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Exercise notesMicrowave Semiconductor Devices and Circuits I

Winter Term 2008 / 2009

Contents

1 Planar microwave circuits 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Waves on transmission lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Waveguide geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3.1 Stripline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3.2 Coplanar waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3.3 Slotline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3.4 Finline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3.5 Microstripline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Substrate materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5 Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Transmission line theory and discontinuities 62.1 Transmission lines and terminations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.1 Matching with / 4 transmission line . . . . . . . . . . . . . . . . . . . . . . . 82.2 Discontinuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.1 Microstrip open circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.2 Step in width of a microstripline . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.3 Rectangular junction and microstrip bends . . . . . . . . . . . . . . . . . . . . . 122.3 CAD-Exercise: Matching Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.1 Design of 4 matching networks using ideal transmission lines . . . . . . . . . . 122.3.2 Design of 4 matching network using microstrip transmission lines . . . . . . . . 13

3 Microwave circuits using coupled transmission lines 143.1 Open-circuited two-conductor line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2 Bandpass element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4 Systematic lter design 184.1 Insertion-loss method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.2 Bandpass lter design equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.3 CAD-Exercise: Design of Chebyshev bandpass lter . . . . . . . . . . . . . . . . . . . 224.3.1 Filter specication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.3.2 Computation of even- and odd mode impedances . . . . . . . . . . . . . . . . . 234.3.3 Simulation using ideal transmission lines and couplers . . . . . . . . . . . . . . 234.3.4 Simulation using microstrip transmission lines and couplers . . . . . . . . . . . 234.3.5 Optimization of the lter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5 Signal ow graphs 25

References 30

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1 Planar microwave circuits 1

1 Planar microwave circuits

1.1 Introduction

The use of waveguide components for microwave circuits leads in general to large circuit sizes and inte-gration of different components (like transistors, diodes etc.) is difcult. Integrated circuits at microwavefrequencies can be easily built up in a planar technology like low-frequency multilayer PCBs (PrintedCircuit Boards) so integration of discrete components is simple. These circuits are called MICs (Mi-crowave Integrated Circuit) if the integration is realized in a hybrid way (transmission lines, passive andactive components on the same carrier substrate, see Fig. 1). A monolithic integration is called MMIC(Monolithic Microwave Integrated Circuit, see Fig. 2). Here, all functions are realized on a single waver.

Figure 1: Oscillator MIC using dielectric resonators [6].

Figure 2: Single-sideband modulator MMIC with GaAs FET [6].

1.2 Waves on transmission lines

This section introduces the properties of waves on transmission lines. A detailed description of theunderlying theory can be found in [1]. A short piece of a TEM transmission line can be described by theequivalent circuit shown in Fig. 3 with the distributed circuit elements L, C , R, und G. In this way ageneralized voltage and current can be dened:

V = V + e j z + V e j z, (1)

I = I + e j z I e j z. (2)V + , I + are waves in positve and V , I are waves in negative direction. The propagation constant canbe computed from L, C , R, und G:

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Z ,0 g, l

z

ZL

Zg

Vg

L'dz R'dz

C'dzG'dz

V(z)

I(z) I(z+dz)

V(z+dz)

Figure 3: Equivalent circuit of TEM transmission line (length dz ).

=

( R+ j L)(G+ j C ) = + j . (3)

is the attenuation of the transmission line and the phase constant at angular frequency . From thosethe following parameters can be dened:

k 0 = 0 0 = 2 0 = c0

, (4)

and

e f f = k 0

2

= 0

wg

2

=c0v p

2

, (5)

e f f is the effective dielectic constant. ( 0) and (c0) are the free-space wavelength and the speed of light, respectively.

wgand v

pare the wavelength and the phase velocity on the transmission line. In the

TEM-case, the characteristic impedance can be computed from (zero losses):

Z 0 = LC = C = e f f c0C (6)TEM-waves are not supported by some waveguides since a TEM-wave requires at least two conductorsand a homogeneous and loss-free medium of propagation. Practical geometries do not satisfy this con-dition, so the axial components of the electric and magnetic eld do not vanish. The waves are TM, TEor even hybrid. However, at microwave frequencies the transversal eld components are dominant, sothe waves can be treated as quasi-TEM. The following denitions of the characteristic impedance arecommon:

Z 0 =P

2 I 2, (7)

Z 0 =V I

. (8)

P is the power of the wave, which can be computed from the poynting-vector.

1.3 Waveguide geometries

The most common planar waveguide geometries are described in this section.

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electromagnetic shielding an additional ground plane can be used. The lateral dimensions of the groundsmetallizations should be at least 5 b.

Advantages: Low dispersion since the ratio of the power, which is transported within the air and thesubstrate is independent of the frequency of operation. Therefore, the characteristic impedance canbe adjusted simply by the ratio of ab . Close packed circuit geometries are possible with this kind of waveguide. Short circuits can easily be realized.Drawbacks: The additional ground plane may cause problems due to the excitation of parallel platemodes. This increases the losses on the transmission line.Effective dielectric constant and characteristic impedance for coplanar waveguide (substrate extends toinnity at lower z-direction) [3]:

e f f = r + 1

2, (11)

Z 0 = 0K (k )

4 e f f K (k ) . (12)k = ab and k = 1 k 2 . The computation can be carried out again by the method of conformal mapping.

