chapter 10 filters

7/29/2019 Chapter 10 Filters

1/90

CHAPTER 10

Filters

with Wael Fathelbab

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506

10.2 Singly and Doubly Terminated Networks . . . . . . . . . . . . . . . . . . . . . . 508

10.3 The Lowpass Filter Prototype .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 513

10.4 The Maximally-Flat (Butterworth) Lowpass Approximation . . . . 514

10.5 The Chebyshev Lowpass Approximation . . . . . . . . . . . . . . . . . . . . . 518

10.6 Element Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521

10.7 Butterworth and Chebyshev Filters .. . . . . . . . . . . . . . . . .. . . . . . . . . 527

10.8 Impedance and Admittance Inverters . .. .. .. . . . . . . . . . . . . . . . . . . 532

10.9 Filter Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541

10.10 Cascaded Line Realization of Filters . .. .. .. . . . . . . . . . . . . . . . . . . . 550

10.11 Richards Transformation .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . 552

10.12 Coupled Line Configurations . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 558

10.13 Inverter Network Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563

10.14 Combline Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566

10.15 Parallel Coupled Line Filters in an Inhomogeneous Medium . . 580

10.16 Design of a Bandstop Filter . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . 581

10.17 Alternative Bandpass Filter Topologies . . . . . . . . . . . . . . . . . . . . . . . 584

10.18 Active Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586

10.19 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591

10.20 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591


2/90

506 MICROWAVE AND RF DESIGN: A SYSTEMS APPROACH

10.1 Introduction

Filters are the most fundamental of signal processing circuits using energystorage elements such as capacitors, inductors, and transmission linesto obtain frequency-dependent characteristics. Some of the importantobjectives of a filter are (1) controlling noise by not allowing out-of-bandnoise to propagate in a circuit; (2) keeping signals outside the transmit band,especially harmonics, from being transmitted; (3) presenting only signals ina specified band to active receive circuitry; and (4) keeping a large transmitsignal from the receive circuitry. At RF, a filter can consist solely of lumpedelements, solely of distributed elements, or comprise a mix of lumpedand distributed elements. The distributed realizations can be transmissionline-based implementations of the components of lumped-element filterprototypes or, preferably, make use of particular frequency characteristics

found with certain distributed structures. For example, coupled lines haveparticular frequency selective characteristics that can be exploited. Lossin lumped elements, particularly above a few gigahertz, means that theperformance of distributed filters nearly always exceeds that of lumped-element filters. However, since the basic component of a distributed filter isa one-quarter wavelength long transmission line, distributed filters can beprohibitively large at a few gigahertz and lower in frequency.

Only a few basic types of responses are required of most RF filters asfollows:

(a) Lowpass, providing maximum power transfer, or passing, at frequen-cies below what is called the corner frequency, fc. Above fc, transmis-sion is blocked. See Figure 10-1(a).

(b) Highpass, passing signals at frequencies above fc. Below fc,transmission is blocked. See Figure 10-1(b).

(c) Bandpass, passing signals at frequencies between lower and uppercorner frequencies and blocking transmission outside the band. (Thefrequency span between the lower and upper corner frequenciesdefines the passband.) This is the most common type of RF filter. SeeFigure 10-1(c).

(d) Bandstop (or notch), which blocks signals between lower and uppercorner frequencies. See Figure 10-1(d).

(e) Allpass, which equalizes a signal by adjusting the phase generally tocorrect for phase distortion in broadband circuits. See Figure 10-1(e).

The above list is not comprehensive, as actual operating conditions maymandate specific frequency profiles. For example, IEEE 802.11a (WiFi)WLAN systems operating at 2.45 GHz are susceptible to potentially largesignals being transmitted from nearby cellular phones operating in the


3/90

FILTERS 507

Lowpass, LPF Highpass, HPFT f( ) ( )T f

(a) (b)

Bandpass, BPF Bandstop, BSF

( )T f( )T f

(c) (d)

Allpass

( )T f ( )T f

(e)

Figure 10-1 Ideal filter responses: (a) lowpass network response; (b) highpass

network response; (c) bandpass network response; (d) bandstop network response;

and (e) allpass network response.

1700 to 2300 MHz range. Thus a front-end filter for a 2.45 GHz systemmust ensure very high levels of attenuation over the 1700 to 2300 MHzrange while also having low insertion loss in-band at 2.45 GHz. Thus, for agiven complexity, the optimum solution probably will have an asymmetricalfrequency response with high rejection below the passband obtained byaccepting lower rejection on the other side of the passband.

In the 1960s an approach to RF filter design and synthesis was developedand this is still followed. The approach is to translate the response of theRF filter (often a bandpass filter) from that of a lowpass filter. A filter withthe desired lowpass response is then synthesized using lumped elementsand the resulting filter is called a lowpass prototype. The lowpass filter isthen transformed so that the new filter has the desired response, such ashighpass or bandpass. In the case of a bandpass filter, each inductor andcapacitor of the lowpass prototype becomes a resonator that is coupled toanother resonator. In distributed form, this basic resonator is a one-quarterwavelength long transmission line.


4/90


(s)

NETWORK

(s)

(s) (s)

LOSSLESS

PASSIVE2V

RS

RL

Vg

1I I2

Zin,1

1V

Zin,2

(a)

PASSIVE

NETWORK

(s)(s)

(s)(s)

LOSSLESS V2V1

Z

gV

2II

in,1

1

L

Zin,2

R

(b)

Figure 10-2 Terminated networks: (a) a doubly terminated network; and (b) a singlyterminated network.

Filter synthesis is the origin of a systematic approach to realizing circuitswith desired frequency characteristics. A filter can also be thought of asa hardware implementation of a differential equation producing a specificimpulse, step, or frequency response. It is not surprising that a filter can bedescribed this way, as the lumped reactive elements, L and C, describe first-order differential equations and their interconnection describes higher-orderdifferential equations.

The synthesis of a filter response begins by identifying the desireddifferential equation response. The differential equations are specified usingthe Laplace variable, s, so that in the frequency domain (with s = j) annth-order differential equation becomes an nth-order polynomial expressionin s. If possible, and this is usually the case, a lowpass version of a filtercan be developed. The conversion to the final filter shape, say a bandpassresponse, proceeds mathematically through a number of stages. Each stagehas a circuit form and each stage is called a filter prototype.

The other aspect of filter design is the identification of particularstructures that inherently have the desired frequency-shaping attributes.Filter design is both science and art matching synthesis with instinctive

knowledge of physical and circuit structures. However, once a suitable filterprototype has been identified it can be optimized using optimization toolsincorporated in circuit analysis packages.

Since the beginnings of circuit synthesis, particularly since the 1950s,it has become common to present circuit concepts using examples, veryoften because the steps in realizing a filter may be too difficult to specifyalgorithmically and there are many steps which require intuition. Thisprocedure is followed here.

10.2 Singly and Doubly Terminated Networks

Filters are generally two-port networks and they provide maximum powertransfer from a load to a source over a specified frequency range whilerejecting the transmission of signals over other ranges. Two possible filter

networks are shown in Figure 10-2. The network in Figure 10-2(a) is knownas a doubly terminated network, as both ports are resistively terminated.The network in Figure 10-2(b) is called a singly terminated network, as only


5/90

FILTERS 509

one port is terminated in a resistor. Most synthesis of singly terminatedfilter networks requires an analogous procedure to that presented here. Itis just that the doubly terminated network is much closer to the type ofnetwork required at RF where loads and source impedances are finite. Asingly terminated network is applicable in some RFIC applications wherevery little RF power is involved and often when feedback is used. In suchcases the output of an RFIC amplifier can approximate an ideal voltagesource, as the Thevenin equivalent source impedance is negligible.

10.2.1 Doubly Terminated Networks

The established synthesis procedure for doubly terminated filter networksfocuses on realizing the input reflection coefficient, which is related to thetransmission coefficient by the characteristic function. The input reflection

coefficient of the doubly terminated network (Figure 10-2(a)) is

1 (s) =Zin,1(s) RSZin,1(s) + RS

, (10.1)

where the reference impedance is the source resistance, RS, and s is theLaplace variable, so the ideal response in the passband is when the reflectioncoefficient is approximately zero. There are several other parameters usedwith filters and these will be introduced now. The Transducer Power Ratio(TPR) is defined as

TPR =Maximum power available from the source

Power absorbed by the load

=

12 (Vg(s)/2)

2/RS12 V

22 (s)/RL =

12RLRS Vg(s)V2(s) 2

. (10.2)

The transmission coefficient, T(s), of the network is

T(s) =1

TPR(s)= 2

RSRL

V2(s)

Vg(s), (10.3)

so that the insertion loss (IL) (or transducer function) is:

IL(s) = TPR(s) = |1/T(s)|2 . (10.4)Note that this development assumes that the filter is lossless. Thenext step in development of the synthesis procedure is introduction ofthe characteristic function, defined as the ratio of the reflection and

transmission coefficients:

K(s) =1 (s)

T (s)=

N(s)

D (s), (10.5)


6/90


where N is the numerator function and D is the denominator function.Ideally the filter network is lossless and so, from the principle of energyconservation, the sum of the magnitudes squared of the transmission andreflection coefficients must be unity:

|T (s)|2 + |1 (s)|2 = 1 (10.6)or

|T(s)|2 = 1 |1(s)|2 . (10.7)Dividing both sides of Equation (10.7) by |T(s)|2 results in

1 =1

|T (s)|2 |1 (s)|2|T (s)|2 , (10.8)

or1 = |IL(s)|2 |K(s)|2 . (10.9)

Rearranging Equation (10.9) leads to

|IL(s)|2 = 1 + |K(s)|2 (10.10)and so

|T(s)|2 = 11 + |K(s)|2

(10.11)

|1(s)|2 = |K(s)|2

1 + |K(s)|2 .(10.12)

It is seen that both the reflection and transmission coefficients are governedby the characteristic function of the two-port network. A milestone hasbeen reached. In an RF filter, the frequency-dependent insertion loss ortransmission coefficient is of most importance, as these are directly relatedto power flow. Equation (10.11) shows that this can be expressed in termsof another function, K(s), which, from Equation (10.5), can be expressed asthe ratio ofN(s) and D(s). For lumped-element circuits, N(s) and D(s) arepolynomials. Sometimes it is better to focus design effort on realizing 1(s),

but the quantities are interrelated.

