lect1

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1

ECE 637 : MIC and Monolithic

Microwave Integrated Circuits

(MMIC)

By Dr. Ghanshyam Kumar Singh

ECE Department, SET

Sharda University

Objectives

� To introduce the design methodology of

Microstrip Lines.

� To acquire knowledge of various typs of

dielectric materials and their properties.

� To describe the frequency properties of

MMIC.

� To explain the fundamentals of various

types of MMIC devices.

Course OutcomesAfter completing this course students will

be able

� design Microstrip lines.

� analyse different types of dielectric

materials and their effects.

� apply optimization technique for

frequency behaviour of MMIC.

� solve the problems related to MMIC Designing.

� analyse various types of MMIC devices.

Lecture 1� TEM, TE and TM Waves

� Coaxial Cable

� Grounded Dielectric Slab Waveguides

� Striplines and Microstrip Line

� Design Formulas of Microstrip Line

Lecture 1

� An Approximate Electrostatic

Solution for Microstrip Line

� The Transverse Resonance

Techniques

� Wave Velocities and Dispersion

TEM, TE and EM Waves� transmission lines and waveguides

are primarily used to distribute microwave wave power from one

point to another

� each of these structures is

characterized by a propagation

constant and a characteristic impedance; if the line is lossy,

attenuation is also needed

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2

TEM, TE and EM Waves� structures that have more than one

conductor may support TEM waves

� let us consider the a transmission

line or a waveguide with its cross section being uniform along the z-

direction

b

aρ

TEM, TE and EM Waves

� the electric and magnetic fields can

be written as

� Where and are the transverse components and and are the

longitudinal components

E x y z e x y ze x y e

H x y z h x y zh x y e

t zj z

t zj z

( , , ) $ ( , ) $ ( , )] ( )

( , , ) $ ( , ) $ ( , )] ( )

==== ++++ −−−− −−−−

==== ++++ −−−− −−−−

−−−−

−−−−

[e

[e

t

t

ββββ

ββββ

1

2

et ht

et ht


� in a source free region, Maxwell’s

equations can be written as

� Therefore,

∇∇∇∇ ×××× ==== −−−− ∇∇∇∇ ×××× ====E j H H j Eωµωµωµωµ ωεωεωεωε,

∂∂∂∂

∂∂∂∂ββββ ωµωµωµωµ

ββββ∂∂∂∂

∂∂∂∂ωµωµωµωµ

∂∂∂∂

∂∂∂∂

∂∂∂∂


E

yj E j H

j EE

xj H

E

x

E

yj H

zy x

xz

y

y xz

++++ ==== −−−−

−−−− −−−− ==== −−−−

−−−− ==== −−−−

, ( )

, ( )

, ( )

3

4

5

∂∂∂∂

∂∂∂∂ββββ ωεωεωεωε


∂∂∂∂ωεωεωεωε

∂∂∂∂

∂∂∂∂

∂∂∂∂


H

yj H j E

j HH

xj E

H

x

H

yj E

zy x

xz

y

y xz

++++ ====

−−−− −−−− ====

−−−− ====

, ( )

, ( )

, ( )

6

7

8


� each of the four transverse

components can be written in terms

of and , e.g., consider Eqs. (3) and (7):

∂∂∂∂


E

yj E j Hz

y x++++ ==== −−−−

−−−− −−−− ====j HH

xj Ex

zyββββ

∂∂∂∂


TEM, TE and EM Waves� each of the four transverse

components can be written in terms of and , e.g., consider Eqs. (3) and

(7):∂∂∂∂


E

yj E j Hz

y x++++ ==== −−−−

−−−− −−−− ====j HH

xj Ex

zyββββ

∂∂∂∂



−−−− −−−− ==== −−−− −−−−

−−−− ==== −−−−

==== −−−− −−−− −−−− −−−− −−−− −−−− −−−− −−−−

==== −−−−

j HH

xj j H

E

yj

k H jE

yj

H

x

Hj

k

E

y

H

x

k k

xz

xz

xz z

x

c

z z

c


∂∂∂∂ωεωεωεωε ωµωµωµωµ

∂∂∂∂

∂∂∂∂ββββ

ββββ ωεωεωεωε∂∂∂∂


∂∂∂∂

∂∂∂∂

ωεωεωεωε∂∂∂∂


∂∂∂∂

∂∂∂∂

ββββ

( ) / ( )