1.3.3 Slotline

Figure 6: Slotline.

This structure (Fig. 6) is built up from two metallizations separated by a single slot with the width w.This waveguide is used in the millimeter-wave range. The guided wave is quasi TE. The expressions of the transmission line parameters are complicated and will not be repeated here.

1.3.4 Finline

A slotline (Fig. 7) inserted into a waveguide is called nline. This transmission line is used for tran-

sitions between waveguides and planar circuits. There are serveral different congurations of a nline(unilateral, bilateral and antipodal).

1.3.5 Microstripline

This structure (Fig. 8) consists simply of a metal stripe of width w and thickness t above a groundmetallization with a substrate of dielectric constant r and height h in between. The guided waves arehybrid TE/TM-waves. This structure is dispersive, so a quasistatic calculation can be carried out onlyapproximately.Advantages: Simple design. Junctions and open circuits can be easily realized. Discrete componentscan be inserted with no difculty.

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Figure 7: Finline.

Figure 8: Microstripline.

Drawbacks: Short circuits are more difcult to realize (via-hole required). Especially at higher frequen-cies parastic elements of this vertical interconnect may cause deviations from an "ideal" short circuit.

The mircostripline is the most common transmission line for MICs. Therefore, this exercise has a specialfocus on it. However, the design procedures presented in the following can also be used for differenttypes of transmission lines.The effective dielectic constant and characteristic impedance in the quasistatic case (TEM) can becomputed by the following formulas [3]:

e f f = r + 1

2+

r 12

F (13)

F =1

1 10hw(14)

and

Z L(wh

, r ) =Z L0

e f f . (15)

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2 Transmission line theory and discontinuities 6

Z L0 = Z L( r = 1) = 60ln8hw

+w4h

(16)

The thickness of the metal stripe has been neglected in the formulas given above. However, all commonCAD tools account for the thickness, so the more complicated formulas will not be repeated here.

1.4 Substrate materials

Substrates should have low losses at RF frequencies. Materials with high dielectric constant can beused for circuit miniaturization since the guided wavelength is reduced. The drawback of these sub-strates when used in combination with microstriplines is a eld concentration within the substrate andsubsequently higher losses. Antenna substrates should exhibit a low dielectric constant to reduce eldconcentration within the substrate. Antenna efciency is increased by the way. Good mechanical stabil-

ity and easy fabrication of the material are other important factors. If active components are used a goodthermal conductivity of the substrates is important, too.Standard FR4 material, which is used for most low-frequency PCBs, can not be used for high-frequencyapplications since the dielectic constant and the thickness are subject to large deviations (up to 20 %)and the losses are relatively high. Resonant applications like lters are very sensitive to these parametersand can not be realized on such substrates. Some suitable materials are listed in [5].A common material used for microwave substrates is PTFE (Teon). Substrates with PTFE as basematerial exhibit dielectric constants of 2 to 10. The high dielectric constant materials are lled withceramic material.Another standard material is alumina (Al2O3) with good thermal conductivity and very low losses. Butthis substrate is expensive and is difcult to machine.The substrates are plated on both sides with a conductor (Cu or Ag).

1.5 Fabrication

The following is a description of the fabrication process at the Arbeitsbereich. A mask with the mi-crowave circuit structure is drawn with a CAD tool on a PC. The mask data are sent to a photo mask manufacturer. The substrates are coated with a photoresist. The mask is attached to the substrate and isexposed by a UV-light source. The exposed photoresist is developed and so the structure is transferredonto the substrate. Subsequently, the substrate is etched and the remaining photoresist removed. Thiscompletes the fabrication.

2 Transmission line theory and discontinuities

2.1 Transmission lines and terminations

Fig. 9 shows a transmission line of propagation constant , characteristic impedance Z 0 and length lterminated with the impedance Z L. The input impedance of the transmission line is:

Z 1 = Z 0 Z L + Z 0 tanh ( l) Z 0 + Z Ltanh ( l)

. (17)

Zero losses:

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Z ,0 g, l

z

ZL

Zg

Vg

Figure 9: General termination at the end of a transmission line.

Z 1 = Z 0 Z L + jZ 0 tan ( l) Z 0 + jZ L tan ( l)

. (18)

Two important cases will be considered in the following: transmission lines with a length of / 4 and / 2. These correspond to resonant circuits from Fig. 10.

Zg

Vg

L

C

Zg

VgL

C

Figure 10: Series- and parallel resonant circuits for comparison with / 4 transmission lines.