10.2.2 Lowpass Filter Response

As an example of the relationship of the various filter responses considerthe typical lowpass filter responses shown in Figure 10-3. This filter has

a lowpass response with a corner frequency expressed in radians as c =1 rad/s. Ideally T(s) for s j (note s = j) would be one, and for s > j,T(s) would be zero. It is not possible to realize such an ideal response, andthe response shown in Figure 10-3(a) is typical of what can be achieved.


7/90

FILTERS 511

21.50

1

1

0.8

0.50

3.532.5

0.2

0.6

0.4

(rad/s)

|

(

)|

T

1.000

0.999

0.998

0.997

0.996

0.995

|

(

)|

T

2

(a) (d)

3.532.521.5

1

1

0.8

0.6

0.5

0.4

0.2

00

(rad/s)

2

|G(

)|

(b) (e)

(c) (f)

Figure 10-3 An example of a lowpass filter in terms of various responses: (a) transmission coefficient; (b)reflection coefficient response; and (c) characteristic function response. Detailed responses are shown in

(d), (e), and (f), respectively.


8/90


( )

( )

( )

( )

sX

ty

sY

txFILTER

INPUT OUTPUT

Figure 10-4 Generic filter.

With the filter being lossless, the reflection coefficient is as shown in Figure10-3(b), with the characteristic function K(s) shown in Figure 10-3(c). If

|K(s)|2 is expressed as the ratio of two polynomials (i.e, as N(s)/D(s)), thenit can be seen from Figures 10-3(e) and 10-3(f) that the zeros of the reflection

coefficient, |1(s)|2, are also the zeros of N(s). Also, it is observed that thezeros of the transmission coefficient are also the zeros of D(s), as shown inFigures 10-3(a) and 10-3(c).

The first objective in lumped-element filter design is development of thetransfer function of the network in the frequency domain or, equivalentlyfor the purposes here, the s domain.1 The input-output transfer function ofthe generic filter in Figure 10-4 is

T(s) = Y(s)/X(s) (10.13)

and the design procedure is to match this response to a response defined bythe ratio of polynomials:

T(s) =N(s)

D(s)=

amsm + am1sm1 + + a1s + a0

sn + bn1sn1 + + b1s + b0 . (10.14)

Here N stands for numerator and D for denominator and are not the same asthose in Equation (10.5) (where they also were just labels for numerator anddenominator). Both N(s) and D(s) are polynomials in the Laplace variable,s. The filter response using a pole-zero description can be synthesized sothe design process begins by rewriting Equation (10.14) explicitly in termsof zeros, zm, and poles, pn:

T(s) =N(s)

D(s)=

am(s + z1)(s + z2) (s + zm1)(s + zm)(s +p1)(s +p2) (s +pn1)(s +pn) . (10.15)

Since only the frequency response is of interest, s = j, where , the angularfrequency, is measured in radians per second, thus simplifying the analysisfor sinusoidal signals. However, the development here is general and s can

be complex, which would correspond to an exotic filter response which is ofno interest here.

1 Care is being taken in the use of terminology as s can be complex with some filter types, butthese filters are generally only realized using digital signal processing.


9/90

FILTERS 513

1(rad/s)

0

1

2T( )

(a)

01 /

(rad/s)

T( )

1

2

3

2

(b)

Figure 10-5 Lowpass filter response: (a) ideal brick-wall response; and (b) response template for a

realizable filter.

The poles and zeroes can be complex numbers and can be plotted on thecomplex s plane. Conditions imposed by realizable circuits require that D(s)

be a Hurwitz polynomial which ensures that its poles are located in the left-half plane. N(s) determines the location of the transmission zeros of thefilter, and the order of N(s) cannot be more than the order of D(s). That is,n m so that the filter has finite or zero response at infinite frequency.

Two strategies can be employed in deriving the filter response. The first isto derive the polynomials N(s) and D(s) in Equation (10.14). This seemslike an open-ended problem, but it turns out that in normal situationsthere are only a few types of useful responses which are described bya few polynomials, including Butterworth, Bessel, Chebyshev, and Cauerpolynomials.

10.3 The Lowpass Filter Prototype

Most filter design is based on the synthesis of a lowpass filter equivalentcalled a lowpass prototype. Transformations are then used to correct forthe actual source and load impedances, frequency range, and the desiredfilter type, such as highpass or bandpass. An ideal lowpass response isshown in Figure 10-5(a), which shows a transmission coefficient of 1 up toa normalized frequency of 1 rad/sec. This type of response defines what iscalled a brick-wall filter, which unfortunately cannot be physically realized.Instead, the response is approximated and specified in terms of a responsetemplate (see Figure 10-5(b)).

The template specifies a passband response that is between 1 and 2 atfrequencies below the corner radian frequency ( = 1), and below 3 above

a radian frequency

= 1. The lowpass filter is harder to realize the closerthe specified response is to the ideal shown in Figure 10-5(a), that is, when

(2 1) 0, 3 0, ( 1) 0. (10.16)


10/90


Several polynomials have particularly interesting characteristics thatmatch the requirements of the response template. At first it may seemsurprising that polynomial functions could yield close to ideal filterresponses. However, as will be seen, there is something special about thesepolynomials: they are natural solutions to extreme conditions. For example,the nth-order Butterworth polynomial has the special property that the firstn derivatives at s = 0 are zero. The other polynomials have similar extremeproperties, and filters that are synthesized from expressions incorporatingthem also have extreme properties. As well as the maximally flat propertyresulting from Butterworth polynomials, filter responses obtained usingChebyshev polynomials have the steepest skirts (or fastest transitions fromthe frequency region where signals are passed to the region where they are

blocked). Usually it is one of these extreme cases that is most desirable.A streamlining of the filter synthesis procedure is obtained by first

focusing on the development of normalized lowpass filters having a 1rad/sec corner frequency and 1 reference impedance. Reasonably simpletransformations transform a lowpass filter prototype into another filterhaving the response desired. These transformations, as will be seen, giverise to symmetrical responses.

10.4 The Maximally Flat (Butterworth) LowpassApproximation

Butterworth filters have a maximally flat response (Figure 10-6) which in thetime domain corresponds to a critically damped system. Butterworth filtershave the transfer function

T(s) =N(s)

D(s) =k

sn + bn1sn1 + + b1s + b0 . (10.17)

This is an all-pole response. The characteristic polynomial of theButterworth filter is

|K(s)|2 =s2n = 2n, (10.18)

since s = j and where n is the order of the function. Thus the transmissioncoefficient is

|T(s)|2 = 11 + |K(s)|2 =

1

1 + |s2n| =1

1 + 2n, (10.19)

which is often written as

|T(s)|2 = 11 + |K(s)|2 =

1

Bn(s)Bn(s) . (10.20)


11/90


12/90


Table 10-1 Factors of the Bn(s) polynomial.

n Factors ofBn(s)

1 (s + 1)2 (s2 + 1.4142s + 1)3 (s + 1)(s2 + s + 1)4 (s2 + 0.7654s + 1)(s2 + 1.8478s + 1)5 (s + 1)(s2 + 0.6180s + 1)(s2 + 1.6180s + 1)6 (s2 + 0.5176s + 1)(s2 + 1.4142s + 1)(s2 + 1.9319s + 1)7 (s + 1)(s2 + 0.4450s + 1)(s2 + 1.2470s + 1)(s2 + 1.8019s + 1)8 (s2 + 0.3902 + 1)(s2 + 1.1111s + 1)(s2 + 1.6629s + 1)(s2 + 1.9616s + 1)

coefficient at = 0 and = 1 are independent of the degree of the

characteristic polynomial. This is shown in Figure 10-6, where the responsesof Butterworth filters are plotted for three different orders.

10.4.1 Construction of the Transfer Function

Often the characteristic function, K, is given in the variable , and thus forsynthesis purposes the response must be transformed back to the s domain.As an example, consider the maximally flat function of third order with acharacteristic polynomial equal to 3 (from Equation (10.19)):

K() = 3 and |T ()|2 = 11 + 6

. (10.25)

Note that the first six derivatives of |K(s)|2 with respect to s are zero ats = 0. So, from Equation (10.7),

|1 ()|2 = 1 11 + 6

=6

1 + 6. (10.26)

At real frequencies (or frequencies lying on the imaginary axis in the s plane)

s = j, (10.27)

and so to transform from to s,

= s/j = js (10.28)is used. Thus for the example under consideration, the reflection coefficient

becomes

|1 (j)|2 = |1 (j (js))|2 = (js)6

1 + (

js)6. (10.29)

That is,

|1 (s)|2 = s6

1 s6 = 1 (s) 1 (s) . (10.30)


13/90

FILTERS 517

Factoring the denominator polynomial yields

1 s6 = (1 s) (1 + s) s2 + s + 1 s2 s + 1 . (10.31)By choosing those factors with roots only in the left-half plane (required fora realizable network), the zeros of the denominator are obtained. Now thenumerator of|1 (s)|2 is easily factored, and since this is third order it onlyhas three reflection zeros at DC. The resulting function is

1 (s) =s3

(s + 1) (s2 + s + 1), (10.32)

and so

T(s) =1

(s + 1)(s2 + s + 1). (10.33)

This example illustrates how 1(s) and T(s) can be obtained from K(s)specified as a Butterworth polynomial. This procedure is generalized in thenext section.

10.4.2 nth-Order Reflection Approximation

Following the procedure outlined in the previous section, generalization ofthe Butterworth response leads to the nth-order response

|1(s)|2 = 1 (s) .1 (s) (10.34)

=

s2n1 + (s2)n =

s2nn

i=1(s si) .