( )

( ) ( )

( )

2 2

2

2 2 2

9

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� Similarly, we have

� is called the cutoff wavenumber

Hj

k

E

x

H

y

Ej

k

E

x

H

y

Ej

k

E

y

H

x

y

c

z z

x

c

z z

y

c

z z

====−−−−

++++ −−−− −−−− −−−− −−−−

====−−−−

++++ −−−− −−−− −−−− −−−−

==== −−−− ++++ −−−− −−−− −−−−

2

2

2

10

11

12

( ) ( )

( ) ( )

( ) ( )

ωεωεωεωε∂∂∂∂


∂∂∂∂

∂∂∂∂



∂∂∂∂

∂∂∂∂



∂∂∂∂

∂∂∂∂

kc


� Transverse electromagnetic (TEM)

wave implies that both and are zero (TM, transverse magnetic,

=0, 0 ; TE, transverse electric,

=0, 0)

the transverse components are also zero unless is also zero, i.e.,

Ez Hz

Hz Ez ≠≠≠≠

Ez Hz ≠≠≠≠

kc

k2 2 2==== ====ωωωω µεµεµεµε ββββ


� now let us consider the Helmholtz’s

equation

� note that and therefore, for TEM wave, we have

( ) ,∇∇∇∇ ++++ ==== ++++ ++++ ++++

====2 2

2

2

2

2

2

2

20 0k Ex y z

k Ex x∂∂∂∂

∂∂∂∂

∂∂∂∂

∂∂∂∂

∂∂∂∂

∂∂∂∂

2

2

2

zββββ−−−−→→→→

∂∂∂∂

∂∂∂∂

∂∂∂∂

∂∂∂∂

∂∂∂∂

∂∂∂∂

2

2

2

20

x yEx++++

====


� this is also true for , therefore,

the transverse components of the electric field (so as the magnetic

field) satisfy the two-dimensional

Laplace’s equation

Ey

∇∇∇∇ ==== −−−− −−−− −−−− −−−−t te2 0 13( )


� Knowing that and

, and we have

� while the current flowing on a

conductor is given by

∇∇∇∇ ====t te2 0

∇∇∇∇ •••• ==== •••• ====D et tε∇ε∇ε∇ε∇ 0

∇∇∇∇ ====t x y2

0ΦΦΦΦ( , )

V E dl12 1 2 12 14==== −−−− ==== −−−− •••• −−−− −−−− −−−− −−−−∫∫∫∫ΦΦΦΦ ΦΦΦΦ ( )

I H dl

C

==== ••••∫∫∫∫ −−−− −−−− −−−− −−−−( )15


� this is also true for , therefore, the

transverse components of the electric field (so as the magnetic

field) satisfy the two-dimensional

Laplace’s equation

Ey

∇∇∇∇ ==== −−−− −−−− −−−− −−−−t te2 0 13( )

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� Knowing that and

, we have

� the voltage between two conductors is given by

� while the current flowing on a conductor is given by

∇∇∇∇ ====t te2 0 ∇∇∇∇ •••• ==== •••• ====D et tε∇ε∇ε∇ε∇ 0

∇∇∇∇ ====t x y2 0ΦΦΦΦ( , )

V E dl12 1 2 12 14==== −−−− ==== −−−− •••• −−−− −−−− −−−− −−−−∫∫∫∫ΦΦΦΦ ΦΦΦΦ ( )

