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INTRODUCTION TO

MICROWAVE FILTERS

18.1 IMPLEMENTATION OF FILTERS

The passive (LC) filters discussed previously work quite well at frequencies up to a few hun-dred megahertz. Beyond this range, components deviate significantly from anything close toideal. Parasitics start to dominate, and component values become impractical, while capac-itors become inductors and vice versa. Distances between components turn important, andtraces on a PC board introduce unwanted capacitance and inductance. The methodology dis-cussed in this section concerns using PC board traces to create transmission lines by con-

trolling their properties, and then configuring these transmission lines into an architectureresulting in filters. The resulting filters are then based on distributed parameters rather thanlumped inductors and capacitors. At submicrowave frequencies, this approach is not feasi-ble since the dimensions based on fractions of a wavelength become impractical.

For low-power applications, stripline and microstrip filters are extensively used becauseof their low cost and repeatability. For high-power requirements, waveguide structures areutilized. Waveguides are covered in great detail in Matthaei (see Bibliography) and will notbe covered here.

This chapter is introductory and is by no means intended as a design guide for microwavefilters. The reader should instead refer to the multiple references cited in the Bibliography

for further information on this topic.

18.2 MICROSTRIP AND STRIPLINETRANSMISSION LINES

The two basic forms of PC board transmission lines are microstrip and stripline. Themicrostrip approach uses a track (or tracks) on one side of a PC board and a ground planeon the other. A stripline is similar but the trace is sandwiched between two ground planes,one on either side of the board. This is shown in Figure 18-1.

One of the advantages of stripline over microstrip is that the sandwiching of the sig-nal trace between two ground planes contains the EMI fields. Another advantage is that thisform of construction results in a more uniform ground plane surrounding the center con-ductor than microstrip. The microstrip approach is more difficult to analyze because ofleakage fields from the PC track since it is not fully surrounded by a ground plane. Theseleakage fields reduce the effective dielectric constant of the board, which raises theimpedance. This is because some of the field lines pass through air, which has a dielectricconstant of 1 rather than 4.7, a figure typical of PC board material FR4.

CHAPTER 18

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Source: ELECTRONIC FILTER DESIGN HANDBOOK

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720 CHAPTER EIGHTEEN

Transmission lines have a characteristic impedanceZO and an electrical length l1. The

calculations involved in determining these equivalent electrical properties of microstripand stripline transmission lines are based on the width and thickness of the traces, the dis-tance from the ground plane and the dielectric constant eR (sometimes called relative per-mittivity), and are quite complex. A number of programs are available that can calculate theresulting electrical properties of microstrip and stripline implementations. An extremelyuseful one is included on the CD-ROM. Its called txline, and is from Applied WaveResearch, Inc., a world leader in electronic design automation (EDA) software.

A program called Filter Solutions is included on the CD-ROM. This program is limitedto elliptic functionLCfilters (up toN 10) and is a subset of the complete program whichis available from Nuhertz Technologies (www.nuhertz.com). Many microwave filter designcapabilities are included in the full version, such as microstrip and stripline low-pass, high-pass, bandpass and bandstop filters, diplexers, multi-conductor microstrip, and stripline band-pass filters and lumped tubular filters.

18.3 RICHARDS TRANSFORMATION

If a transmission line is shortedat the opposite end and is of a wavelength in length (l/4),an applied sinusoidal signal will be reflected back to the input, exactly in phase with theinput, preventing any current from flowing. This would be analogous to the infinite imped-ance of a parallel resonant circuit, at resonance.

If a transmission line is open at the opposite end and is of a wavelength in length (l/4),an applied sinusoidal signal will be reflected back to the input out of phase with theinput and cancel the signal.

Hence an attenuation pole will occur at this frequency analogous to a series resonant cir-cuit, at resonance, in parallel with the source

Below resonance, a parallel resonant circuit appears inductive and a series resonant cir-

cuit is capacitive. Transmission lines can then be used to implement inductors and capaci-tors for microwave filters providing the wavelengths are below l/4. Typically, a filters

180

1@4

1

@4

FIGURE 18.1 Microstrip and stripline construction.