2n

=n+1(s s)

.

So |1(s)|2 has n roots (these are the sis) lying in the left-halfs plane, and nroots (the ss) lying in the right-halfs plane. Note that is used as an indexand j (j, without the dot) represents

1. It is reasonable to group all of theleft-half plane roots together (a circuit could not be synthesized if there werea right-half plane root) so that

1(s) = (s)n

ni=1

(s si). (10.35)

Solving for the roots of the denominator of Equation (10.35) (i.e., finding theroots of1 +

s2n = 0) yields the following roots in the left-halfs plane:si = expj (2i 1 + n)

2n i = 1, 2,...,n, (10.36)

where exp(j) = cos ()+j sin(). There are also roots in the right-half plane,but these are just the negative of the left-half plane.


14/90


EXAMPLE 10. 1 Reflection Coefficient Derivation for Butterworth Filter

Develop the reflection coefficient from the roots of the third-order (n = 3) Butterworthlowpass filter prototype.

Solution: From Equation (10.36) the three roots are

s1 = exp

jj (2 1 1 + 3)

2 3ff

= exp

jj

2

3

ff(10.37)

s2 = exp

jj (2 2 1 + 3)

2 3ff

= exp {j } (10.38)

s3 = exp

jj (2 3 1 + 3)

2

3

ff= exp

jj

4

3

ff(10.39)

and the input reflection coefficient of the filter is

1 (s) =s3

s + 12j

3

2

(s + 1)

s + 1

2+j

3

2

= s3(s + 1) (s2 + s + 1)

. (10.40)

This is the same as Equation (10.32).

10.5 The Chebyshev Lowpass Approximation

The maximally flat approximation to the ideal lowpass filter response isbest near the origin but not so good near the band edge. Chebyshev filtershave better responses near the band edge, with lower insertion loss nearthe edges, but at the cost of ripples in the passband. Chebyshev filters are

also known as equiripple all-pole lowpass filters. Example reflection andtransmission responses are shown in Figure 10-7 for a seventh-order and asixth-order Chebyshev lowpass filter. The general form of the Chebyshevtransmission coefficient is

|T(s)|2 = 11 + 2|K(s)|2 , (10.41)

where is an extra design parameter and defines the passband ripple(PBR):

PBR =1

1 + 2or PBR |dB = 10log10

1

1 + 2

. (10.42)

For the nth-order Chebyshev (lowpass filter) approximation the square ofthe characteristic function is

|Kn ()|2 = cos2

n cos1 ()

, (10.43)


15/90

FILTERS 519

N = 7

21.510.50(rad/s) ( )s j=

|

(

)|

T

s

2

1

0.8

0.6

0.4

0.2

0

|

(

)|

s

2

1

0.8

0.6

0.4

0.2

0

(a)

N = 61

0.8

0.6

0.4

0.2

021.510.50

(rad/s) ( )s j=

1

0.8

0.6

0.4

0.2

0

|

(

)|

s

2

|

(

)|

T

s

2

(b)

Figure 10-7Chebyshev lowpass filter approximation: (a) seventh-order filter; and(b) sixth-order filter.


16/90


which can be expressed as a polynomial. For example, with n = 3,

K3() = cos [3 cos1()] = 43 3, (10.44)

and it is surprising that the trigonometric expression has such a simplepolynomial equivalence. From Equation (10.11) the transmission coefficientis

|T()|2 = 11 + 2 cos2 [n cos1 ()]

(10.45)

and the reflection coefficient is

|1()|2 =2 cos2

n cos1()

1 + 2 cos2 [n cos1 ()]

. (10.46)

Factorizing the denominator of either Equation (10.45) or Equation (10.46)yields the following roots (of the denominator of1(s) and T(s)):

si = sin

(2i 1)

2n

sinh

1

nsinh1

1

+ j cos

(2i 1)

2n

cosh

1

nsinh1

1

i = 1, 2, 3 . . . .(10.47)

The roots of the numerator of1(s) in the s plane are

s = j cos(2 1)

2n = 1, 2, ..,n . (10.48)

Equations (10.47) and (10.48) may be utilized to obtain the reflection andtransmission coefficients directly in the s domain.

10.5.1 Chebyshev Approximation and Recursion

It is possible to obtain the characteristic function of the Chebyshevapproximation from the recursion formula,

Kn () = 2Kn1 () Kn2 () , (10.49)

with

K1 () = ; K2 () = 22 1. (10.50)

For example, with n = 3,

K3 () = 2K31 () K32 () (10.51)= 2

22 1 = 43 2 = 43 3.


17/90

FILTERS 521

10.6 Element ExtractionIn the previous two sections the mathematical responses of Butterworth andChebyshev filters were derived for various orders. In this section it will beshown how these filters can be implemented with inductors and capacitorsusing what is called ladder synthesis [139]. To obtain the element valuesyielding the desired transfer function, an impedance or admittance functionmust first be obtained. The impedance or admittance function can be readilyobtained from the input reflection coefficient of a network, but for now thefocus is on synthesizing a given impedance function. A general impedancefunction can be expressed as

Z(s) =an

s2 + 21

s2 + 23

s2 + 25

...

bms (s2 + 22) (s2 + 24) (s

2 + 26) ..., (10.52)

where an and bm are constants. This can be realized by extracting L and Celements, provided that the degree of the numerator and denominator differ

by unity |mn| = 1. The lumped-domain elements are inductors, capacitors,and transformers. The poles and zeroes of Z(s) correspond to zeroes of thetransmission coefficient. A pole of Z(s) at DC requires the extraction of aseries capacitor of value

C0 =1

sZ(s)

s=0

, (10.53)

while a pole at infinity requires the extraction of a series inductor of value

L =Z(s)

s

s=

. (10.54)

Another possibility is a pole at a finite frequency (call this 0

), whichrequires the extraction of a series parallel LC block, as shown in Figure 10-9(e), with a capacitor of value

Ci =s

(s2 + 20) Z(s)

s=jo

(10.55)

and an inductor of value

Li =1

20.Ci. (10.56)

The extraction process can also be carried out on an admittance basis.First,

Y(s) =bms

s2 + 22

s2 + 24

s2 + 2s

. . .

an (s2 + 21) (s2 + 23) (s

2 + 25) . . .. (10.57)

Now a pole at zero requires the extraction of a shunt inductor of value

L0 =1

sY (s)

s=0

(10.58)


18/90


i i+1

X

in, Zin,Z

Figure 10-8 Extraction of a network X to reduce an impedance Zin,i to a lower-

order impedance Zin,i+1.

and a pole at infinity requires the extraction of a shunt capacitor of value

C =Y(s)

s

s=

. (10.59)

A pole at a finite frequency requires the extraction of a shunt-series LCblock(as shown in Figure 10-9(f)) with values

Li =s

(s2 + 2o) Y (s)

s=o

(10.60)

and

Ci =1

2o .Li. (10.61)

The extractions are summarized in Figure 10-9. After the extraction ofan element or a pair of elements an impedance, Zrem, or admittance, Yrem,remains which can be similarly simplified. For example, and referring toFigure 10-9(a), Z(s) = 1/(sC) + Zrem. In general, the synthesis procedure

involves extracting a network X from Zin,i, leaving a reduced-orderimpedance to Zin,i+1 (see Figure 10-8).

Many aspects of filter synthesis can seem abstract when presented in fullgenerality. Consequently it is common to illustrate concepts using examples.Following this time-honored tradition, an example is now presented. Veryoften a general approach is developed from a specific example. However,this time a general approach was possible, and this was presented above.

EXAMPLE 10. 2 Element Extraction for a Third-Order Lowpass Filter

A third-order maximally flat filter has the reflection coefficient

1(s) =s3

(s + 1) (s2 + s + 1). (10.62)

Synthesize this filter as a doubly terminated network.


19/90

FILTERS 523

IMPEDANCE BASIS ADMITTANCE BASIS0

remZ

C

Zin = ZC + Zrem

ZC =1

sC0

YL0 rem

Yin = YL + Yrem

YL =1

sL0

(a) (b)

L

Zrem

Zin = ZL + Zrem

ZL = sLC remY

Yin = YC + Yrem

YC = sC

(c) (d)

remZ

Ci

L i

Zin = ZLC + Zrem

ZLC =sLi

s2LiCi + 1

iremY

Ci

L

Yin = ZLC + Yrem

YLC =sCi

s2LiCi + 1

(e) (f)

Figure 10-9 Synthesis of impedance and admittance functions. Starting with an impedance function

Z(s): (a) extraction of a series capacitor; (c) extraction of a series inductor; and (e) extraction of a series

parallel LC block i. Starting with an admittance function Y(s): (b) extraction of a shunt inductor; (d)

extraction of a shunt capacitor; and (f) extraction of a shunt series LC block.

Solution:

The reflection coefficient function (Equation (10.62)) has all its poles located at infinity, sothe corresponding network realization must be made of simple L or C elements. Hence,referring to Figure 10-2,

Zin,1(s) =1 + 1(s)

1 1(s) (in a 1 system) =2s3 + 2s2 + 2s + 1

2s2 + 2s + 1. (10.63)

Note that the input impedance approaches infinity as the frequency goes to infinity, hence aseries inductor must be extracted. The value of this inductor is

L1 =Zin,1(s)

s

s=

= 1 H. (10.64)

So the filter is developed by extracting one element at a time. Following the extraction ofeach element, the stage impedance is left. The stage impedance function here is

Zin,2(s) = Zin,1(s) sL1 = 2s3

+ 2s2

+ 2s + 12s2 + 2s + 1 sL1


20/90


=2s3 + 2s2 + 2s + 1 s `2s2 + 2s + 1

2s2 + 2s + 1=

s + 1

2s2 + 2s + 1.