I H dl

C

==== ••••∫∫∫∫ −−−− −−−− −−−− −−−−( )15


� we can define the wave impedance for theTEM mode:

� i.e., the ratio of the electric field to themagnetic field, note that the components

must be chosen such that E x H is pointingto the direction of propagation

ZE

HTEM

x

y

==== ==== ==== ==== −−−− −−−− −−−−ωµωµωµωµ

ββββ

µµµµ

εεεεηηηη ( )16


� for TEM field, the E and H are related

by

h x yZ

z e x yTEM

( , ) $ ( , ) ( )==== ×××× −−−− −−−− −−−− −−−−1

17

why is TEM mode desirable?

� cutoff frequency is zero

� no dispersion, signals of differentfrequencies travel at the same

speed, no distortion of signals

� solution to Laplace’s equation is

relatively easy


� a closed conductor cannot support

TEM wave as the static potential is either a constant or zero leading to

� if a waveguide has more than 1

dielectric, TEM mode cannot exists as cannot be zero in all regions

et ==== 0

k kci ri==== −−−−( )/εεεε ββββ2 2 1 2


� sometime we deliberately want to have a cutoff frequency so that a

microwave filter can be designed

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TEM Mode in Coaxial Line

� a coaxial line is shown here:

� the inner conductor is at a potentialof Vo volts and the outer conductoris at zero volts

b

aρ

V=0V=Vo


� the electric field can be derived from

the scalar potential ΦΦΦΦ; in cylindricalcoordinates, the Laplace’s equation

reads:

� the boundary conditions are:

�

1 1

ρρρρ

∂∂∂∂

∂ρ∂ρ∂ρ∂ρρρρρ

∂Φ∂Φ∂Φ∂Φ

∂ρ∂ρ∂ρ∂ρ ρρρρ

∂∂∂∂

∂φ∂φ∂φ∂φ

++++ ==== −−−− −−−− −−−− −−−−

2

2

20 18

ΦΦΦΦ( )

ΦΦΦΦ ΦΦΦΦ( , ) ( ), ( , ) ( )a V boφφφφ φφφφ==== −−−− −−−− −−−− ==== −−−− −−−− −−−− −−−− −−−−19 0 20


� use the method of separation of variables, welet

� substitute Eq. (21) to (18), we have

� note that the first term on the left only

depends on ρ while the second term onlydepends on φ

ΦΦΦΦ( , ) ( ) ( ) ( )ρρρρ φφφφ ρρρρ φφφφ==== −−−− −−−− −−−− −−−−R P 21

)22(02

P21RPR

−−−−−−−−−−−−−−−−====∂φ∂φ∂φ∂φ

∂∂∂∂++++

∂ρ∂ρ∂ρ∂ρ∂∂∂∂ρρρρ



� if we change either ρ or φ, the RHS

should remain zero; therefore, eachterm should be equal to a constant

)25(02k2k),24(2k2

P2

1

)23(2kR

P

R

−−−−−−−−−−−−−−−−====φφφφ

++++ρρρρ−−−−−−−−−−−−−−−−φφφφ

−−−−====∂φ∂φ∂φ∂φ

∂∂∂∂

−−−−−−−−−−−−ρρρρ−−−−====∂ρ∂ρ∂ρ∂ρ∂∂∂∂ρρρρ



� now we can solve Eqs. (23) and (24) inwhich only 1 variable is involved, the final

solution to Eq. (18) will be the product ofthe solutions to Eqs. (23) and (24)

� the general solution to Eq. (24) is

P A k B k( ) cos( ) sin( )φφφφ φφφφ φφφφφφφφ φφφφ==== ++++


� boundary conditions (19) and (20)

dictates that the potential isindependent of φ, therefore must be

equal to zero and so as

� Eq. (23) is reduced to solving

kρρρρ

kφφφφ

∂∂∂∂

∂ρ∂ρ∂ρ∂ρρρρρ

∂∂∂∂

∂ρ∂ρ∂ρ∂ρ

R

==== 0

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� the solution for R(ρ) now reads