PC board withdielectricconstant of

PC board withdielectricconstant of

Microstrip

Stripline

Trace

Trace

Ground plane

Ground plane

Ground plane

R

R

1The characteristic impedanceZO of a line can be defined as the impedance seen at the input of that line if it were

infinitely long, while the electrical length is expressed as a multiple or submultiple of the wavelength lof a periodic

signal traveling through a medium. The wavelength of a signal is the distance it travels during one cycle. In free space,

it is l c (speed of light) /f(frequency). In another medium, it is reduced by the multiplication factor .1/!R



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cutoff frequency vC is chosen at l/8. The attenuation poles then occur at 2vC(l/4). The

behavior of open and shorted transmission line stubs is shown in Figure 18-2.Richards transformation (see Bibliography) enables the conversion of lumped element

filters into distributed filters using PC board traces functioning as transmission lines.Shorted and open transmission line stubs approximate the behavior of inductors andcapacitors over a finite frequency range. Each stub has an electrical length ofl/8 at a cut-off frequency vC. For a low-pass filter, vCis the 1 radian/sec cutoff in the normalized case.At 2vC, attenuation poles will occur where the shorted stub behaves like a parallel resonantcircuit at resonance, and the open stub behaves like a series resonant circuit at resonance.

INTRODUCTION TO MICROWAVE FILTERS 721

FIGURE 18-2 Shorted and open transmission line stub equivalents: (a) shorted;length l/4, (b) shorted; length l/4, (c) open; length l/4, and (d) open;length l/4.

(a)

(b)

< /4

< /4

At resonance

At resonance

(c)

(d)

/4

/4




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Line with Short Circuit at Output. The impedance seen at the input of a lossless trans-mission line terminated in a short circuit is

(18-1)

where g is the propagation constant for the lossless line equal to and l is the unitlength. (L and Care the distributed inductance and capacitance of the transmission line per unitlength.) Therefore, we can express the input impedance with a short circuit at the output as

(18-2)

Note that theL and Cin Equation (18-2) are the distributed parameters of the transmis-sion line per unit length l.

The reactance of an inductor is given by the well-known expression

(18-3)

So if we can establish an equivalency between Equations (18-2) and (18-3), we can sub-stitute a shorted transmission line stub for an inductor. Richards transformation results inthe following expression:

(18-4)

Equation (18-4) implies that a shorted transmission line having a characteristic imped-

ance ZO

has an equivalent inductance ofZO

over the frequency range between v and vQ

,

where vQ corresponds to the wavelength frequency. For v vQ, the value ofjvL is infi-

nite (tan p/2 or tan 90 is infinity). We have mapped the S jvfrequency plane into a new

compressed frequency plane bounded by vQ, which would correspond to v infinity in the

original frequency plane.

For a shorted line: L ZO over the frequency domain of 0 v vQ

Line with Open Circuit at Output. In a similar manner, for a lossless transmission lineopen at the far end, Richards transformation results in an expression for the susceptance(1/XC) of a capacitor :

(18-5)

Equation (18-5) then implies that an open transmission line having a characteristicadmittance YO (1/ZO) has an equivalent capacitance ofYO over the frequency range betweenv and vQ, where vQ corresponds to the wavelength frequency.

For an open line: C YO over the frequency domain of 0 v vQ

These results imply that we can implement a lumped element low-pass filter by replac-

ing each inductorL by a shorted transmission line having aZO equal toL, and each capac-itor C by an open transmission line, having a YO equal to C. The lines must have awavelength ofl/4 (90 or p/2) at the highest frequency of operation where an attenuationpole will occur. The cutoff frequency vCshould correspond to l/8.

It should be recognized that as we go above vC, the impedance of the stubs do notexactly track the original lumped element impedances, so deviations from the response ofthe theoretical low-pass prototype will occur. Also, it is important to be aware of the factthat the response isperiodic and repeats every 4vC. This periodicity is a general character-istic of distributed-element filters and should be anticipated.