Note that the stage impedance above, Zin,2, approaches zero as the frequency goes toinfinity. However, the stage admittance function,

Yin,2(s) =1

Zin,2(s)=

2s2 + 2s + 1

s + 1(10.65)

goes to infinity as the frequency approaches infinity and thus a shunt capacitor can beextracted:

Yin,3(s) = Yin,2(s) sC2 = 1s + 1

, (10.66)

where

C2 = 2 F. (10.67)

So sometimes it is more convenient to consider extraction of an admittance and sometimesit is better to consider extraction of an impedance.

By examining the remaining stage impedance, it is seen that a pole exists at infinity,implying that a series inductor, L3, must be extracted. The value of this inductor comesfrom

Zin,3 =1

Yin,3= s + 1 (10.68)

and so the inductor value is

L3 =s + 1

s

S=

= 1 H. (10.69)

The final step is to extract a load of value 1 as follows:

Zin,4 = Zin,3 sL3 = 1 . (10.70)

This example synthesized a doubly terminated network. The resulting network, called aladder circuit, is shown in Figure 10-10. The left-most 1 resistor is part of the source. Thisis inherently behind the conversion, in Equation (10.63), of the input reflection coefficient,1(s), to an input impedance that used a 1 reference impedance.

Note that this circuit has a dual form which consists of two shunt capacitors separated bya series inductor. The dual would be obtained if the admittance function was obtained fromthe reflection coefficient and then synthesized. Alternative network extraction techniquesare presented in Scanlan and Levy [139,140] and Matthaei et al. [141].

Lowpass Butterworth or Chebyshev filters can be realized by laddernetworks of simple series and shunt combinations of individual inductorsor capacitors. Such networks can be used, following a number of

transformations, to realize highpass filters, and symmetrical bandpass andbandstop filters. Sometimes asymmetric filters with asymmetric responsesare required, for example, to provide additional suppression of a large


21/90

FILTERS 525

(s)

1

1

1

H1 H1

F2

Figure 10-10 Synthesized maximally flat network with a third-order lowpass

reflection response.

interferer (e.g., a radar), and then it is necessary to synthesize more compleximpedance functions.

The input impedance functions of lumped-element circuits can beexpressed as the ratio of two polynomials in s. An important property of

the impedance function is that the order of the numerator and that of thedenominator polynomial can differ by at most one [140]. If the orders differ

by one, then a single inductor or capacitor can be extracted, however, theremaining impedance function may not be realizable. This indicates thata more complex LC (and possibly R) combination is required. To be ableto extract arbitrarily complex circuits a long list of possible functions, suchas those shown in Figure 10-9, are required. For most of the circuits ofinterest here the series and shunt LCcombinations shown in Figure 10-9 aresufficient. The next example describes impedance function extraction thatrequires an LC combination.

EXAMPLE 10. 3 Element Extraction of an Impedance Function

Realize the impedance function

Zw =4s3 + 4s2 + 2s + 2

4s2 + 2s + 1.

Solution:

The order of the numerator is 1 greater than the order of the denominator and this indicatesthat a series inductor is perhaps present. The series inductance is

L1 =Zw(s)

(s)

z=

= 1 H. (10.71)

The remaining impedance is

Zin,2 = Zw sL1 = 4s3 + 4s2 + 2s + 2

4s2 + 2s + 1 s = 2s

2 + s + 2

4s2 + 2s + 1. (10.72)

The numerator and denominator of Zin,2 have the same order and the impedance is notresistive. Therefore a simple L or C element cannot be used to reduce the complexity of theimpedance function. Thus the extraction must backtrack and an initial series inductor is notthe right choice.


22/90


Figure10-9 shows several element combinations that can be used to reduce the complexityof an impedance function. Insight into which alternative to choose comes from factoring zw ,and real roots are required. So consider

Zw =4s3 + 4s2 + 2s + 2

4s2 + 2s + 1(10.73)

=(2s2 + 1)(2s + 2)

4s2 + 2s + 1(10.74)

and the denominator of zw does not have real roots. Examining Figure 10-9, a ready fit toZw is not found. Instead consider the admittance function

Yw =1

Zw=

4s2 + 2s + 1

(2s2 + 1)(2s + 2). (10.75)

So the reduction shown in Figure 10-9(f) looks like the right candidate. The general choicefor the element is

yx =as

bs2 + 1. (10.76)

Choosing b = 2 is the best choice for reducing complexity (since part of the factoreddenominator ofYw occurs), so

Yw =as

2s2 + 1+

4s2 + 2s + 1

(2s2 + 1)(2s + 2) as

2s2 + 1

(10.77)

=as

2s2 + 1+

(4 2a)s2 + (2 2a)s + 1

(2s2 + 1)(2s + 2)

. (10.78)

Choose a = 1,

Yw =s

(2s2

+ 1)

+2s2 + 1

(2s2

+ 1)(2s + 2)

(10.79)

=s

2s2 + 1+

1

2s + 2(10.80)

=s

2s2 + 1+ Yin,2. (10.81)

So C1L1 = b = 1, C1 = a = 1 F, L1 = 2H, and

Yin,2 = 1/(2s + 2) (10.82)

or

Zin,2 =1

Yin,2= 2s + 2. (10.83)

The final network is as follows:

2

2 H

1 F2 H


23/90

FILTERS 527

+1

( odd)

( even)

g2 g4

g1 g3

gn

g0

Vg

gn n

gn n

(a)

+1

( odd)

even)(

3gg1

g2

g0

gn

gn n

gn n

Vg

(b)

Figure 10-11 Chebyshev filter prototypes in the Cauer topology: (a) of type 1; and

(b) of type 2. Here n is the order of the filter.

10.7 Butterworth and Chebyshev Filters

The nth-order lowpass filters constructed from the Butterworth andChebyshev polynomials have the ladder circuit forms of either Figure 10-11(a) or Figure 10-11(b). Figure 10-11 uses several shorthand notationscommonly used with filters. First, note that there are two prototype formsdesignated Type 1 and Type 2, and these are referred to as duals of eachother. The two prototype forms have identical responses with the samenumerical element values g1, , gn. Consider the Type 1 prototype ofFigure 10-11(a). The right-most element is the resistive load, which is alsoknown as the (n + 1)th element. The next element to the left of this is either

a shunt capacitor (of value gn) if n is even, or a series inductor (of value gn)ifn is odd. So for the Type 1 prototype, the shunt capacitor next to the loaddoes not exist if n is odd. The same interpretation applies to the circuit inFigure 10-11(b).

10.7.1 Butterworth Filter

A generalization of the example of the previous section leads to a formulafor the element values of a ladder circuit implementing a Butterworthlowpass filter. For a maximally flat or Butterworth response the elementvalues of the circuit in Figures 10-11(a) and 10-11(b) are

gr = 2 sin

(2r 1)

2nr = 1, 2, 3, . . . , n . (10.84)

Equation (10.84) provides the values ofg1 to gn in Figure 10-11, with g0 andgn+1 both being 1. Table 10-2 lists the coefficients of Butterworth lowpassprototype filters up to ninth order.


24/90


Table 10-2 Coefficients of the Butterworth lowpass prototype filter normalized to

a radian corner frequency of 1 radian/s and a 1 system impedance.

Order, n 2 3 4 5 6 7 8 9

g1 1.4142 1 0.7654 0.6180 0.5176 0.4450 0.3902 0.3473g2 1.4142 2 1.8478 1.6180 1.4142 1.2470 1.1111 1g3 1 1 1.8478 2 1.9318 1.8019 1.6629 1.5321g4 1 0.7654 1.6180 1.9318 2 1.9615 1.8794g5 1 0.6180 1.4142 1.8019 1.9615 2g6 1 0.5176 1.2470 1.6629 1.8794g7 1 0.4450 1.1111 1.5321g8 1 0.3902 1g9 1 0.3473

g10 1

L LC C1 2 3 4

1

1Vg

L1 = 0.765369 H C2 = 1.847759 F

L3 = 1.847759 H C4 = 0.765369 F

Figure 10-12 Fourth-order Butterworth lowpass filter prototype.

EXAMPLE 10. 4 Fourth-Order Butterworth Lowpass Filter

Derive the fourth-order Butterworth lowpass prototype of type 1.

Solution:

From Equation (10.84),

g1 = 2sin[/(2 4)] = 0.765369 H (10.85)g2 = 2 sin [3/(2 4)] = 1.847759 F (10.86)g3 = 2 sin [5/(2 4)] = 1.847759 H (10.87)g4 = 2 sin [7/(2 4)] = 0.765369 F. (10.88)

Thus the fourth-order Butterworth lowpass prototype circuit with a corner frequency of1 radian/s is as shown in Figure 10-12.


25/90

FILTERS 529

10.7.2 Chebyshev FilterFor a Chebyshev response, the element values of the lowpass prototypeshown in Figure 10-11 are found from the following recursive formula [141]:

g0 = 1 (10.89)

gn+1 =

1, n oddtanh2(/4), n even

(10.90)

g1 =2a1

(10.91)

gk =4ak1akbk1gk1

, k = 2, 3, . . . , n (10.92)

ak = sin (2k 1)2n

, k = 1, 2, . . . , n (10.93)and

= sinh

2n

, (10.94)

where

bk = 2 + sin2

k

n

k = 1, 2, . . . , n (10.95)

= ln

coth

RdB

17.37

(10.96)

RdB = 10log

2 + 1

, (10.97)

where n is the order of the filter and is the ripple expressed in absolute

terms. RdB is the ripple expressed in decibels (the ripple is generallyspecified in decibels).

An interesting point to note here is that the source resistor, the value ofwhich is given by g0, and terminating resistor, the value of which is given bygn+1, are only equal for odd-order filters. For an even-order Chebyshev filterthe terminating resistor, gn+1, will be different and a function of the filterripple. Because it is generally desirable to have identical source and loadimpedances, Chebyshev filters are nearly always restricted to odd order.Thus the odd-order Chebyshev prototypes are as shown in Figure 10-13.

