employed to minimize sensitivities. Leaving as a freeparameter, let us compute the sensitivity of again:

SCi

ωo 12---–=

ωoseconds 1–

0.5ωo

rK Q 20=

K 0.5r2 1+> Q r 1=

Q K 1–2K 3–----------------=

Q 20= K Ko 1.5128= = QK

SKQ K

Q----

K∂∂Q K–

K 1–( ) 2K 3–( )---------------------------------------

Ko

115–= = =

0.25% Q28% 0.25%

QK 1.5= 0.85% Ko

Q K 1.5<0.5%

57%± Q36%– 140%

rQ



(2.42)

Extracting from (2.34) and substituting in (2.42):

(2.43)

Fig. 2.13 shows a representation of the absolute value of the sensitivity as afunction of . Observe that was an unfortunate selection. Sensitivitydecreases with increasing ’s. A better selection is a larger , e. g. that leads to . This is a great enhancement in sensitivity, obtainedat the expense of a larger range of values for the resistances.

The sensitivity of decreases with increasing ’s too:

(2.44)

Therefore, the circuit of Fig. 2.12 is good if the selection of the lementsis correct. Considering and choosing these results follow:

SKQ rQ K

K 1–( )2--------------------–=

1 K 1–( )⁄

SKQ 1 4

r2----- 1

rQ-------– 2Q

r------- 1 2

r2-----+

–+=

r r 1=r r r 6=

SKQ 5.9–=

Figure 2.13: Sensitivity as a function of r.

SKQ r( )

r0

50

100

150

200

250

300

350

400

0 1 2 3 4 5 6 7

Q r

SrQ 4Q

r------- 1–=

r 6= C 5nF=



(2.45)

The amplifier gain has increased but is still easily realizable.

2.3.- Consider the circuit of the previous exercise. The amplifier will beimplemented with the non-inverting configuration of Fig. 2.14.Suppose that an OPAMP of open-loop dc gain andvariability is employed. Obtain the variability of thequality factor for both values of : 1 and 6.

Solution:From Fig. 2.14 the closed-loop gain obtained is:

(2.46)

and the sensitivity with respect to the OPAMP’s gain:

(2.47)

Using equations (2.42) and (2.34) yields the following:

R11

12πfoC------------------ 1.768KΩ= =

R2 r2R1 63.66KΩ= = K 22.18=

A 104=dA A⁄ 40%=

∆Q Q⁄ r

Figure 2.14: Finite gain non-inverting amplifier.

A

RK 1–( )R

V2V–

µV2V –------- K

1 K A⁄–--------------------

K A«K≅= =

SAµ K A⁄

1 K A⁄–--------------------– µ

A---– K

A----–≅= =



(2.48)

For the suggested values:

(2.49)

Although has been reduced from to , does not im-prove too much. The gain-sensitivity product results:

(2.50)

which has the minimum around .

Proposed problems

2.4.- a) Obtain the sensitivity of the network function in thecircuit of the figure with respect to each of the elements of the circuit.

b) Obtain the sensitivities of and with respect to , and .c) Obtain the variability of for a

i) change in . ii) change in .

∆QQ

-------- SµQSA

µdAA

------- rQ KK 1–------------- 2dA

A2------- rQ 1 2

r2----- 1

rQ-------–+

2dA

A2-------= = =

∆QQ

-------- 0.7% r 1=0.5% r 6=

≅

SKQ 115– 5.9– ∆Q Q⁄

ΓµQ rQ 1 2

r2----- 1

rQ-------–+

2=

r2 6=

Vo s( ) Vi s( )⁄

ωo Q R L Cωo

+5% C+5% R

R

L C

+

Vo

−

+

Vi

−



Appendix 2.1: Sensitivity of the network function

The network function of a filter can be expressed as:

(2.51)

If the coefficients depend on element , the sensitivity of withrespect to is:

(2.52)

As a function of the coefficients, it can be written as:

(2.53)

It is interesting to obtain the sensitivity for a network function expressedin magnitude and phase:

(2.54)

thus:

(2.55)

The real part of the sensitivity of the network function corresponds tothe sensitivity of the magnitude, while the imaginary part corresponds to thesemirelative sensitivity of the phase.

The variation in the coefficients leads to shifts in the position of thepoles and zeros. It is interesting to study the effect of these changes in .Taking the logarithm of the network function expressed in terms of its zerosand poles:

H s( ) N s( )D s( )-----------

ajsj

j∑

bisi

i∑--------------= =

x H s( )x

SxH Sx

N SxD– x 1

N----

x∂∂N 1

D----

x∂∂D–

= =

SxH x

N----

x∂∂ajsj

j∑

xD----

x∂∂bisi

i∑–=

H jω x,( ) H jω x,( ) ejφ ω x,( )=

SxH jω( ) Sx

H jω( ) jQxφ ω( )+=

H s( )



(2.56)

where . Computing the derivatives and multiplying by :

(2.57)

As it is expected, these shifts are better perceived at the proximities ofthe poles and zeros locations. The sensitivity of the network function will behaigh in the vicinity of the poles and zeros. In sinusoidal steady state

, the network function sensitivity will tend to at the transmissionzeros, that lie at the -axis, and at poles with high .

It can be observed also that has poles in all the poles and zeros of. Equation (2.57) is a partial fraction expansion of . The

semirelative sensitivities of the poles and zeros are y the residuesof the poles and zeros of .

Hln Kln s zi–( )ln s pi–( )lni 1=

n

∑–i 1=

m

∑+=

K am bn⁄= x

SxH Sx

K Qxzi

s zi–------------

i 1=

m

∑–Qx

pi

s pi–------------

i 1=

n

∑+=

s jω= ∞jω Q

SxH

H s( ) SxH

Qxpi Q– x

zi

pi zi SxH

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FUNDAMENTAL CONCEPTS IN FILTER DESIGN · FUNDAMENTAL CONCEPTS IN FILTER DESIGN 3º Telecomunicación Escuela Superior de Ingenieros, Universidad de Sevilla 2 - 2 2.1. Filters: concept