/ 4 transmission line:l =

4

l = 2

. (19)

For small deviations of frequency = 0 + : l = 2 + with = 2

0 . The input

impedance can be calculated from eq. (18):

Z 1 Z 0

jZ 0

Z L

jZ L Z 0 . (20)A / 4 transmission line terminated with an open circuit ( Z L = ) has the input impedance:

Z 1 Z 0

= j . (21)

The impedance of a series resonant circuit is:

Z s = j L+1

j C (22)

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=1 2 LC

j C (23)

=1 1 + 0

2

j( 0 + )C (24)

j2 L , (25)Higher order terms have been neglected. 0 = 1 LC . Comparison with eq. (21) yields:

C =4

1 0 Z 0

, (26)

L =

4

Z 0

0. (27)

The short circuited transmission line can be modelled by an equivalent parallel resonant circuit:

C = 4

1 Z 0 0

, (28)

L =4

Z 0 0

. (29)

/ 2 transmission line:The calculation is essentially the same; the series- and parallel resonant circuit are interchanged inthis case.

impedance- and admittance inverterA / 4 transmission line has following input impedance:

Z 1 =Z 20 Z L

. (30)

The termination Z L is "inverted". Therefore, a / 4 transmission line can be used as an impedanceinverter. These elements are required in lter theory and can also be used for transformation of a

series circuit into a parallel circuit.

2.1.1 Matching with / 4 transmission line

The most simple matching network is a single / 4 transmission line (Fig. 11) [1]. The impedancetransfomation function with tan = t is:

Z in = Z 1 Z L + jZ 1t Z 1 + jZ Lt

. (31)

The reection coefcient vanishes if Z 1 = Z 0 Z L is chosen for a particular frequency. For the reectioncoefcient follows:

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Z0 Z , l1 b = p/2Z

L

Z0

Vg

Zin

Figure 11: Single stage matching network with / 4 transmission line.

=Z in Z 0 Z in + Z 0

(32)

=Z 1 Z L + jZ 21 t Z 0 Z 1 Z 0 Z Lt Z 1 Z L + jZ 21 t + Z 0 Z 1 + Z 0 Z Lt

(33)

=Z 1( Z L Z 0) + jt ( Z 21 Z 0 Z L) Z 1( Z L + Z 0) + jt ( Z 21 + Z 0 Z L)

(34)

=Z L Z 0

Z L + Z 0 + 2 jt Z 0 Z L (35)

For broadband applications the input bandwidth is an important gure of merit. For the calculation themagnitude of the reection coefcient is evaluated:

= | Z L Z 0| ( Z L + Z 0)2 + 4t 2 Z 0 Z L

(36)

= ( Z L Z 0)2 Z 2l + Z 20 + ( 4 s2c2 + 2) Z 0 Z L (37)= ( Z L Z 0)2( Z L Z 0)2 + Z 1 Z L 1c2 (38)=

1

1 + 2 Z 0 Z L

Z L Z 02

1

c2

. (39)

s = sin and c = cos . All impedances are considered as real quantities. For a maximum input reectionm follows the maximum / minimum electrical length of the matching transmission line m:

m = cos1 2m Z 0 Z L

( Z L Z 0) 1 2m. (40)

The bandwidth for TEM transmission lines with = l = f m f 0 2 and f 0 as frequency of operation is:

f = 2( f 0 f m) = 2( f 0 2 f 0

m). (41)

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The usable bandwidth of a matching network is relatively small. In order to increase the usable bandwidtha multi stage matching network can be used. An exemplary network is calculated in the following. Fig.

12 shows a network of cascaded / 4 transmission lines.

Z0 Z , l1 b = Q Z , l2 b = Q Z , lN b = Q ZL

Zg

Vg

G0G1 G2 G N

Figure 12: Multi stage matching network of cascaded / 4 transmision lines.

For the reection coefcients at each transmission line follows:

n =Z n+ 1 Z n Z n+ 1 + Z n

. (42)

Z n are the characteristic impedances of each transmission line (Fig. 12). n is only the reection coef-fection at the respective boundary and not the total reection coefcient. For the following calculationit is assumed that Z L > Z 1 and all characteristic impedances are real quantities (for Z L < Z 1 the signs of the reection coefcients must be inverted). A maximally at reection coefcient in the passband isdesired. The function of the reection coefcient vs. frequency shall:

vanish at f 0 like

all N 1 derivatives.Such a function is given by:

= A(1 + e j2 ) N ; (43)the magitude of the reection coefcient is = A2 N cos N .The impedances of each single stage and the constant A must be calculated:

= A(1 + e j2 ) N = A N

n= 0

N n

e2 jn (44)

with the binomial coefcients:

N n

=N !

( N n)!n!. (45)

If the difference of the impedances is not too large, multiple reections can be neglected (rst orderreections only). This assumption leads to:

= N

n= 0

ne j2n . (46)

Comparison with eq. (44) yields:

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n = A N n

= AN

N

n

= N n . (47)

The equation is symmetrical: n = N n .The missing expressions for the characteristic impedances and the coefcient A can be calculated fromthe reection at f = 0 ( = 0). In this case the reection is simply:

( = 0) = 2 N A =Z L Z 0 Z L + Z 0

12

lnZ L Z 0

. (48)

The approximation is based on the power series of the ln-function. The reections at the single transmis-sion lines can be calculated by the same approximation:

n =Z n+ 1 Z n Z n+ 1 + Z n

12

lnZ n+ 1 Z n

. (49)

with eq. (47) - (49) follows:

lnZ n+ 1 Z n

=1

2 N N n

lnZ L Z 0

. (50)

The impedances can be calculated with this equation (starting with Z 1).For the maximum specied input reection m the bandwidth is (cf. eq. (43)):

= | | = | A(1 + e j2 ) N |= (51)

= | A|2 N |cos N | (52)

m = cos1 2m

ln Z L Z 0

1 N

(53)

f f 0

= 2 4

cos1 2mln Z L Z 0

1 N

. (54)

This bandwidth is considerably larger than that of the single stage network.