Also, for an odd-degree function (n is odd) there is a perfect match at DC,

|T (0)|2 = 1, (10.98)

while for an even-degree function a mismatch exists of value

|T(0)|2 = 4RL(RL + 1)

2 =1

1 + 2(10.99)


26/90


( odd)

g2 g4

g1 g3n1 n3

n2 n2

Vg 1

gn n1

(a)

1

( odd)

3gg1

gnn1

n2

n2

n3

Vg

g21

1

gn n

(b)

Figure 10-13 Odd-order Chebyshev lowpass filter prototypes in the Cauer

topology: (a) of type 1; and (b) of type 2. Here n is the order of the filter.

so that

RL = gn+1 =

+

(1 + 2)2

. (10.100)

Coefficients of several Chebyshev lowpass prototype filters with differentlevels of ripple and of odd orders up to ninth order are given in Table 10-3.

10.7.3 Summary

A Butterworth filter has a monotonic response without ripple but a

relatively slow transition from the passband to the stopband. A Chebyshevfilter has rapid transition but has ripple in either the stopband or passband.Butterworth and Chebyshev filters are special cases of elliptical filters,which are also called Cauer filters. An elliptical filter has equiripple (orequalized ripple) in both the stopband and the passband. The level of theripple can be selected independently in each band. With zero ripple in thestopband, but ripple in the passband, an elliptical filter becomes a TypeI Chebyshev filter. With zero ripple in the passband, but ripple in thestopband, an elliptical filter is known as a Type II Chebyshev filter. Withno ripple in either band the elliptical filter becomes a Butterworth filter.With ripple in both the passband and stopband, the transition betweenthe passband and stopband can be made more abrupt or alternatively thetolerance to component variations increased.

Another type of filter is the Bessel filter which has maximally flat groupdelay in the passband, which means that the phase response has maximumlinearity across the passband. The Legendre filter (also known as theoptimum L filter has a high transition rate from passband to stopband


27/90

FILTERS 531

Table 10-3 Coefficients of Chebyshev lowpass prototype filter normalized to a

radian corner frequency of 1 radian/s and a 1 system impedance.

Order n = 3Ripple 0.01 dB 0.1 dB 0.2 dB 1.0 dB 3.0 dB

g1 0.6291 1.0315 1.2275 2.0236 3.3487g2 0.9702 1.1474 1.1525 0.9941 0.7117g3 0.6291 1.0315 1.2275 2.0236 3.3487


g1 0.7563 1.1468 1.3394 2.1349 3.4817g2 1.3049 1.3712 1.3370 1.0911 0.7618

g3 1.5573 1.9750 2.1660 3.0009 4.5381g4 1.3049 1.3712 1.3370 1.0911 0.7618g5 0.7563 1.1468 1.3394 2.1349 3.4817


g1 0.7969 1.1811 1.3722 2.1664 3.5182g2 1.3924 1.4228 1.3781 1.1116 0.7723g3 1.7481 2.0966 2.2756 3.0934 4.6386g4 1.6331 1.5733 1.5001 1.1736 0.8039g5 1.7481 2.0966 2.2756 3.0934 4.6386g6 1.3924 1.4228 1.3781 1.1116 0.7723g7 0.7969 1.1811 1.3722 2.1664 3.5182


g1 0.8144 1.1956 1.3860 2.1797 3.5340g2 1.4270 1.4425 1.3938 1.1192 0.7660g3 1.8043 2.1345 2.3093 3.1215 4.6692g4 1.7125 1.6167 1.5340 1.1897 0.8118g5 1.9057 2.2053 2.3728 3.1747 4.7272g6 1.7125 1.6167 1.5340 1.1897 0.8118g7 1.8043 2.1345 2.3093 3.1215 4.6692g8 1.4270 1.4425 1.3938 1.1192 0.7660g9 0.8144 1.1956 1.3860 2.1797 3.5340

for a given filter order, and also has a monotonic frequency response (i.e.,without ripple). It is a compromise between the Butterworth filter withmonotonic frequency response but slower transition and the Chebyshevfilter, which has a faster transition but ripples in the frequency response.

More in-depth discussions of a large class of filters along with coefficienttables and coefficient formulas are available in Matthaei et al. [141], Hunter[142], Daniels [143], Lutovac et al. [144], and in most other books dedicated


28/90


K

Z

ZL

in

K= K0Z

(a) (b) (c)

Figure 10-14 Impedance inverter (of impedance K): (a) represented as a two-port;

(b) the two-port terminated in a load; and (c) the two-port implemented as a one-

quarter wavelength long line of characteristic impedance Z0.

solely to filters.

10.8 Impedance and Admittance Inverters

Inverters are two-port networks in the prototype networks of many RFand microwave filters. The input impedance of an inverter terminated inan impedance ZL is 1/ZL. Impedance and admittance inverters are thesame network, with the distinction being whether siemens or ohms areused to define them. An inverter is sometimes called a unit element (UE).At frequencies of a few hundred megahertz and below an inverter can

be realized using operational and transconductance amplifiers. However,these are not readily available at RF and microwave frequencies. At RF,inverters are of particular interest, as they can be approximately realized

by lumped elements, transmission line stubs, and coupled transmissionlines. The transmission line realizations are better approximations; theyhave broader bandwidth than the lumped-element implementations. As aresult, inverters are available in RF and microwave filter design and theirmain application is converting a series element into shunt elements and

inverters. It is much easier to realize shunt elements in distributed circuitsthan series elements. Similar circuit transformations enable inductors to bereplaced by capacitors and inverters.

RF filter design could be presented using only impedance inverters,however, admittance inverters are often used as design manipulationsmore often involve putting elements in shunt (which is preferred). Theuse of admittance simply unclutters the derivations. One or the other, andsometimes both, are used in various books on filter design and in journalsand conference publications. Both will be used here.

The schematic representation of an impedance inverter is shown in Figure10-14(a) where the distinguishing parameter is K which is in ohms. Theconstitutive property of the inverter is that the input impedance of theterminated impedance inverter in Figure 10-14(b) is

Zin = K2

ZL, (10.101)

and so the inverter both inverts the load impedance and scales it. Similarly,


29/90

FILTERS 533

K

/4

= K0Z

/4

= K0ZinZ

LZ

(a) (b) (c)

Figure 10-15 Inverter equivalence: (a) two-port impedance inverter (of impedance

K): (b) a quarter-wave transmission line of characteristic impedance Z0 = 1/K;

a terminated one-quarter wavelength long transmission line of characteristic

impedance Z0.

if Port 1 was terminated in ZL the input impedance at Port 2 is Zin as definedabove. Equation 10.101 does not fully define an inverter which is a two-port.An impedance inverter is fully defined by its ABCD parameters:

T =

0 jK

j/K 0

, (10.102)

and this is done in the next subsection where equivalence of ABCDparameters is used.

An impedance inverter has the value K, which is in ohms, and sometimesK is called the characteristic impedance of the inverter. Sometimes K iscalled just the impedance of the inverter.

An admittance inverter is exactly the same element, but now J is usedand is called the characteristic admittance of the inverter, and sometimes

just the admittance. They are related as J = 1/K. At low frequencies theycan be implemented as transformers or using operational amplifiers. At RFand microwave frequencies inverters are of such great interest as they can

be realized using combinations of transmission lines of reasonable length.Recall that in Section 4.8.4 on Page 218 it was shown that a /4 long line witha load has an input impedance that is the inverse of the load, normalized

by the square of the characteristic impedance of the line. So an invertercan be realized at microwave frequencies using a one-quarter wavelengthlong transmission line (see Figure 10-15(b)). For the configuration shown inFigure 10-15(c),

Zin =K2

ZL. (10.103)

One of the principal advantages of inverters in developing networks isthat they can be used to replace series inductors and capacitors by shuntelements, which are far more easily implemented. This will be seen in thenext few sections.


30/90


10.8.1 Properties of an Impedance Inverter

At low frequencies it is necessary to use either transformers or active circuitsbased on operational amplifiers. In this section the impedance inverter canbe treated as an abstraction. A two-port lossless, reciprocal, and frequency-independent impedance inverter has the ABCD matrix

T =

0 jK

j/K 0

, (10.104)

where K is called the characteristic impedance of the inverter. With a loadimpedance, ZL (at Port 2), the input impedance (at Port 1) is

Zin(s) =A.ZL + B

C.ZL + D=

jK

(j/K) ZL=

K2

ZL. (10.105)

Thus the impedance at the input port is proportional to the inverse of theload impedance. (Note that K is in ohms.)

Now the ABCD matrix of the transmission line of Figure 10-14(c) iscos jZ0 sin jZ0

sin cos

, (10.106)

which is identical to Equation (10.104) when the electrical length is = /2(i.e., when the line is /4 long). The inverter is shown in Figure 10-14(a)as a two-port and implemented as a /4 long line in Figure 10-14(c). The

bandwidth over which the line realizes an impedance inverter is limited,however, as it is an ideal inverter only at the frequency at which it is /4long and when it is (n/2 + /4) long.

10.8.2 Replacement of a Series Inductor by a ShuntCapacitor

A series inductor can be replaced by a shunt capacitor surrounded by a pairof inverters followed by a negative unity transformer (i.e., an inverter withK = 1). This equivalence is shown in Figure 10-16 and will now be shownmathematically.

From Table 6-1, the ABCD matrix of the series inductor shown in Figure10-16(a) (which has an impedance ofsL) is

TL =

1 sL0 1

(10.107)

and the ABCD matrix of a shunt capacitor (which has an admittance ofsC)is, from Table 6-1,

T1 =

1 0

sC 1

. (10.108)


31/90

FILTERS 535

L

(a)

C

1:1

KK

(b)

C

1:1

KK

(c)

Figure 10-16 Equivalent realizations of a series inductor: (a) as a two-port; (b) its realization using a

capacitor, inverters of characteristic impedance K, and a negative unity transformer; and (c) an alternative

realization. C = L/K2 .