R C D

A B

( ) ln

( , ) ln

ρρρρ ρρρρ

ρρρρ φφφφ ρρρρ

==== ++++

==== ++++ΦΦΦΦ

ΦΦΦΦ

ΦΦΦΦ

ΦΦΦΦ

( , ) ln

( , ) ln ln

/ ln( / )

( , )ln( / )

(ln ln ) ( )

a V A a B

b A b B B A b

A V b a

V

b ab

o

o

o

φφφφ

φφφφ

ρρρρ φφφφ ρρρρ

==== ==== ++++

==== ==== ++++ ⇒⇒⇒⇒ ==== −−−−

====

==== −−−− −−−− −−−− −−−−

0

26


� the electric field now reads

� adding the propagation constant back,

we have

eV

b at t

o( , ) ( , ) $ $ $ln /

ρρρρ φφφφ ρρρρ φφφφ ρρρρ∂Φ∂Φ∂Φ∂Φ

∂ρ∂ρ∂ρ∂ρφφφφ

ρρρρ

∂Φ∂Φ∂Φ∂Φ

∂φ∂φ∂φ∂φρρρρ

ρρρρ==== −∇−∇−∇−∇ ==== −−−− ++++

====ΦΦΦΦ

1

E e eV e

b at

j z oj z

( , ) ( , ) $

ln /( )ρρρρ φφφφ ρρρρ φφφφ ρρρρ

ρρρρ

ββββββββ

==== ==== −−−− −−−− −−−−−−−−−−−−

27


� the magnetic field for the TEM mode

� the potential between the two

conductors are

HV e

b a

oj z

( , ) $

ln /( )ρρρρ φφφφ ρρρρ

ρηρηρηρη

ββββ==== −−−− −−−− −−−−

−−−−28

)29(eVd),(EV zjo

b

aab −−−−−−−−−−−−====ρρρρφφφφρρρρ∫∫∫∫==== ββββ−−−−

ρρρρ


� the total current on the inner conductor

is

� the surface current density on the outer

conductor is

I H adb a

V ea oj z==== ∫∫∫∫ ==== −−−− −−−− −−−−−−−−

φφφφ

ππππββββρρρρ φφφφ ρρρρ

ππππ

ηηηη0

2 230( , )

ln( / )( )

J H bz

b b aV es o

j z==== −−−− ×××× ====−−−− −−−−

$ ( , )$

ln( / )ρρρρ φφφφ

ηηηη

ββββ


� the total current on the outer conductor

is

� the characteristic impedance can becalculated as

I J bdb a

V e Ib sz oj z

a==== ∫∫∫∫ ====−−−−

==== −−−−−−−−φφφφππππ

ηηηη

ππππββββ

0

2 2

ln( / )

ZV

I

b ao

o

a

==== ==== −−−− −−−− −−−−ηηηη

ππππ

ln( / )( )

231


� higher-order modes exist in coaxial line but isusually suppressed

� the dimension of the coaxial line is controlled sothat these higher-order modes are cutoff

� the dominate higher-order mode is mode,the cutoff wavenumber can only be obtained bysolving a transcendental equation, the

approximation is often used inpractice

TE11

k a bc ==== ++++2 / ( )

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Surface Waves on a

Grounded Dielectric Slab

� a grounded dielectric slab will generatesurface waves when excited

� this surface wave can propagate a longdistance along the air-dielectric interface

� it decays exponentially in the air region whenmove away from the air-dielectric interface