2.2 Discontinuities

Discontinuities occur when one transmission line parameter changes abruptly. If these changes are smallcompared with the wavelength discontinuities can be described by an equivalent circuit model of discretecomponents. Some important discontinuities and their equivalent circuit models will be described in thefollowing. The additional parasitic effects which occur are:

distortion of the electric and magnetic eld and of the current distribution on the transmission line.Higher order modes are excited.

a fraction of the power may be radiated or surwave waves may be excited.Most CAD-Tools include accurate models of the most common discontinuities.

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2.2.1 Microstrip open circuit

An abrupt ending of the microstripline. This should result in an impedance of Z L = . However, there isan electrical eld around edge of the ending which results in a capacitive load C L of the microstripline.This effect can also be described by an additional piece of transmission line l L. The equivalent circuitelements are given by [3]:

l L =C LC

= C L Z 0c0

e f f . (55)

l L = 0, 412 h( e f f + 0.3)( wh + 0.26)

( e f f 0.258 )( wh + 0.813 ). (56)

The error is smaller than 5%.

2.2.2 Step in width of a microstripline

A step in width of the microstripline from w1 to w2 changes not only the characteristic impedance butalso introduces parasitic effects. There is a distortion of the current densitiy which has an inductiveeffect ( Ls). The stray electric eld has a capacitive effect C p or can be modelled by an additional pieceof transmission line ls1 (cf. microstrip open circuit). The equivalent circuit elements are given by [3]:

Lsh

=2 0

ln 1/ sin ( 2

Z L1,0 Z L2,0

) , (57)

C p = ( e f f 1 Z L1c0 0 r

w1h

)(w1 w2)/ 2, (58) lS1 =

1.35 r

+ 0.44 (1 w1w2

)h, (59)

Z Li,0 is the impedance of width i without dielectric (air). The above formulas are valid for w1 > w2 .

2.2.3 Rectangular junction and microstrip bends

A rectangular junction of three tranmission lines has the following parasitics: distortion of the currentdensity and distortion of the electric and magnetic elds. Again, the current distortion can be modelledby an inductance and the distortion of the electric eld can be modelled by an capacitance. A commonlyused bend is the optimal right-angle mitered bend. This bend compensates for the parasitics:

Lw

= 2(1.04 + 1.3e1.35 wh ), (60)b = 2w

L2

. (61)

2.3 CAD-Exercise: Matching Networks

2.3.1 Design of 4 matching networks using ideal transmission lines

Task: Design and comparison of single and multiple stage matching networks (up to fourth order) at f = 2 GHz and Z 0 = 50 matched to Z 1 = 100 .

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computation of characteristic impedances (eq. 50)here: 50 matched to 100

N=1:n = 0 Z 1 = Z 0 EXP ( 12 10 ln ( 100 50 )) =

N=2:n = 1 Z 1 = Z 0 EXP ( 14 20 ln ( 100 50 )) = n = 2 Z 2 = Z 1 EXP ( 14 21 ln ( 100 50 )) =

N=3:n = 1 Z 1 = Z 0 EXP ( 18 30 ln ( 100 50 )) = n = 2 Z 2 = Z 1 EXP ( 18 31 ln ( 100 50 )) = n = 3 Z 3 = Z 2 EXP ( 18 32 ln ( 100

50 )) =

N=4:

n = 1 Z 1 = Z 0 EXP ( 116 40 ln ( 100 50 )) = n = 2 Z 2 = Z 1 EXP ( 116 41 ln ( 100 50 )) = n = 3 Z 3 = Z 2 EXP ( 116 42 ln ( 100 50 )) = n = 4 Z 4 = Z 3 EXP ( 116 43 ln ( 100 50 )) =

drawing of the schematic and simulation of the network Draw the networks in ADS schematicwindow and start the simulation. The bandwidth can be obtained from the simulated scatteringparameters.

B1 = MHz B4 = MHz

2.3.2 Design of 4 matching network using microstrip transmission lines

Task: Design and comparison of single and multiple stage matching networks (up to fourth order) at f = 2 GHz and Z 0 = 50 matched to Z 1 = 100 .

computation of microstrip line widths and lengths (ADS-Tool LineCalc) A substrate with fol-lowing parameters is used:

h = 1 mm r = 3 tan = 0.001 hcu = 35 m

w1 = mm l1 = mmw2 = mm l2 = mm

w3 = mm l3 = mmw4 = mm l4 = mm

drawing of the schematic and simulation of the network Draw the networks in ADS schematicwindow and start the simulation. The bandwidth can be obtained from the simulated scatteringparameters.

B1 = MHz B4 = MHz

Due to the different microstrip line widths additional parasitic effects are introduced. Insert anappropriate simulation model (step in width) between each transmission line and compare theresults with the ideal case.