1@4

jvC jYOtanBp2 , vvQR

1@4

jvL jZO tanBp2 , vvQR

XL jvL

ZSC jZO tan(v2LCl)

jv!(LC)

ZSC jZO tan(gl)





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Example 18-1 Design of a Microstrip Low-Pass Filter

Required:

Design a 0.5 dBN 5 Chebyshev Low-Pass filter normalized to 1 radian/sec using amicrostrip-distributed transmission line.

Result:

Figure 18-3a illustrates a n 5 0.5 dB Chebyshev low-pass filter obtained fromTable 11-30 normalized for a 3 dB cutoff of 1 radian/sec and 1 . The circuit of Figure18-3b replaces each capacitor Cby a parallel stub open-circuited transmission line hav-ing a characteristic impedanceZequal to the reciprocal of the capacitor C. Each induc-torL is realized by a series stub short-circuited transmission line having a characteristicimpedanceZequal toL.


FIGURE 18-3 A normalized n 5 0.-dB Chebyshev low-pass filter with(a) lumped constant values from Table 11-30, and (b) a microstrip-distributedtransmission line equivalent.

SC SC

OCOCOC

Z1

Z2

Z3

Z4

Z5

Y1 = C1 = 1.8068Z2 = L2 = 1.3025Y3 = C3 = 2.6914Z4 = L4 = 1.3025

Y5 = C5 = 1.8068

N = 5 0.5 dB Chebyshev

C1 C3 C5

C1 = 1.8068L2 = 1.3025C3 = 2.6914L4 = 1.3025C5 = 1.8068

(a)

Z1 = 1/Y1 = 0.5535Z3 = 1/Y3 = 0.3716Z5 = 1/Y5 = 0.5535

1

L2 L4

1

1

1

(b)




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18.4 KURODAS IDENTITIES

The implementation of Figure 18-3 requires series stubs with a short circuit at the far end.These shorted stubs are extremely difficult to implement on a PC board without using otherapproaches such as coaxial transmission lines. The physical geometry of the PC board con-taining these series shorted stubs in addition to the parallel open circuited stubs becomesawkward and impractical.

Kurodas identities allow the transformation of series stubs into shunt stubs and viceversa. This is an exact transformation and not an approximation. This transformation isaccomplished by introducing a building block called a Unit Element or UE, which is atransmission line having a length ofl/8 at vC, and a characteristic impedance of 1- in thenormalized case.

Figure 18-4 shows the application of Kurodas identities to make this conversion.

Series to Shunt Stub. To transform a series stub (inductive) to a shunt stub (capacitive),as shown in Figure 18-4a, first compute n as follows:

(18-6)n 1 Z1

ZO


FIGURE 18-4 Kurodas identities: (a) a series to shunt stub, and (b) a shunt to series stub.

Unitelement

Unitelement

Unitelement

Z0

Z1

Z1

Unitelement

(a)

(b)

n = 1 +Z0

Z1

ZC = n Z0

ZE =n Z0n 1

ZC

ZE

ZE =n 1

n

n = 1 +Z1

Z0

Z0

ZC = n

ZCZ0

ZE

Z0




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then

(18-7)

(18-8)

Shunt to Series Stub. To transform a shunt stub (capacitive) to a series stub (inductive),shown in Figure 18-4b, first compute n as follows.

(18-9)

then:

(18-10)

(18-11)

Combining Richards Transformation and Kurodas Identities to Design a Low-PassFilter. The following example is a continuation of Example 18-1 converting all seriesstubs to parallel stubs using Kurodas identities.

Example 18-2 Implementing the Microstrip Filter of Example 18-1

Required:

Implement the filter of Example 18-1 using a microstrip approach for a cutofffc of1GHz and a source and load impedance of 50-.

Result:

(a) The normalized low-pass filter is shown in Figure 18-5a. Replace each capacitor Cwith a shunt open-circuit stub having a characteristic impedance of 1/Cand eachinductorL with a series shorted stub having a characteristic impedance ofL. This is

shown in Figure 18-5b.