Surface Waves on a

Grounded Dielectric Slab� while it does not support a TEM mode, it excites

at least 1 TM mode

� assume no variation in the y-direction whichimplies that

� write equation for the field in each of the tworegions

� match tangential fields across the interface

x

zε

r

d

∂∂∂∂ ∂∂∂∂/ y →→→→ 0

Surface Waves on a


� for TM modes, from Helmholtz’s equationwe have

� which reduces to

∂∂∂∂

∂∂∂∂

∂∂∂∂

∂∂∂∂

∂∂∂∂

∂∂∂∂

2

2

2

2

2

2

20

x y zk Ez++++ ++++ ++++

====

∂∂∂∂

∂∂∂∂εεεε ββββ

∂∂∂∂


2

2

2 2

2

2

2 2

0 0

0

xk E x d

xk E d x

r o z

o z

++++ −−−−

==== ≤≤≤≤ ≤≤≤≤

++++ −−−−

==== ≤≤≤≤ ≤≤≤≤ ∞∞∞∞

,

,

Surface Waves on a


� Define

(((( )))) (((( ))))k k h kc r o o2 2 2 2 2 2==== −−−− ==== −−−− −−−−εεεε ββββ ββββ,

∂∂∂∂

∂∂∂∂

∂∂∂∂

∂∂∂∂

2

2

2

2

2

2

0 0 32

0 33

xk E x d

xh E d x

c z

z

++++

==== ≤≤≤≤ ≤≤≤≤ −−−− −−−− −−−−

−−−−

==== ≤≤≤≤ ≤≤≤≤ ∞∞∞∞ −−−− −−−− −−−−

, ( )

, ( )

Surface Waves on a


� the general solutions to Eqs. (32) and

(33) are

� the boundary conditions are

� tangential E are zero at x = 0 and x

� tangential E and H are continuous at x = d

e x y A k x B k x x d

e x y Ce De d x

z c c

zhx hx

( , ) sin cos ,

( , ) ,

==== ++++ ≤≤≤≤ ≤≤≤≤

==== ++++ ≤≤≤≤ ≤≤≤≤ ∞∞∞∞−−−−

0

→→→→ ∞∞∞∞

Surface Waves on a


� tangential E at x=0 implies B =0

� tangential E = 0 when x implies

C = 0

� continuity of tangential E implies

� tangential H can be obtained from Eq.

(10) with

→→→→ ∞∞∞∞

A k d Dechd

sin ( )==== −−−− −−−− −−−−−−−− 34

Hz ==== 0

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Surface Waves on a


� tangential E at x=0 implies B =0

� tangential E = 0 when x impliesC = 0

� continuity of tangential E implies

→→→→ ∞∞∞∞

A k d Dechd

sin ( )==== −−−− −−−− −−−−−−−− 34

Surface Waves on a


� continuity of tangential H implies

� taking the ratio of Eq. (34) to Eq. (35)

we have

εεεεr

cc

hd

kA k d

h

hDecos ( )====

−−−−

−−−−−−−− −−−− −−−−−−−−

235

k k d hc c rtan ( )==== −−−− −−−− −−−−εεεε 36

Surface Waves on a


� note that

� lead to

� Eqs. (36) and (37) must be satisfied

simultaneously, they can be solved for bynumerical method or by graphical method

(((( )))) (((( ))))k k h kc r o o2 2 2 2 2 2==== −−−− ==== −−−− −−−−εεεε ββββ ββββ,

k h kc r o2 2 21 37++++ ==== −−−− −−−− −−−− −−−−( ) ( )εεεε

Surface Waves on a


� to use the graphical method, it is more

convenient to rewrite Eqs. (36) and (37)into the following forms:

k d k d hdc c rtan ( )==== −−−− −−−− −−−−εεεε 38

( ) ( ) ( )( ) ( )k d hd k d rc r o2 2 2 21 39++++ ==== −−−− ==== −−−− −−−− −−−−εεεε

Surface Waves on a


� Eq. (39) is an equation of a circle with a

radius of , each interceptionpoint between these two curves yields a

solution

( )εεεεr ok d−−−− 1

π/2 π kcd

hd

r

Eq.(38)

Eq. (39)