The ABCD matrix of an inverter with K = 1 (generally units are droppedand SI is assumed) is

T2 =

0 jKj/K 0

, (10.109)

and finally, the ABCD matrix of a negative unity transformer, N = 1, is,from Table 6-1,

T3 =

1 00 1

. (10.110)

The ABCD matrix of the cascade shown in Figure 10-16(b) is

TC = T2T1T2T3

= 0 jK

j/K 0 1 0

sC 1 0 jK

j/K 0 1 00

1

=

0 jK

j/K 0

1 0

sC 1

0 jK

j/K 0

=

jsCK jK j/K 0

0 jK

j/K 0

=

1 sCK2

0 1

. (10.111)

Thus TC = TL if L = CK2 (compare Equations (10.107) and (10.111)). Aseries inductor can be replaced by a shunt capacitor with an inverter beforeand after it and with a negative unity transformer. The unity transformermay also be placed at the first port, as in Figure 10-16(c). Thus the two-portsshown in Figure 10-16 are all electrically identical, with the limitation being

the frequency range over which the inverter can be realized. An interestingand important observation is that as a result of the characteristic impedanceof the inverter (e.g., 50 ), a small shunt capacitor can be used to realize alarge series inductance value.


32/90


C K K

1:1

L

(a) (b)

Figure 10-17 A series capacitor: (a) as a two-port; (b) its realization using a shunt

inductor, inverters and negative unity transformer. L = CK2

EXAMPLE 10. 5 Inductor Synthesis Using an Inverter

Consider the network of Figure 10-16(c) with inverters having a characteristic impedance of50 . What value of inductance is realized using a 10 pF capacitor?

Solution:

K = 50, so L = CK2 = 1011 2500 = 25 nH.

10.8.3 Replacement of a Series Capacitor by a ShuntInductor

Analogous to the replacement demonstrated in the previous section, aseries capacitor can be replaced by a shunt inductor plus inverters and anegative transformer (see Figure 10-17). The ABCD parameters of the seriescapacitor in Figure 10-17(a) are

T = 1 1/sC0 1 , (10.112)

and here it is shown that the cascade in Figure 10-17(b) has the same ABCDparameters. Choosing the value of the shunt inductor in Figure 10-17(b)as C and using unity impedance inverters, the cascade has the ABCDparameters

T =

0 jK

j/K 0

1 0

1/sL 1

0 jK

j/K 0

1 00 1

=

0 jK

j/K 0

1 0

1/sL 1

0 jK

j/K 0

=

jK/sL jK j/K 0

0 jK

j/K 0

= 1 K2/sL

0 1

. (10.113)

So the series capacitor, C, can be realized using a shunt inductor, L,inverters, and a negative unity transformer, and C = L/K2. It is unlikely


33/90

FILTERS 537

+1

( odd)

( even)

g2 g4

g1 g3

gn

g0

Vg

gn n

gn n

(a)

1

Vg

g1 g2 g3 g4 g5

1 1 1 1 1 1

1:1 1:1 1:1

(b)

1

Vg

g1 g2 g3 g4

1 1

g5

1 1 1

(c)

Figure 10-18 Ladder prototype filters using impedance inverters: (a) lumped-element prototype; (b) first

stage in transformation using inverters; and (c) final stage.

that this would be desirable, as much better lower loss capacitors can berealized at RF than inductors.

10.8.4 Ladder Prototype with Impedance Inverters

The transformations discussed in the previous two sections can be usedto advantage in simplifying filters. In this section the transformations areapplied to the prototype of the lowpass ladder filter shown in Figure 10-18(a). The inductors in the ladder circuit are a particular problem as theyhave considerable resistance at radio and microwave frequencies and limitthe frequency selectivity that can be obtained from a filter. Using the conceptof inverters, the series inductors can be replaced by a circuit with capacitors,inverters, and transformers, as shown in Figure 10-18(b). This simplifiesfurther to the realization shown in Figure 10-18(c), as the negative unitytransformers only affect the phase of the transmission coefficient. So alowpass ladder filter can be realized using just capacitors and inverters.

10.8.5 Lumped-Element Realization of an Inverter

The admittance inverter is functionally the same as the impedance inverter(see Figure 10-14(a)) and the schematic is the same (see Figure 10-19(a)).As will be shown, this can be realized using frequency-invariant lossless


34/90


1J=K _ _

+

= jBy =

jBy =

jB __j/J j/

j/J

(a) (b) (c)

Figure 10-19 Admittance inverter: (a) as a two-port; (b) generalized inverter

realized using lumped elements (identical if B = J); and (c) lumped equivalentcircuit of the admittance inverter of (a) (the element values in (c) are impedances).

elements (i.e., elements whose reactance or susceptance do not vary withfrequency) using the network of Figure 10-19(b). J is generally used withadmittance inverters and K with impedance inverters. If not specified by

the context, the inverter (with value specified by a number) defaults to beingan impedance inverter. Alternatively units can be used to indicate whichtype of inverter is being used. The function of the inverter is the same inany case; both can be realized by one-quarter wavelength long lines, forexample. For the remainder of this chapter it will be more convenient to usethe admittance inverter, as many calculations will be in terms of admittancessince it is much easier to realize shunt elements in distributed form (e.g.,using stubs) than to realize series elements.

Now it will be shown that the lumped-element network of Figure 10-19(b) realizes an inverter. To do this we must show that the inverter andthe lumped-element network have the same two-port parameters. First, theABCD matrix of an inverter of characteristic admittance J is

TJ = 0 j/J

jJ 0

. (10.114)

Referring to Table 6-1, the circuit of Figure 10-19(b) has the ABCD matrix

T =

1 0

jB 1

1 1/(jB)0 1

1 0

jB 1

=

1 1/(jB)

jB 0

1 0jB 1

=

0 /B

jB 0

, (10.115)

where B is the susceptance of the frequency-invariant elements. Equation(10.115) is identical to Equation (10.114) if B = J. More practicalequivalents of the circuit of Figure 10-19(b) can be derived, as shown later.

For completeness, the lumped-element equivalent of the impedanceinverter is shown in Figure 10-20 (derived from Figure 10-19 with J = 1/K).


35/90

FILTERS 539

K_ _jK jK

jK

(a) (b)

Figure 10-20 Impedance inverter: (a) as a two-port; and (b) its lumped equivalent

circuit (the element values in (b) are impedances).

+_

L

_

iny = y

jBy =jBy =

y

jB=

in 3 2

1

Ly

y

yy

(a) (b)

Figure 10-21 Terminated lumped-element admittance inverter.

EXAMPLE 10. 6 Lumped Inverter Analysis

Demonstrate that the lumped-element inverter model in Figure 10-19(b) is an admittanceinverter.

Solution:

Terminating the network in Figure 10-19(b) results in the network shown in Figure 10-21(a).This is relabeled for convenience in Figure 10-21(b) where the elements are admittances.Then

yin = y3//(y1$(y2 + yL)) , (10.116)

where // indicates in parallel with and $ indicates in series with. These are common

shorthand notations in circuit calculations. Continuing on from Equation (10.116),

yin = y3 +

1

y1+

1

y2 + yL

1= y3 +

y2 + yL + y1y1(y2 + yL)

1. (10.117)

Substituting y1 = jB, y2 = y3 = jB, this becomes

yin = jB + jB(yL jB)yL

=jByL +jByL + B2

yL=

B2

yL. (10.118)

Thus the lumped-element of Figure 10-19(b) is an admittance inverter of value B (inSiemens).

10.8.6 Narrowband Realization of an Inverter Using

Transmission Line StubsIn this section it will be shown that an impedance inverter can beimplemented using short- and open-circuits stubs. The match is good over


36/90


K _ _jK jK

jK

jKjKZ0

Z0

Z0

STUB 3STUB 1

STUB 2

jK

(a) (b) (c)

Figure 10-22 Narrowband inverter equivalents at frequency f0: (a) impedance

inverter with characteristic impedance K; (b) lumped-element equivalent network;

(c) inverter realized by short- and open-circuited stubs.

a narrow band centered at frequency f0. An impedance inverter is shownin Figure 10-22(a) and its equivalent lumped-element network in Figure10-22(b). A stub-based implementation is shown in Figure 10-22(c), wherethere are short- and open-circuited stubs of characteristic impedance Z0. Theinput impedance of the stubs is shown at the inputs of the stubs. The stubshave an electrical length at f0, and the stubs are one-quarter wavelengthlong (i.e., is resonant) at what is called the commensurate frequency, fr.

Now it will be shown that the network of Figure 10-22(c) is a goodrepresentation of the inverter at f0. This is done by matching ABCDparameters. The ABCD parameters of an inverter are

T =

0 jK

j/K 0

(10.119)

and, at frequency f0, the ABCD parameters of the stub circuit of Figure 10-22(c) are

T =

1 01/[jZ0 tan()] 1

1 jZ0 tan()0 1

1 01/[jZ0 tan()] 1

=

1 jZ0 tan()

1/[jZ0 tan()] 0

1 01/[jZ0 tan()] 1

=

0 jZ0 tan()

j/[Z0 tan()] 0

. (10.120)

Thus, equating Equations (10.119) and (10.120), the stub network is a goodrepresentation of the inverter if

K = Z0 tan() , (10.121)

and so the required characteristic impedance of each stub at frequency f0 is

Z0 =K

tan()=

K

tan2f0fr

. (10.122)


37/90

FILTERS 541

Special Case fr = 2f0

The most commonly used designs choose the stub resonant frequency, thecommensurate frequency, to be twice that of the center frequency of thedesign, f0. So with fr = 2f0:

Z0 =K

tan2

f02f0

=

K

tan /4

= K. (10.123)

10.9 Filter Transformations

So far the discussion has centered around lowpass filters. Filter designtechnology has developed so that the design of lowpass filters is theessential first step and these are used as prototypes to derive filters withother characteristics. The three most important transformations are listedhere:

1. Impedance scaling: A lowpass filter prototype is referenced to astandard impedance. Usually 1 is used, so the reference sourceand load resistances are also 1 . To reference to a higher or lowerimpedance, scaling of the impedance of all elements of the filter isrequired.