ZEn 1

n ZO

ZC

ZOn

n 1 ZO

Z1

ZE

nZO

n 1

ZC nZO


1

1

N = 5 0.5 dB Chebyshev

(a)

C1 = 1.8068

L2 = 1.3025 L4 = 1.3025

C3 = 2.6914 C5 = 1.8068

FIGURE 18-5 The design development process of Example 18-2: (a) a low-pass prototype; (b) convertingLCelements into series and shunt stubs; (c) addingunit elements UE1 and UE2 at ends; (d) moving shunt stubsZ1 andZ5 to the leftand right respectively of UE1 and UE2, and converting them to series stubs;(e) adding unit elements EU3 and UE4; (f) shifting series stubsZ1 andZ2 to theleft, andZ4 andZ5 to the right, converting them into shunt stubs; and (g) a finalimplementation using Microstrip traces on PC board.




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FIGURE 18-5 (Continued)

Z1 = 1/1.8068 = 0.5535

1

1

Z3 = 1/2.6914 = 0.3716 Z5 = 1/1.8068 = 0.5535

Z4 = 1.3025Z2 = 1.3025

(b)

1

1

ZUE1

= 1

UE1 UE2

Z1

= 0.5535 Z5

= 0.5535Z3

= 0.3716

Z2 = 1.3025 Z4 = 1.3025

ZUE2

= 1

(c)




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(d)

UE1 UE2

Z21

Z2 = 1.3025 Z4 = 1.3025

Z3 = 0.3716

Z1 = 0.6437 Z5 = 0.6437

ZUE1 = 0.3563 ZUE2 = 0.3563

1

(e)

UE1

1

1 UE2 UE4UE3

ZUE1 = 0.3563 ZUE2 = 0.3563ZUE3 = 1ZUE4 = 1

Z1 = 0.6437 Z5 = 0.6437Z2 = 1.3025

Z3 = 0.3716

Z4 = 1.3025

(f)

Z3 = 0.3716Z2 = 0.4538Z1 = 2.5535

UE1

1

1

ZUE1 = 1.6588 ZUE2 = 1.6588ZUE4 = 1.6437ZUE3 = 1.6437

UE3 UE2 UE4

Z4 = 0.4538 Z5 = 2.5535




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50 50

128 128

82.2 82.2

22.7 22.7

82.9 82.9

18.6

(g)

(b) The objective is to convert all series stubs to parallel stubs. First, add unit elements,UE1 and UE2, at each end. This is shown in Figure 18-5c.

(c) Move shunt stubsZ1 andZ5 to the left and right respectively of UE1 and UE2, con-verting them to series stubs, as shown in Figure 18-5d. The values ofZ1,Z5, UE1, andUE2 will change in accordance with Equations (18-9) through (18-11) (Figure 18-4b)using Kurodas identities.

(d) Add unit elements EU3 and UE4, as shown in Figure 18-5e.

(e) Shift series stubsZ1 andZ2 to the left, andZ4 andZ5 to the right, converting them intoshunt stubs. The values of UE1 through UE4 andZ1,Z2,Z4, andZ5 will all change asthe series stubs are shifted across the unit elements and converted into shunt stubs inaccordance with Equations (18-6) through (18-8) (Figure 18-4a) using Kurodas

identities. The final circuit is shown in Figure 18-5fconsisting of all shunt stubs.

( f) The microstrip implementation is illustrated in Figure 18-5g, where the design isimpedance-scaled to 50-ohms. This figure actually shows the PC board traces repre-senting the characteristic impedances as indicated. Each section should have an electri-cal wavelength ofl /8 at anfc of 1 GHz. At 2 GHz, the wavelength is l /4, wheremaximum attenuation will occur since each stub has an attenuation pole at this frequency.Keep in mind the presence of a ground plane separated by the PC board thickness.

The dimensions can be computed using the txline program included on the CD-ROM. Forexample, for a PC board of fiberglass resin (FR4 material), use a dielectric constant of

4.7 and copper as the conductor. Enter the cutoff frequency, impedance, and wavelength(l/8 or 45). The dimensions are indicated on the right after clicking the arrow ( ).