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3 Microwave circuits using coupled transmission lines 14

3 Microwave circuits using coupled transmission lines

Beside simple transmission line circuits like 4

-transformators coupled transmission lines are widelyused. These are simple multiconductor ciruits. However, for the operating frequency range the coupledline sections can be transformed to noncoupled lines. If the operating frequency range is sufcientlynarrow, an equivalanet circuit model of lumped elements exists. This is illustrated in the following bytwo coupled open-circuited transmission lines.

3.1 Open-circuited two-conductor line

The following calculation is carried out for a homogeneous and symmetrical two-conductor transmissionline (Fig. 13, 14).

w

w

s

(a) (b)

Even mode

Odd mode

I1

I1

I1

-I1

Conductor (a)

Conductor (b)

Figure 13: Two coupled microstrip lines (a) conductors, (b) eld-plot of even- and odd mode.

The generalized currents and voltages on the transmission line can be described by forward- and back-ward travelling waves:

V a ( z) = V +e e j z + V e e j z + V +o e j z + V o e j z, (62)

I a ( z) =V +e Z e

e j z V e Z e

e j z +V +o Z o

e j z V o Z o

e j z, (63)

V b( z) = V +e e j z + V e e j z V +o e j zV o e j z, (64) I b( z) =

V +e Z e

e j z V e Z e

e j z V +o Z o

e j z + V o Z o

e j z (65)

e: even mode, o: odd mode, + : forward-travelling wave, backward-travelling wave, 1: conductor 1,2: conductor 2, Z e : even mode impedance, Z o : odd mode impedance.

The coupled transmission lines can be calculated with the above formulas.For circuit synthesis it is convenient to have an equivalent circuit model without coupled transmissionlines (see Fig. 15).For comparison of the two circuits the results for short-circuit and open-circuit terminations are com-puted. For unsymmetrical networks this has to be done on both sides (input and output). Since thenetwork is symmetrical one can immediately set Z 1 = Z 3 for the equivalent circuit model in Fig 15.

The following boundary conditions are obtained:

port 1 ( z = 0) - xed voltage V 1 :

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w

w

s

Conductor (a)

Conductor (b)

Port 1

Port 2

Figure 14: Two coupled open-circuited tranmission lines.

Figure 15: Equivalent tranmission line circuit model for two coupled open-circuited transmission lines.

V a (0) = V 1 = V +e + V e + V +o + V o . (66)

open circuit at the end of conductor a ( z = l):

I a (l) = 0 = Y eV +e e j l Y eV e e j l + Y oV +o e j l Y oV o e j l . (67)

open circuit at the end of conductor b ( z = 0):

I b(0) = 0 = Y eV +e Y eV e Y oV +o + Y oV o . (68)

open-circuit termination of port 2 ( z = l):

I b(l) = 0 = Y eV +e e j l Y eV e e j l Y oV +o e j l + Y oV o e j l . (69)

short-circuit termination of port 2:

V b(l) = 0 = V +e e j l + V e e j l V +o e j l V o e j l . (70)This is a 4 4 system of equations, which has to be solved for the unknown forward- and backwardtravelling waves. The impedance of port 1 for the open-circuit termination of port 2 is:

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Z oc =

j Z e + Z o

2cot l (71)

For the short-circuit termination the impedance is:

Z sc = j( Z e Z o)22( Z e + Z o)

tan l jZ e Z o

Z e + Z ocot l (72)

The impedances (open/short-circuit) for the equivalent circuit model (Fig. 15) are:

Z oc = j( Z 1 + Z 2) cot (73)and

Z sc = jZ 21

Z 1 + Z 2tan j

Z 22 + 2 Z 1 Z 2 Z 1 + Z 2

cot (74)

Comparison leads to:

Z 1 + Z 2 =Z e + Z o

2, (75)

Z 21 Z 1 + Z 2

=( Z e Z o)22( Z e + Z o)

(76)

The impedances of the equivalent cirfcuit model can now be calculated:

Z 1 = Z o = Z 3 , (77)

Z 2 =Z e Z o

2(78)

This demonstrates the equivalence of both circuits (coupled transmission lines and transmission line withstubs). The equivalent circuit model can be analyzed by conventional transmission line theory.

3.2 Bandpass element

A bandpass can be realized with multiple coupled transmission lines. We use two coupled transmissionlines in series connection (Fig. 16). For a small frequency range this circuit can be further simnplied.We assume an electrical length of the tranmission lines of = 2 (

4 -transmission lines). The open-

circuited stub lines can be described by a LC-serial resonant circuit. The inductance is L = Z 4 0 and thecapacitance C = 4

1 0 Z . 0 is the angular frequency (for =

2 ) and Z the chracteristic impedance of the

transmission line. The resulting equivalent circuit model is shown in Fig.16.This model uses impedance inverters which are described by the length of the connecting transmissionline ( 4 ) and the parameter K =

Z e Z o2 .The impedances in the reference planes (Fig. 18) are:

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Figure 16: (a) Bandpass element, (b) equivalent circuit model.

Figure 17: Simpied equivalent circuit model for the bandpass element (impedance inverter K ).

