Lesson 1:Introduction and Backgrounds

on Microwave Circuits

Giuseppe MacchiarellaPolytechnic of Milan, Italy

Electronic and Information Department

“A microwave filter is a 2-port junction exhibiting a selective frequency behavior in the transmission from the input port to the output port”

Passband(s): Frequency band(s) which is transferred from input to output without attenuation (ideally) Stopband(s): Frequency band(s) where the transmission is blocked (signal is reflected at input port)

A very general definition

Definition of transmission

Inputport

Outputport

Reference sections

Zc Zc

Physical StructureEquivalent

Representation

12-port

network

2

Zc Zc

Both the real structure and the equivalent circuit are characterized through the Scattering Parameters (Smatrix) defined with respect to Zc.

S11, S22 = Reflection at the portsS21=S12=Transmission (<1 for lossless junctions)

Attenuation (dB) = -20 log10(|S21|) >0 for lossless junctions

Basic Classification

Filter are classified according to the number and location of passbands and stopbands. The most basic classification considers 4 filters classes:

Low-pass, High-pass, Band-pass, Band-stop

Passband

Stopband

Low-pass High-pass Band-pass Band-stop

Stopband Stopband Stopband Stopband

Passband Passband Passband Passband

Att

enua

tion

Frequency

Specification of filter requirements: the Attenuation Mask

StopbandStopband

Passband

fp1 fp1

Ap

As2

As1

As3

As4

fs2fs1 fs3 fs4

A(dB)

Freq.

Attenuation and matching

Ideally, a filter is a lossless network. In this case the attenuation in the passband is produced only by the not perfect match at input/output ports. In the real world the filter components introduce dissipation, which becomes the main contribution to passband attenuation.

Passband Losses affect very few (in general) the port matching

In addition to the attenuation mask (which refer to overall attenuation) must be then specified also

the matching requirements at the ports

Effect produced by losses on attenuation shape

1840 1850 1860 1870 1880 1890 1900 1910 1920-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

Frequency (MHz)

dBS11-S12 Magnitude (dB)

1800 1820 1840 1860 1880 1900 1920 1940 1960 1980 2000-140

-120

-100

-80

-60

-40

-20

0

Frequency (MHz)

dBS11-S12 Magnitude (dB)

Q0=3000

Requirements on transmission phase

To have a distortion-less two port, the phase of transmission parameter should be linear in the passband

Often, the requirement is given on the group delay, which is defined as the derivative of transmission phase with respect the radian frequency. The group delay should be ideally constant in the passband

Phase linearity is generally assessed a posteriori, after the attenuation requirements are satisfied.

If the requirements on phase linearity is not verified, a phase equalizer may be required.

In alternative, complex transmission zeros can be introduced in the response for phase equalization

Approaches to the design of microwave filters

Image parameters methods (old technique based on attenuation produced by basic blocks, which are suitably interconnected).

Synthesis of networks composed by commensurate transmission line sections and stubs (suited for broadband filters; poses strong bounds on the configuration of the filter structure)

Equivalent Synthesis method, based on the equivalence between the real (distributed components) structure and a lumped component filter network (suitable for narrow and moderate bandwidth passband filters. The most used technique today

Advantages of the Equivalent Synthesis Method

Precise control of the curve representing the filter response

The synthesis is developed in the lumped-element world, where well established, analytical and numerical procedures are available

The results of the synthesis can be expressed by means of universal parameters which maintain the same meaning both for lumped-element circuit and distributed microwave networks

A first-order dimensioning of the physical structure can be easily performed using these universal parameters

Recalls on Microwave Circuits for Filter Design

General physical structure of a microwave filter

CAVITY 4

CAVITY 5

CAVITY 1

CAVITY 2

CAVITY 3

k14

k12

k45

k42 k25

k23

k53

IN OUT

Basic components:- Cavities (resonating on a specific mode)- Coupling structures (represented as )

Not always cavities and couplings can be identifies separately

Basic equivalent circuit of the cavity

LeqCeqReq

LeqCeq

Geq

Series or parallel resonator

Characteristic parameters:- Mode Resonant frequency (f0)- Equivalent slope parameter (Leq, Ceq)- Loss parameter (Req, Geq)