18.5 BANDPASS FILTERS

There are various permutations of using open and shorted stubs and transmission line sec-tions for designing filters. Matthaei (see Bibliography) covers numerous design techniquesusing stubs in great detail, along with a wide range of other microwave filter design meth-ods. The mathematics for the following filters is beyond the intent of this chapter and the

reader should refer to Matthaei and the other references listed.

Bandpass Filters Using Shorted Parallel Stubs. An implementation for bandpass filtersusing shorted l/4 stubs connected by transmission lines l/4 long at the bandpass centerfrequency is shown in Figure 18-6a. Figure 18-2a indicates that an open stub having awavelength ofl/4 at a particular frequency is equivalent to a parallel resonant circuit atresonancethat is, exhibiting infinite impedance. The l/4 sections connecting the threeparallel stubs transform the center parallel stub into a series impedance that is a series

S





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resonant circuit. The equivalent circuit is shown in Figure 18-6b. This approach is mosteffective when applied to narrow-band bandpass filters. If the stubs were of the open-circuit type, a band-stop filter would result.

Bandpass Filters Using Edge-Coupled Half-Wavelength Lines. The filter shown inFigure 18-7a is a narrow band edge coupled n 5 bandpass filter. This approach is lim-ited to relatively narrow bandwidths. The analogousLCcircuit is shown in Figure 18-7b


(a)

/4/4

/4/4

/4

FIGURE 18-6 (a) A bandpass filter using l/4 shorted parallelstubs and interconnecting l/4 transmission lines, and (b) theequivalent bandpassLCcircuit.

(b)

(a)

(b)

FIGURE 18-7 An edge-coupled bandpass filter: (a) a microstrip filter, and (b) the equivalent circuit.




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and is similar to the narrow-band capacitive coupled resonators covered in Section 5.1.Other versions involve laying the sections end-to-end rather than using parallel cou-pling.

18.6 ADDITIONAL DESIGN METHODS USINGPC BOARD TRACES

The following section discusses how PC board traces can be used as filter elements ratherthan the use of discrete components.

Using PC Board Traces to Replace Inductors and CapacitorsRather than using a transmission line approach with open stubs, an alternate method usestraces on the PC board interacting with the ground plane to replace inductors and capaci-tors. The length of these traces must be much less than l/4 in the passband. A long nar-row trace can replace an inductor and a wide trace can replace a capacitor. An intuitiveexplanation for this is that the wide capacitor trace has more surface area and hence morecapacitance to the ground plane. The length of the narrow trace is proportional to theinductance. Figure 18-8 illustrates what form this type of implementation would take. Thereader should refer to the Winder citation (see the Bibliography) for the computationaldetails.

Inductors can be made more efficiently as far as PC board utilization goes by takingforms other then straight traces. Figure 18-9 shows inductors in a spiral format. Designequations are can be found by researching the Wadell reference (see Bibliography).Removal of ground planes under traces will increase inductance.


N = 5 0.5dB Chebyshev3dB = 2GHz

50

L2 = 5.18nH

C1 = 2.88pF C3 = 4.28pF C5 = 2.88pF

L4 = 5.18nH

50

C1 C3 C5

L2 L4

FIGURE 18-8 A microstrip 2 GHz filter using PC board traces.




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BIBLIOGRAPHY

Kuroda, K. Some Equivalent Transformations in Ladder-Type Networks with Distributed Constants.Inst. Elec. Commun. Engr. Monograph Series on Circuit Theory (February, 1957) (in Japanese).

Matthaei, G. L., Young, L., and E. M. T. Jones.Microwave Filters, Impedance-Matching Networks,and Coupling Structures. Massachusetts: Artech House, 1980.

Rhea, R. W. HF Filter Design and Computer Simulation. Georgia: Noble Publishing Company, 1994.

Richards, P. I. Resistor Transmission Line Circuits.Proceedings of the IRE36 (February, 1948):

217.

Wadell, B. C. Transmission Line Design Handbook. Massachusetts: Artech House, 1991.

Winder, S.Analog and Digital Filter Design. Massachusetts: Elsevier Science, 2002.


FIGURE 18-9 Spiral inductors.




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