2. Cutoff frequency scaling: The cutoff frequency of a lowpass filter

prototype is normalized to 1 radian/s. To reference to anotherfrequency, the values of the elements must be altered so that they havethe same impedance at the scaled frequency.

3. Filter type transformation: These transformations enable the circuit ofa lowpass filter to be converted to a circuit with a bandpass, bandstop,or highpass response. The concept is that the response at DC is to bereplicated at infinite frequency for the highpass (so capacitors becomeinductors, etc.); to be replicated at the center of the passband for a

bandpass filter (so capacitors become a shunt LC circuit); and to beinverted at the center of the bandstop filter (so capacitors become aseries LC circuit).

The transformations can be performed in any order. The key point is thatthe complexity of the lowpass circuit design is minimal since the number ofelements does not grow until the filter type transformation is performed, sothis step is often done last.


38/90


LL xZ0

(a)/CC Z

0

(b)

K Z0

K

(c)

Figure 10-23 Impedance transformations: (a) for an inductor; (b) for a capacitor;

and (c) for an inverter.

10.9.1 Impedance Transformation

The lowpass prototypes discussed up to now have been referred to a 1 system. The system impedance can be changed to any level by simplyscaling the impedances of all the circuit elements in a filter by the sameamount, as shown in Figure 10-23. The impedance transformation followsthe same procedure forall filter types (the procedure is the same for lowpass,highpass, bandpass, and bandstop filters, for example). It should also benoted that impedance scaling of a transmission line is simply multiplicationof the lines characteristic impedance by the scale factor.

EXAMPLE 10. 7 Lowpass Filter Design

Consider the lowpass filter, with an inverter, shown in Figure 10-24(a). This filteris referenced to 1 , as the source and load impedances are both 1 . Redesign thefilter so that the same frequency response is obtained with 50 source and loadimpedances.

Solution:

It is necessary to impedance transform from 1 to 50 . The resulting filter isshown in Figure 10-24(b). Each element has an impedance (resistance or reactance)that is 50 times larger than it had in the prototype.

10.9.2 Frequency Transformation: Lowpass

Frequency transformations differ depending on the type of the filter. So itis normal to frequency transform the lowpass prototype before convertingthe prototype to another form (such as bandpass). The lowpass prototypesnormally have a band-edge or cutoff frequency at an angular frequency of


39/90

FILTERS 543

K 1

1

C1 C2 C3

L1

gV

(a)

/50 /50

x50

/50

50 50K

50

C3C1 C2

L1

gV

(b)

Figure 10-24 Impedance transformation of an example filter: (a) filter normalized

to 1 ; and (b) impedance transformed to a reference of50 .

=1(rad/s)

j)s=j(

(T s)2

(rad/s)

c

0

j)(s=j

(s)T2

(a) (b)

Figure 10-25 Frequencytransformation of a lowpass filter from, (a) onenormalized

to a corner frequency of 1 rad/s, to (b) one with a radian corner frequency of c.

unity,2 that is, 1 rad/s. The band edge frequency can be transformed fromunity to an arbitrary frequency c, as shown in Figure 10-25, by scalingthe reactive elements, as shown in Figure 10-26, so that they have thesame impedance at the transformed frequency as they did at the originalfrequency. The inverter is unchanged, as are the source and load resistances,since these are frequency independent.

2 In this book (and in most treatments) the band-edge frequency is normalized to 1 rad/s.Some books and papers use a 1 Hz normalization, so be wary.


40/90


C/L L

(a)/

CC C

(b)

Figure10-26 Frequencytransformations for(a) an inductor;and (b)a capacitor. The

impedance of the new (scaled) element at the new (scaled) frequency is the same as

it was at the original frequency.

=1(rad/s)

j)s=j(

(T s)2

=1(rad/s) )(s=j

j

(s)T2

(a) (b)

Figure 10-27 Lowpass to highpass transformation.

10.9.3 Lowpass to Highpass Transformation

The transformation of a lowpass filter prototype to a highpass filter is showndiagrammatically in Figure 10-27. Mathematically the transformation is

|T ()|2 = 11 + 2 |K()|2 (10.124)

1

(10.125)T

1

2

=1

1 + 2K 1 2 (10.126)

so that reactive elements are transformed as shown in Figure 10-30. Hereinductors are transformed into capacitors and capacitors into inductors.

For example, if the odd-order lowpass filters are of the types shown inFigure 10-13 then the transformation to highpass is shown in Figure 10-28.

10.9.4 Lowpass to Bandpass TransformationUnderstanding the transformation of the lowpass filter into its correspond-ing bandpass forms requires that the lowpass filter be considered in both its


41/90

FILTERS 545

( odd)

Vg

1

1

n n

2

1 3

2

n

n n

nC C

L L

1 3 C

42

(a)

1

( )odd

Vg

n1

n2

n2

n3

3

1

1

C

gn n

L L

n2

1

C

(b)

Figure 10-28 Odd-order Chebyshev highpass filter prototypes in the Cauer

topology: (a) of type 1; and (b) of type 2. Here n is the order of the filter.

positive and negative frequency response, as shown in Figure 10-29(a).This response is shifted in frequency to obtain the bandpass response

shown in Figure 10-29(b). Mathematically the radian frequency, , in theresponse function is replaced by its bandpass form:

o o

. (10.127)

This separately maps the 1 and +1 band-edge radian frequencies of thelowpass response to the bandpass frequencies 1 and 2:

1

1o

o1

(10.128)

+1

2o

o2

. (10.129)

Solving the above equations simultaneously yields the center frequency 0and the band-edge frequencies 1 and 2 with

o =

12 (10.130)

and the so-called transformation constant

=o

2

1

. (10.131)

The resulting element conversions are given in Figure 10-30.As an example, a lumped-element Type 2 Cauer bandpass filter is shown

in Figure 10-31. The shunt LC combination and the series LC combinations


42/90


+1(rad/s)

j)(s=j

1

(s)T 2

(a)

(rad/s)wj

)(s=j

1

0

2

T s( )2

(b)

Figure 10-29 Frequency responses in lowpass to bandpass transformation: (a) two-

sided lowpass filter response; and (b) bandpass filter approximation.

are resonators resonant at the center frequency of the filter. Here thefilter is normalized to a Z0 source and load impedances. As these are thesame, this filter topology only applies for an odd-order filter. Combiningtransformations, the element values of a lumped bandpass filter with centerradian frequency 0 = 2f0 and radian bandwidth BW = 2(f2

f1) are

as follows:

Cr =

grBWZ0

r = odd

BW20grZ0

r = even

(10.132)

Lr =

BWZ020gr

r = odd

grZ0BW

r = even

. (10.133)

10.9.5 Lowpass to Bandstop TransformationAgain, consider both the positive and negative frequency responses ofthe lowpass filter prototype, as shown in Figure 10-32(a). This response is


43/90

FILTERS 547

LOWPASS PROTOTYPE TO LOWPASS TRANSFORMATION

LOWPASS LOWPASS REACTANCE TRANSFORMATION

PROTOTYPE ELEMENT 0 = corner frequency

C0 C1

C1 = C0/0

L 0 L1L1 = L0/0

LOWPASS PROTOTYPE TO HIGHPASS TRANSFORMATION

LOWPASS HIGHPASS REACTANCE TRANSFORMATION

PROTOTYPE ELEMENT 0 = corner frequency

C0 L1 L1 = 1/(0C0)

L 0 C1

C1 = 1/(0L0)

LOWPASS PROTOTYPE TO BANDPASS TRANSFORMATION

LOWPASS BANDPASS REACTANCE TRANSFORMATION

PROTOTYPE ELEMENT 0 = center of passband

C0 1

1L

C

C1 = C0/0 L1 = 1/(C00)

L 0 C1 1LL1 = L0/0 C1 = 1/(L00)

LOWPASS PROTOTYPE TO BANDSTOP TRANSFORMATION

LOWPASS BANDSTOP REACTANCE TRANSFORMATION

PROTOTYPE ELEMENT 0 = center of stopband

C0 C1 1LL1 = /(C00) C1 = C0/(0)

L 0 1

1L

CL1 = L0/(0) C1 = /(L00)

Figure10-30 Transformations of theelementsof a prototypelowpass filter to obtain

specific filter types. The corner frequency of the lowpass prototype is 1 radian/s. In

the transformations to bandpass and bandstop filters, 0 = 1/

L1C1 =

12;

1 and 2 are the band-edge frequencies; and is the transformation constant, = 0/(2 1). The corner frequency of the lowpass prototype is 1 radian/s.


44/90


1 1

nL 1 C1

L 2 C2

L 3 C3 CnL n

Z0

Z0

L n Cn

Vg

1n

2n

3n

2

Figure 10-31 Lumped-element odd-order (nth-order) Chebyshev bandpass filter

prototypes in the type II Cauer topology.

+1(rad/s)

j)(s=j

1

(s)T2

(a)

j

T(s)

s=j( )(rad/s)

1

2

2

0

(b)

Figure 10-32 Frequency responses in lowpass to bandstop transformation: (a) two-

sided lowpass filter response; and (b) bandstop filter approximation.

shifted in frequency to obtain the bandstop response shown in Figure 10-32(b). Mathematically the frequency, , in the response function is replaced

by its bandstop form:

o o

1

. (10.134)

The center frequency (corresponding to DC of the response of the lowpassprototype) is

o =

12 (10.135)


45/90

FILTERS 549

1

1

L 1 L 3 C3 CnL nC1

Z0

Z0

Vg

1n

2n

3n

2n

L 2 L n

C2 Cn

Figure 10-33 Lumped-element odd-order (nth-order) Chebyshev bandpass filter

prototypes in the type II Cauer topology.

and the transformation constant is

=

o

2 1 , (10.136)where 1 and 2 are the band-edge frequencies. The resulting elementconversions are given in Figure 10-30.