Basic equivalent circuit of the coupling

JYL

2

inL

JYY

KZL

2

inL

KZZ

Couplings are generally modeled with impedance or admittance inverters

The inverter has two basic functions:- Operate as impedance transformer- Change the nature of the load

The inverter is an ideal component which can be approximated by real elements in a limited frequency band

S parameters of the Impedance Inverter

The impedance inverter is a symmetrical and reciprocal 2-port network. It behaves as an ideal /2 phase shifter.Imposing the lossless condition and the K value it is not sufficient to identify univocally the network (the ± sign of 12 remains undetermined)

2

0 2 20 0

11 11 2 2 20

00

11 12 12

Real (positive or negative)

2 2

K ZZ K ZS SK K ZZZ

Equivalent circuit for the impedance inverter

1

2

tan 2

1 2tan2 2

1

C

C

C

C C

K Z

XZ

K ZXZ K Z

Zc

/4

K=Zc

-jX -jX

jX

K=X

-jB -jB

jB

J=1/K=B

jXZC ZC

jBYC YC

1

2

tan 2

1 2tan2 2

1

C

C

C

C C

J Y

BY

J YBY J Y

Modeling of a microwave junction with a lumpedequivalent circuit (1-port)

Reference Section

Microwave Junction

ZJ=RJ()+jXJ()≈ ZJ=RJ()+jXJ()

Goal: model of the junction in a defined frequency range B around f0 (B<<f0)

The model is represented by an equivalent lumped-element impedance Zeq= Req+ Xeq obtained by imposing:

• Constant real part: Req=RJ(0)• Same value at 0 of the imaginary part: Xeq(0)=XJ(0)• Same value of derivative at 0: Xeq(0)/ = XJ(0) /

Equivalent impedance Zeq

The most elementary network exhibiting Zeq is a series resonator: Leq

Zeq

Ceq

Req=RJ(0)

1eq eq eq

eq

Z R j LC

Imposing the previous conditions:

00

0 0 00

20

1

1

eq eq Jeq

eq Jeq

eq

X L XC

X XL

C

0 0

0 02

0 0 0

1 1 1, 2 2

J J J Jeq

eq

X X X XL

C

Equations for the Equivalent Admittance

Yeq

1eq eq eq

eq

Y G j CL

Imposing the previous conditions:

00

0 0 00

20

1

1

eq eq Jeq

eq Jeq

eq

B C BL

B BC

L

0 0

0 02

0 0 0

1 1 1, 2 2

J J J Jeq

eq

B B B BC

L

LeqCeq

Geq=GJ(0)

Remark on the equivalent network

The equivalence between the microwave junction and the lumped-element equivalent circuit is exact only at f0

The deviation between ZJ (YJ) and Zeq (Yeq) becomes larger and larger with the increase of B

f0=1000 MHzX(f0)=188.4

800 900 1000 1100 1200Frequency (MHz)

0

100

200

300

400

Im(Z(1,1))Junction

Im(Z(1,1))Equivalent Circuit

XJ

Xeq

Special case: Resonant Junction (Cavity)

In case of resonant junctions (XJ(0)=0, BJ (0)=0), the previous equations become:

0

0

20

20

1 1 2

1 1 2

Jeq eq

eq

Jeq eq

eq

XL C

L

BC L

C

These equations define an equivalent circuit for a large class of cavity resonators. The type of resonator depends on the resonant modeand on the reference section

Example: TEM cavity realized with short-circuited/open-circuited transmission line

open Zc

L

short Zc

L

LeqCeq

Leq

Ceq

LeqCeq

Leq

Ceq

Leq

Ceq

LeqCeq

L=0/2

L=0/4

L=0/2

L=0/4

02C

eqZ

L

04C

eqY

C

04C

eqZ

L

02C

eqY

C

Capacity-loaded coaxial resonator

z

L

Tuning screw

short Yc

L<0/4

CsReference section

Reference section cottot s CB C Y L

Resonance condition: 0 0 00 cottot s CB C Y L

Equivalent capacitance:

0

00 0 2

0

00 0

0

1 1 cot2 2 sin

21 cot 1 12 sin 2 2 2

toteq C

CC eq

B LC Y L

L

L YY L C

L

L=

Losses in the cavity: the unloaded Q

0 0Energy stored in the junction

Power dissipated in the junctionQ Cavity

LeqCeq

Req=Rp(0)