Z A =K 2

Z L, (79)

Z B = Z S +K 2

Z L, (80)

Z C =K 2

Z S + K 2

Z L

. (81)

The input admittance is Y C = Z SK 2 +1

Z L. If a serial resonant circuit is chosen for Z S = j L 1 j C this

circuit is transformed to a parallel resonant circuit ( C = LK 2 , L = 1K 2C ). The resulting equivalent circuitmodel is shown in Fig. 19.This is a standard bandpass circuit element with following parameters:

L = Z o4 0

, (82)

C =4

1 Z o

, (83)

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4 Systematic lter design 18

Figure 18: Impedance inverter reference planes.

Figure 19: Further simplied circuit: standard bandpass.

L = ( Z e Z o)22 0 Z o

, (84)

C = Z o( Z e Z o)

2 0

. (85)

It is now possible to determine the impedances of the coupled transmission lines from the element values.This is necessary for a systematic lter design, which is described in the next section.

4 Systematic lter design

Filters are among the few circuit components which can be synthesized analytically. Starting from thespecications and the lter topology a lter prototype can be designed with the procedure described in the

following. The practical realization needs serveral optimzing steps because of some assumptions whichmust be made in the theory (e.g. losses and dispersion are neglected). Even if the CAD-Tools providemore sophisticated models a design has to be made with care. Numerical optimization of the lterstructure is possible but sometimes the optimization process yields unphysical results. Therefore, anyoptimization has to be closely monitored. Filter design can be carried without complete understanding of the theory by use of lter tables. However, for specialized applications a good knowledge of lter theoryis necessary.After a brief introduction of the insertion-loss method common lter prototypes are presented. An equiv-alent circuit model for edge-coupled nicrostriplines has been developed in the last section. Design for-mulas are given for translation from circuit theory lter prototypes to the edge-coupled lters. The naldesign is carried out in the second CAD-exercise.

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4.1 Insertion-loss method

The lter is treated as a two-port device and described by scattering parameters. The quanity of interestis the reection coefcient from which the power loss ratio

P LR =1

1 2=

11

(86)

(input power / power delivered to load) can be computed ( = | |). For lter synthesis P LR( ) or ( )are given as a function of frequency. Some boundary conditions must be observed:

passive circuits: 1

the normalized impedance Z = Z Z c can be described by

Z ( ) = R( ) + jX ( ) (87)

Z c is the reference impedance (design specication). R( ) is an even function, X ( ) an oddfunction of angular frequency. This requirement follows from a theorem of circuit theory that theresponse of a circuit to a real-time-dependent driving function must also be real. The equivalentcondition for the reection coefcient is ( ( ) = ( )) . The reection coefcient can bewritten as:

( ) =R( ) 1 + jX ( ) R( ) + 1 + jX ( )

. (88)

Therefore, the function 2( ) has as argument 2 .

any lowpass-imepdance function can be described by the ratio of two polynomials M , N :

2( ) =M ( 2)

M ( 2) + N ( 2)=

( R( ) 1)2 + X 2( ))

( R( ) + 1)2 + X 2( )). (89)

The power loss ratio can now be expresed as:

P LR = 1 +M ( 2) N ( 2) . (90)

So far any polynomials can be chosen for lter synthesis. However, there are some standard-polynomials,which are frequently used in microwave engineering. The two most inmportant will be described in thefollowing.

Maximally Flat lter (Butterworth lter)The Butterworth lter is also known as the maximally at lter due to the fact that it has the most atpassband response. This kind of lter is useful if signal distortions in the passband must be kept atminimum. The lter response is less steep compared with other lters. Therefore, this lter is less usefulfor ltering signals which are located closely to each other in the spectrum. The lter polynomials are:

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M ( 2) = k

2 c

2 N

,

N ( 2) = 1,

P LR = 1 + k 2 c

2 N

. (91)

N is the lter order and c is the cutoff frequency.

Chebyshev lterThe Chebyshev lter is also known is the equal-ripple lter due to the occurence of ripples in thepassband. This lter has a much sharper cutoff region separating the passband and stopband comparedwith the Butterworth lter. The power loss ratio of this lter is:

P LR = 1 + k 2T 2 N ( c

). (92)

c is the cutoff frequency.The rst Chebyshev polynomials are:

T N ( c

) = cos N cos1 c

T 0( x) = 1T 1( x) = xT 2( x) = 2 x2 1T 3( x) = 4 x3 3 xT 4( x) = 8 x4 8 x2 + 1 (93)

The maximum reection coefcient in the passband max is specied by the factor k :

max =k

1 + k 2 . (94)Therefore, k 2 is called passband tolerance. It is now possible to design a lowpass-prototype. For re-alization of highpass, bandpass or bandstop lters the lowpass prototype can be used as design basis.Afterwards a frequency transformtation is used:

= fct( ). (95)Thus, the power loss ratio becomes:

P LR = 1 + P ( 2) = 1 + P ( f 2( )) . (96)P can be obtained from (eq. 92). For lowpass to bandpass transformation following formula is used:

= f ( ) = 0

0

0

. (97)

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The bandwidth is = 2 1 and the center frequency:

0 = 2

1. (98)Systematic design of Chebyshev bandpass lters:

1. specication of cutoff frequencies, ripple attenuation and lter order. The lter order can be ob-tained by specication of x dB stopband attenuation at y GHz distance from center frequency.This can be done graphically.