Surface Waves on a

Grounded Dielectric Slab� note that there is always one intersection

point, i.e., at least one TM mode

� the number of modes depends on the radius rwhich in turn depends on the d and

� h has been chosen a positive real number,we can also assume that is positive

� the next TM will not be excited unless

εεεεr ok,

kc

r k dr o==== −−−− ====( )εεεε ππππ1

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Surface Waves on a


� In general, mode is excited if

� the cutoff frequency is defined as

TMn

r k d nr o==== −−−− ≥≥≥≥( )εεεε ππππ1

( )( / )εεεε ππππ ππππr cf c d n−−−− ====1 2 →→→→f

nc

dnc

r

====−−−−

====2 1

0 1 2εεεε

, , , , . . .--- (40)

Surface Waves on a


� once and h are found, the TM field

components can be written as forkc

0 ≤≤≤≤ ≤≤≤≤x d

E A k xe

Ej

kA k xe

Hj

kA k xe

z cj z

xc

cj z

yo r

cc

j z

==== −−−− −−−− −−−− −−−−

====−−−−

−−−− −−−− −−−− −−−−

====−−−−

−−−− −−−− −−−− −−−−

−−−−

−−−−

−−−−

sin ( )

cos ( )

cos ( )

ββββ

ββββ

ββββ

ββββ

ωεωεωεωε εεεε

41

42

43

Surface Waves on a


� For

� similar equations can be derived for TE

fields

d x≤≤≤≤ ≤≤≤≤ ∞∞∞∞

E A k de e

Ej

hA k de e

Hj

hA k de e

z ch x d j z

x ch x d j z

yo

ch x d j z

==== −−−− −−−− −−−− −−−−

====−−−−

−−−− −−−− −−−−

====−−−−

−−−− −−−− −−−− −−−−

−−−− −−−− −−−−

−−−− −−−− −−−−

−−−− −−−− −−−−

sin ( )

sin ( )

sin ( )

( )

( )

( )

ββββ

ββββ

ββββ

ββββ

ωεωεωεωε

44

45

46

Striplines and Microstrip Lines

� various planar transmission line

structures are shown here:

stripline slot line

microstrip coplanar

line line


� the strip line was developed from the

square coaxial

coaxial square coaxial

rectangular line flat stripline


� since the stripline has only 1 dielectric, it supports TEM wave, however, it is difficult to

integrate with other discrete elements and excitations

� microstrip line is one of the most popular types of planar transmission line, it can be fabricated by photolithographic techniques

and is easily integrated with other circuit elements

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10


� the following diagrams depicts the

evolution of microstrip transmission line

+

-

two-wire line

+

-

single-wire above

ground (with image)

+

-

microstrip in air

(with image)

microstrip with

grounded slab


� a microstrip line suspended in air can support TEM wave

� a microstrip line printed on a grounded slab does not support TEM wave

� the exact fields constitute a hybrid TM-TE wave

� when the dielectric slab become very thin (electrically), most of the electric fields are trapped under the microstrip line and the fields are essentially the same as those of the static case, the fields are quasi-static


� one can define an effective dielectric constant so that the phase velocity and the

propagation constant can be defined as

� the effective dielectric constant is bounded by

, it also depends on the slab thickness d and conductor width, W

vc

pe

==== −−−− −−−− −−−− −−−−εεεε

( )47

ββββ εεεε==== −−−− −−−− −−−−ko e ( )48

1 <<<< <<<<εεεε εεεεe r

Design Formulas of Microstrip Lines

� design formulas have been derived for microstrip lines

� these formulas yield approximate values which are accurate enough for most

applications

� they are obtained from analytical expressions for similar structures that are solvable exactly

and are modified accordingly


� or they are obtained by curve fitting

numerical data

� the effective dielectric constant of a

microstrip line is given by

εεεεεεεε εεεε

rr r

d W====

++++++++

−−−−

++++−−−− −−−− −−−− −−−−

1

2

1

2

1

1 1249

/( )