Zeq

00

eq

p

LQ

R

RJ is the sum of two terms: Rp, due to thefinite conductivity of metallic walls and Re to to medium dissipation (dielectric losses). Is has then:

0 0 0

1 1 1

JQ Q Q

p JR R R

J JYC YC

LeqCeq

gp

LeqCeq rp

K KZC ZC

00

eq

p

LQ

r

0 202

3

for 2

eqL p C

dBC

L fQ r K Z

BK Z

Modeling of a cavity coupled to loads

00

eq

p

CQ

g

0 202

3

for eqL p C

dBC

C fQ g J Y

BJ Y

Parameters of loaded cavities

The 3 dB bandwidth (B3dB=f0/QL). For a given cavity (Leqor Ceq), a value for K (J) can be evaluated for obtaining the desired B3dB:

The transducer attenuation between input and output at the resonant frequency

The matching at the input (output) port. For small losses, it can be shown that the bandwidth B for a given value of the input reflection coefficient is related to B3dB as follows:

3dBB B

0 0, 2 2

C eq C eq

L L

Z L Y CK J

Q Q

The loaded Q (QL): determines

Power transmitted to load (transducer attenuation)

Cavity Output PowerInput Power

Dissipated Power

Input Power=

Available Power

LeqCeq rp

K K

ZC

ZC

LeqCeq rp

K K

ZC

ZC

Output Power=

Power on ZcDissipated

Power=

Power on rp

1

0

210log 1 L

dBQ

AQ

Coupling of two cavities: equivalent circuit

LeqCeq

K

Leq Ceq

Characteristic parameter: the coupling coefficient

0 eq

KkL

k determines the coupling bandwidth of the two resonators (the larger is k the larger is the bandwidth)

0

for shunt resonatorseq

JC

Evaluation of k from resonances

Even and odd resonances of two coupled resonators:

LeqCeq

K

Leq Ceq LeqCeq

K

Leq Ceq

LeqCeqLeq Ceq -jK -jK

jK

LeqCeqLeq Ceq -jK -jK

jK

LeqCeq

±jK

Short/Open

2

00 0

12 2e

eq eq

K Kf fL L

2

00 0

12 2o

eq eq

K Kf fL L

Short-jKOpen+jK

00

, e oe o

f ff f f kf

OddResonance(-jK)

EvenResonance(+jK)

Practical evaluation of k

LeqCeq

K

Leq Ceq

R0 R0

0 0eqL R Input/output coupling ≈0

1900 1950 2000 2050 2100Frequency (MHz)

-70

-60

-50

-40

-30

-20

-10

0

2025.2 MHz-24 dB

1975.2 MHz-32.3 dB

Odd Resonance Even Resonance

Xeq=10, R0=0.01fe=2025.2 MHzfo=1975.2

0

0

2000

0.025

e o

e o

f f f MHzf fk

f

Coupled cavities: example 1

InductiveIris (XL)

K Zc ZcjX

12 , tan 2

1C

KX XKZ

Zc Zc

/2

K

/2Cavity 1 Cavity 2L L

example 1 (cont.)

Zc ZcjXL

Zc Zc

INVERTER

1= tan 22 LL X 2 (for )

1L C

C

KX K K ZKZ

Coupling coefficient:

2

002g

eq CL Z

2

0 0

2 gL

eq C

XKkL Z

Equiv. reactance:

jXLZc Zc

L L

Resonators with coupled lines: Interdigital

K

Zm Zm

open

open

Zeven, Zodd

IN

OUT

L=0/4

12

1 12 sin

m even odd

even odd

Z Z Z

K Z ZL

0

4eq mL Z

0

4 4even odd

eq even odd

Z ZKk mL Z Z

short

short

Yeven, Yodd

IN

OUT

L=0/8CS

CS

JYm

CS

Ym

CS

12

1 12 tan

m even odd

even odd

Y Y Y

J Y YL

0

0

1 , 2 2

2 22 2

meq

odd even

eq even odd

YC

Y YJ mkC Y Y

Note: odd even even odd

even odd even odd

Y Y Z Zm

Y Y Z Z

Resonators with coupled lines: Comb