Combining transformations, the element values of a lumped bandstopfilter with center radian frequency 0 = 2f0 and radian bandwidth BW =2(f2 f1) are as follows:

Cr =

grBW20Z0

r = odd

1

BWgrZ0r = even

(10.137)

Lr =

Z0BWgr

r = odd

grBWZ020

r = even

. (10.138)

Similarly a lumped-element type II Cauer bandstop filter is shown inFigure 10-33. The parallel LC combination and the series LC combinationsare resonators resonant at the center frequency of the filter. The parallel LCresonator is an open circuit at the center frequency of the stopband and theseries LC resonators are short circuits.

10.9.6 Transformed Ladder Prototypes

Combining the filter type transformations, and with appropriate use ofinverters, the original lowpass prototype ladder filter and its various filtertype transforms are shown in Figure 10-34.


46/90


g1 g2 g3 g4

RS

Vg

1 1 1 1

(a)

1/1/1/1/ g4g3g2g1

RS

Vg

1 1 1 1

(b)

==

==

==

== ////

/ / / /LC

LC

LC

LC

g1

g2 g3 4g

g1 g2 g3 g4

Vg

RS1 1 1 1

(c)

== 1/ ( )

== 1/ ( )

== 1/ ( )

== 1/ ( )L

C L

C L

C L

C g1

g1

Vg

RS1

g2

g2

g3

g3

g4

g4

1 1 1

(d)

Figure 10-34 Ladder prototype filters: (a) lowpass prototype ladder filter; (b) highpass prototype ladder

filter; (c) bandstop prototype ladder filter; and (d) bandpass prototype ladder filter.

10.10 Cascaded Line Realization of Filters

In this section, filters are presented that use cascaded sections oftransmission linerealizing series inductors and shunt capacitors. Forexample, a short (less than one-quarter wavelength long) line of relativelyhigh-impedance line behaves predominantly as series inductances. Also, avery short (much less than one-quarter wavelength long) length of relativelylow-impedance line acts predominantly as a shunt capacitance. So a Pinetwork of lumped elements can be realized with alternate sections of low-

and high-impedance microstrip lines. Such an inductive line is shown inFigure 10-35(a), which has the two equivalent circuits shown in Figures 10-35(b) and 10-35(c). Basic transmission line theory gives the input reactance


47/90

FILTERS 551

IMPEDANCE

IMPEDANCE LOWIMPEDANCE

HIGH

LOWl

CC ZO

( )C CjX

L

(a) (b) (c)

Figure 10-35 Inductive length of line with adjacent capacitive lines: (a) microstrip form; (b) lumped

equivalent circuit; and (c) lumped-distributed equivalent circuit.

HIGH

LOW

IMPEDANCE IMPEDANCEHIGH

IMPEDANCE

( )C

jX jX

y=jB

(a) (b)

Figure 10-36 Capacitive length of line with adjacent inductive lines: (a) microstrip;

and (b) lumped equivalent (the left-most and right-most series inductors come from

the high-impedance lines).

of the line of length, (with a low-impedance load),

XL = Z0 sin(2/g) , (10.139)

so that the length of this predominantly inductive line is

= g2

sin1

LZ0

. (10.140)

Previously it was shown that a short length of line having a relativelylow characteristic impedance yields a capacitive element and this is shown,together with its equivalent circuit, in Figure 10-36. The predominatingshunt capacitance is determined by first considering the susceptance

B =1

Z0sin

2

g

(10.141)

so that

=g2

sin1 (CZ0) . (10.142)

Thus a cascade of low-impedance and high-impedance lines can thereforerealize (approximately) an LC ladder network.

It is tempting to take filters synthesized in lumped-element form andrealize them in the above manner with transmission lines. The series


48/90


element realization concept presented in this section could be extendedusing shorted and open stubs to better realize shunt elements. The problemwith this approach is that the resulting filters are narrowband and theresponse outside the desired operating range is unpredictable. It is far betterto employ the Richards transformation, considered in the next section,which is a much broader bandwidth technique for realizing filters indistributed form.

10.11 Richards Transformation

10.11.1 Richards Transformation and Transmission Lines

The Richards transformation is a remarkable scheme that takes into accountthe actual properties of transmission lines, yielding broadband transmission

line-based implementations of lumped-element filter prototypes. Considera section of transmission line of electrical length with ABCD parameters

T =

cos() jZ0 sin()

j/Z0 sin() cos ()

. (10.143)

If this line is terminated in a load, ZL, then its input impedance is

Zin() =cos().ZL +jZ0. sin()

/Z0 sin().ZL + cos(). (10.144)

Now examine two extreme conditions. As the load impedance increases,eventually becoming an open circuit, the input impedance of a line withelectrical length is defined in terms of a cotangent of the electrical length:

ZL Zin() = Z0j

. cot() (10.145)

Yin() = jY0. tan(). (10.146)

As the load impedance reduces to become a short circuit, the inputimpedance of a line with electrical length is defined in terms of a tangentof the electrical length:

ZL 0 Zin() = jZ0. tan(). (10.147)

These results lead to the Richards transformation, which replaces theLaplace variable, s, by Richards variable, S, where S = j tan(). Thistransformation is written

s S = j tan(). (10.148)For now and are constants that can be chosen as design variables. ,of course, is the electrical length of the line and is not just 90 for a /4


49/90

FILTERS 553

line. Also, must have the units of impedance and it is the characteristicimpedance of the transmission line.

Applying the Richards transformation to a capacitor, the admittance ofthe element is transformed as follows:

y = sC Y = SC = jCtan(), (10.149)

so that the capacitor is transformed into an open-circuited stub withcharacteristic admittance

Y0 = C . (10.150)

If a lumped-element capacitor with admittance y = sC is to be realizedusing a transmission line, the admittance Y = SC = jCtan() is insteadrealized. There are two parameters to select to realize this admittance. The

first, , is the characteristic admittance of the transmission line (and forany given transmission line topology there is a minimum and maximumcharacteristic admittance or impedance that can be realized), and is theelectrical length of the line.

Applying the transformation to an inductor, the impedance of the elementis transformed as follows:

Z = sL Z = SL = jL tan(), (10.151)

so that the inductor is transformed into a short-circuited stub withcharacteristic impedance

Z0 = L. (10.152)

Thus the Richards transform converts an inductor into a short-circuited

stub and a capacitor into an open-circuited stub.

10.11.2 Richards Transformation and Stubs

There is a duality between stubs and inductors and capacitors; they arecoupled by Richards transformation. One of the important quantities usedin the transformation is the commensurate frequency, fr, which most oftenis just twice the operating frequency, f0. Considering stubs that are one-quarter wavelength long at fr , the duality is as shown in Figure 10-37.

10.11.3 Richards Transformation Applied to a Lowpass Filter

In this section, Richards transformation is used to realize a lumped-elementfilter in distributed form. The design example begins by considering a

Chebyshev lowpass filter with the transmission characteristic

|T(s)|2 = 11 + 2 |K(s)|2 .

(10.153)


50/90


s, Laplace variable S, Richards variableSchematic Equivalence

jX Z0

/4r

Z0

X = Z0fr = 2f0

jXZ0

/4rjZ0

Z0

X = Z0fr = 2f0

jX Z0 jZ0

/4r

Z0X = Z0fr = 2f0

jXZ0

/4rjZ0

Z0

X = Z0fr = 2f0

Figure 10-37 Equivalences resulting from Richards transformation. With fr = 2f0the transmission line stubs are one-eighth wavelength long at f0.

With the Richards transformation

s j j tan() , (10.154)the response becomes

|T (j tan )|2 = 11 + 2 |K(j tan ))|2 . (10.155)

Thus the passband edge at = 1 is mapped to = 1 as

1 tan(1) (10.156) =

1

tan(1). (10.157)

Recalling that a capacitor is transformed into an open-circuited stub (seeEquation (10.150)), Richards transformation applied to the lowpass filterprototype results in a filter with transmission line elements only, as shownFigure 10-38, provided that the inverters are realized using transmissionlines.


51/90

FILTERS 555

12K

C1 C2 C4C3

K23K 34

(a)

ii= 1/( )

12K 23K 34KZ3Z2Z1

C

Z4

Z

(b)

Figure 10-38 Lowpass prototypes: (a) lowpass prototype as a ladder filter with

inverters; and (b) lowpass distributed prototype with open-circuited stubs (the

impedance looking into the stub is indicated).

The implementation of a lowpass filter in distributed form results inpassbands and stopbands that repeat in frequency, as shown in Figure 10-39.This occurs because transmission lines are used to realize lumped elementsand the impedance of a transmission repeats every wavelength (or halfwavelength in some cases). For example, the input impedance of a stub isthe same whether it is one-half wavelength long or one wavelength long.

10.11.4 Richards Transformation Applied to a Highpass Filter

With reference to Figure 10-40(a), the passband edge at = 1 is mapped to1, as shown in Figure 10-40(b). This means that the passband at = 1 ismapped to 1 as

1 tan(1) (10.158) =

1

tan(1). (10.159)

The sequence of steps transforming a lumped lowpass prototype to itsdistributed highpass prototype are shown in Figure 10-41. As previouslydiscussed, all the inverters can be approximated by transmission lines oflength /2 at the corner frequency of the filter.

10.11.5 Kurodas Identities

Kurodas identities embody a number of specific manipulations usingimpedance or admittance inverters. They are particularly useful in imple-menting Richards transformation as they physically separate transmission


52/90


=1(rad/s) )(s=j

j

(s)T2

(a)

31 222

( )T2

(b)

Figure 10-39 Lowpass to distributed lowpass transformation: (a) lowpass filter response; and (b)

distributed lowpass filter response.

=1

)

j

T(j 2

(a)

3

)

1 22

2

T(j 2

chapter 10 filters

Documents