� the characteristic impedance is given by

� for W/d 1

� For W/d 1

≤≤≤≤

Zd

W

W

do

r

==== ++++

−−−− −−−− −−−− −−−−

60 8

450

εεεεln ( )

≥≥≥≥

[[[[ ]]]]Z

W d W do

r

====++++ ++++ ++++

−−−− −−−−120

1 393 0 667 1 44451

ππππ

εεεε / . . ln( / . )( )

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11


� for a given characteristic impedance

and dielectric constant , the W/d

ratio can be found as

for W/d<2

Zo

εεεεr

W de

e

A

A/ ( )====

−−−−−−−− −−−− −−−− −−−−

8

252

2


� for W/d > 2

� Where

� And

W d B B

B

r

r

r

/ [ ln( )

{ln( ) ..

}] ( )

==== −−−− −−−− −−−− ++++−−−−

××××

−−−− ++++ −−−− −−−− −−−− −−−− −−−−

21 2 1

1

2

1 0 390 61

53

ππππ

εεεε

εεεε

εεεε

AZo r r

r r====

++++++++

−−−−

++++++++

60

1

2

1

10 23

0 11εεεε εεεε

εεεε εεεε( .

.)

BZo r

====377

2

ππππ

εεεε


� for a homogeneous medium with a complex dielectric constant, the propagation constant is written as

� note that the loss tangent is usually very small

γγγγ αααα ββββ κκκκ

γγγγ ωωωω µµµµ εεεε εεεε δδδδ

==== ++++ ==== −−−−

==== −−−− −−−−

d c

c o o r

j k

k j

2 2

2 21( tan )

γγγγ δδδδ==== −−−− ++++k k jkc2 2 2 tan


� Note that where x is small

� therefore, we have

( ) //

1 1 21 2++++ ==== ++++x x

γγγγδδδδ

==== −−−− ++++−−−−

−−−− −−−− −−−−k kjk

k kc

c

2 22

2 22

54tan

( )


� Note that

� for small loss, the phase constant is

unchanged when compared to the

lossless case

� the attenuation constant due to

dielectric loss is therefore given by

� Np/m (TE or TM) (55)

j k kcββββ ==== −−−−2 2

ααααδδδδ

ββββd

k====

2

2

tan


� For TEM wave , therefore

� Np/m (TEM) (56)

� for a microstrip line that has

inhomogeneous medium, we multiply Eq. (56) with a filling factor

k ==== ββββ

ααααδδδδ

dk

====tan

2

εεεε εεεε

εεεε εεεεr e

e r

( )

( )

−−−−

−−−−

1

1

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12


� = (57)

� the attenuation due to conductor loss is given by

� (58) Np/m where

� is called the surface resistance of the conductor

ααααεεεε δδδδ

do ek

====tan

2

εεεε εεεε

εεεε εεεεr e

e r

( )

( )

−−−−

−−−−

1

1

ko r e

e r

εεεε εεεε δδδδ

εεεε εεεε

( ) tan

( )

−−−−

−−−−

1

2 1

ααααcs

o

R

Z W====

R s o==== ωµωµωµωµ σσσσ/ ( )2

Rs


� note that for most microstrip substrate,

the dielectric loss is much more significant than the conductor loss

� at very high frequency, conductor loss

becomes significant

An Approximate Electrostatic

Solution for Microstrip Lines

� two side walls are sufficiently far away that

the quasi-static field around the microstrip would not be disturbed (a >> d)