2. lookup of lter coefcients (table). For specialized lter designs the lter coefcients may not befound in a table. For these, the lter coefcients must be calculated analytically.

3. the lter coefcients gn are normalized values (impedance: 1 , cutoff frequency: 1Hz) for lumpedelement lter prototypes and represent either capacitors or inductors.

4. the lter prototype is now complete.

4.2 Bandpass lter design equations

the Chebyshev bandpass lter prototype shall be realized using:

microstrip transmission lines edge-coupled, open-circuited 2 -resonators

In the last section an equivalent circuit model of edge-coupled microstriplines was presented. Now

equations must be found for the inverse problem: synthesis of edge-coupled microstriplines from thevalues of the equivalent circuit model. The design equations are given in the following:

1 = 12 0

,

P sin 1 =K 10

12 tan 1 + K 10 2,

s =1

12 tan 1 + K 10

2 . (99)

The impedance inverters can be calculated (impedances normalized to Z c):

K 10 = K N , N + 1 =1

g0g1 =1

g N g N + 1 ,K k ,k + 1 =

1gk gk + 1 . (100)

N k + 1,k = K 2k ,k + 1 + 14 tan 2 1 (101)With the above equations the even- and odd mode impedances can be computed:

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Z 1e =

Z N + 1e =

Z c(

1+

P sin 1) Z 1o = Z

N + 1o = Z c(1 P sin 1)

Z k + 1e = Z N k + 1e = Z cs( N k + 1,k + K k ,k + 1)

Z k + 1o = Z N k + 1o = Z cs( N k + 1,k K k ,k + 1). (102)

The electrical parameters of the lter prototype using coupled microstriplines are now determined. Thephysical dimensions can be obtained by use of CAD-Tools (e.g. ADS LineCalc).

4.3 CAD-Exercise: Design of Chebyshev bandpass lter

The design is carried out in ve steps:

1. specication of the lter and lookup of lter coefcients for the lter prototype

2. computation of even- and odd mode impedances using the design equations

3. simulation of the lter using ideal transmission lines and couplers

4. simulation of the lter using microstrip transmission lines and couplers

5. optimization of the lter

4.3.1 Filter specication

The lter shall have:

a center frequency of f 0 = 02 = 5.0 GHz

a bandwidth of 0.5 GHz, i.e. the cutoff frequencies are f d = d 2 = 4.75 and 5 .25 GHz, respectively

a stopband attenuation of at least 15 dB at f s = s2 = 4.5 resp. 5 .5 GHz

a ripple attenuation of no more than 0.5 dBFor lookup in lter tables the above values have to be normalized.

normalized angular frequency: = 0 ( s 0 0 s ) =required lter order (table 4.03-7): N =lter coefcients (table 4.05-2(a)): g0=

g1=g2=g3=g4=

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4.3.2 Computation of even- and odd mode impedances

The even and odd mode impedances can be calculated using the design equations (note the symmetry of the lter: Z e/ o1 = Z e/ o4 and Z e/ o2 = Z e/ o3):

1 =

K 1,0 =1

g0g1 =K 2,1 =

1g1g2 =

K 3,2 =1

g2g3 =K 4,3 =

1

g3g4=

P sin 1 =P =

N k + 1,k = K 2k + 1,k + 14 tan 2 1 = N 2,1 =

s =1

0.5tan 1 + K 210=

Z e1 = Z c(1 + P sin 1) = Z e2 = Z c s( N 21 + K 21 ) = Z o1 = Z c(1 P sin 1) = Z o2 = Z c s( N 21 K 21 ) =

4.3.3 Simulation using ideal transmission lines and couplers

Draw a circuit of ideal transmission lines and couplers in ADS (scrollbar: TLINES->IDEAL). Add S-Parameter ports and a simulation control block and simulate the lter.

4.3.4 Simulation using microstrip transmission lines and couplers

The lter shall be realized on a microwave substrate with following parameters:

r = 10 .8h = 0.635 mmtan = 0.0025metal thickness: T = 35 m

Add a substrate parameter block in ADS and enter the values. The physical parameters of the microstriptransmission lines and couplers must be calculated before a simulation can be carried out. Start theLineCalc-Tool (Menu->Tools->LineCalc). Choose the waveguide structure and enter the substrate para-meters. Now enter the electrical parameters of the coupled lines and calculate the physical dimensionsof the coupler.

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Z o. Z e Z o E [] w [mm] s [mm] P [mm] K o. K odd K evenMS(in/out) X 2 X X

CPL1CPL2OST1 X X XOST2 X X X

Draw a new circuit using microstrip elements (TLINES->MICROSTRIP). Simulate the lter (includingopen-end effect of open-circuited microstrip lines).

4.3.5 Optimization of the lter

The center frequency has shifted due to the capacitance at the end of the open-circuited microstrip lines.

The effective length can be computed by the formula of Kammestad:

le f f = 0.412 h e f f + 0.3

e f f 0.258 wh + 0.262wh + 0.813

(103)

Compute the effective transmission line lengths.

w [mm] Z K bzw. e f f le f f [mm] whMSMS

Draw a new circuit and compensate for the open-end capacitance by shortening the transmission lines.Simulate the optimized lter and compare the results with the previous simulations!

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5 Signal ow graphs

microwave n-ports can be described by scattering parameters:

b1b2

bn

= S

a 1a 2

a n

S is the scattering matrix, Si j are the scattering parameters, a 1 a n : incident waves, b1 bn :reected waves

the signal ow graph consists of nodes and directed line segments (see two-port Fig. 20):

Sa

2

b2

a1

b1

Figure 20: Microwave two-port.

b1 = S11 a 1 + S12 a 2b2 = S21 a 1 + S22 a 2

a2

b2

a1

b1

S21

S12

S11

S22

Rules for signal ow graphs a pair of linear equations [ x2 = C 21 x1 ; x3 = C 32 x2] has the graphical representation:

x 2x 1 x 3

c21

c32

x3

x1

c21

c32

Figure 21: Illustration of rule one.

If there are no other inputs at node 2 the node can be omitted. The transmission factor isequal to the product of the separate transmission factors.

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parallel paths can be combined:

x2 = A21 x1 + B21 x1 + C 21 x1= ( A21 + B21 + C 21 ) x1

The graphical representation is:

x2

x1

C21

B21

A21 x

2x

1

A + B21 21 21

+ C

Figure 22: Illustration of rule two.

combination when a node has one output and several inputs:

x4 = C 41 x1 + C 42 x2 + C 43 x3 und x5 = C 54 x4 (104)

The graphical representation is:

x 2

x3

C43

C42

C41

x1

x4

x5

C54

x '4

x2

C43

C42

C41

x1

x3

x5C

54

x '4

x '4

C54

C54

Figure 23: Illustration of rule three.

For the output signals follows:

x5 = C 54 x4 = C 54C 41 x1 + C 54C 42 x2 + C 54C 43 x3

combination when a node has one input and several outputs:

x 1x

3

C 41

C31

C21

x2

x4

x 0

C10

x '1

x 3

C41

C31

C 21x

2

x 4

x 0C

10

x '1

x '1

C10

C10

Figure 24: Illustration of rule four.

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feedback loops should be eliminated to obtain more simple graphs. A feedback loop is de-scribed by the equations:

x2 = c21 x1 + C 23 x3 x3 = C 32 x2 x4 = C 43 x3

First node 3 is duplicated:

x 3x 2

C32

C23

C43C 21

x 1 x 4

x2

C32

C23

C43C 21

x1

x4

C32

x '3

Figure 25: Illustration of rule ve.

The new equations are:

x2 = C 21 x1 + C 23C 32 x2 x2 =C 21

1 C 23C 32 x1

x3 = C 32 x2 ; x4 = C 43 x3

The equations are the same except for node x3. The last reduction yields:

x4

x1

C21

x2

C32 43

C

1-C C23 32

Figure 26: Illustration of rule ve.

Signal ow graphs of common circuit elements:

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a1

b1

1

1

r L

Figure 27: Termination.

Figure 28: Matched transmission line.

a) termination: b1 = La 1b) matched transmission line (no losses): b1 = e j la 2 ; b2 = e j la 1c) generator:

g =a 1b1

=Z g Z 0 Z g + Z 0

Zg

V

a 1

b 1

Gg

V 1

1

Figure 29: Generator.

d) detector:

M =Z

M Z

0 Z M + Z 0

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Z M

a 1

b 1

GM

1

1

M

M

Figure 30: Detector.

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References 30

References

[1] H.G. Unger, Elektromagnetische Wellen auf Leitungen, 3. Auage, Huethig Buch Verlag, Heidel-berg 1991.

[2] B.C. Wadell, Transmission Line Design Handbook, Artech House, Boston, 1991.

[3] R.K. Hofmann, Integrierte Mikrowellenschaltungen, Springer Verlag, Berlin, 1983.

[4] J. Helszajn, Microwave Planar Passive Circuits and Filters, John Wiley & Sons, Chichester, 1994.

[5] R.E. Collin, Foundations for Microwave Engineering, 2nd Edition, McGraw Hill, 1992, 1966.

[6] G.D. Vendelin, A.M. Pavio, U.L. Rohde, Microwave Circuit Design Using Linear and NonlinearTechniques, John Wiley & Sons, New York, 1990.

[7] I.Bahl, P.Bhartia, Microwave Solid State Circuit Design, John Wiley & Sons, 1988.

[8] G.L. Matthaei, L. Young, E.M.T. Jones, Microwave lters, impedance-matching networks, andcoupling structures, McGraw Hill, 1964.

[9] M.W. Medley, Microwave and RF Circuits: Analysis, Synthesis and Design, Artech House, 1993.

[10] K. C. Gupta, R. Garg, I. Bahl, P. Bhartia, Microstrip Lines and Slotlines, Artech House, 1996.

[11] W. Baechtold, Mikrowellentechnik: Kompakte Grundlagen fuer das Studium, Vieweg Verlag, 1999.

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