y

x

a/2

d

-a/2

W

εr



� we need to solve the Laplace’s equation

with boundary conditions

� two expressions are needed, one for

each region

∇∇∇∇ ==== ≤≤≤≤ ≤≤≤≤ <<<< ∞∞∞∞

==== ==== ±±±±

==== ==== ∞∞∞∞

t x y x a y

x y x a

x y y

2 0 2 0

0 2

0 0

ΦΦΦΦ

ΦΦΦΦ

ΦΦΦΦ

( , ) , | | / ,

( , ) , /

( , ) , ,



� using the separation of variables and

appropriate boundary conditions, we write

ΦΦΦΦ

ΦΦΦΦ

( , ) cos sinh ( ),

( , ) cos ( ),

,

,

/

x y An x

a

n y

ay d

x y Bn x

ae d y

nn odd

nn odd

n y a

==== ∑∑∑∑ −−−− −−−− −−−− ≤≤≤≤ ≤≤≤≤

==== ∑∑∑∑ −−−− −−−− −−−− ≤≤≤≤ ≤≤≤≤ ∞∞∞∞

====

∞∞∞∞

====

∞∞∞∞ −−−−

1

1

59 0

60

ππππ ππππ

ππππ ππππ



� the potential must be continuous at y=d

so that

� note that this expression must be true for any value of n

An d

aB en n

n d asinh

/ππππ ππππ==== −−−−

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� due to fact that

� if m is not equal to n

cos cos// m x

a

n x

adxa

a ππππ ππππ−−−−∫∫∫∫ ====2

2 0

ΦΦΦΦ

ΦΦΦΦ

( , ) cos sinh ( ),

( , ) cos sinh ( ),

,

,

( )/

x y An x

a

n y

ay d

x y An x

a

n d

ae

d y

nn odd

nn odd

n y d a

==== ∑∑∑∑ −−−− −−−− −−−− ≤≤≤≤ ≤≤≤≤

==== ∑∑∑∑ −−−−

≤≤≤≤ ≤≤≤≤ ∞∞∞∞

====

∞∞∞∞

====

∞∞∞∞ −−−− −−−−

1

1

61 0

62

ππππ ππππ

ππππ ππππ ππππ



� the normal component of the electric field is discontinuous due to the presence of surface

charge on the microstrip, E yy ==== −−−−∂Φ∂Φ∂Φ∂Φ ∂∂∂∂/

E An

a

n x

a

n y

ay d

x y An

a

n x

a

n d

ae

d y

y nn odd

nn odd

n y d a

==== −−−− ∑∑∑∑ ≤≤≤≤ ≤≤≤≤

==== ∑∑∑∑

≤≤≤≤ ≤≤≤≤ ∞∞∞∞

====

∞∞∞∞

====

∞∞∞∞ −−−− −−−−

1

1

0,

,

( )/

cos cosh ,

( , ) cos sinh ,

ππππ ππππ ππππ

ππππ ππππ ππππ ππππΦΦΦΦ



� the surface charge at y=d is given by

� assuming that the charge distribution is given by on the conductor and

zero elsewhere

ρρρρ εεεε εεεε εεεεs o y o r yE x y d E x y==== ==== −−−−++++ −−−−( , ) ( , )

ρρρρ εεεεππππ ππππ ππππ

εεεεππππ

s o nn odd

rAn

a

n x

a

n d

a

n d

a==== ∑∑∑∑ ++++ −−−− −−−−

====

∞∞∞∞

1

63,

cos (sinh cosh ) ( )

ρρρρs ==== 1

� multiply Eq. (63) by cos mπx/a and

integrate from -a/2 to a/2, we have

ρρρρππππ ππππ

ππππ

εεεεππππ ππππ ππππ

εεεεππππ

εεεεππππ ππππ

εεεεππππ ππππ ππππ

εεεεππππ ππππ

εεεεππππ

saa

W

W

oaa

nn odd

r

o n ra

a

o n r

dxm x

adx

m W a

m a

An

a

n x

a

n d

a

n y

adx

An

a

n d

a

n d

a

m x

a

n x

adx

An

a

n d

a

n d

a

am

−−−−−−−−

−−−−====

∞∞∞∞

−−−−

∫∫∫∫ ==== ∫∫∫∫ ====

∫∫∫∫ ∑∑∑∑ ++++ ====

∑∑∑∑ ++++ ∫∫∫∫ ====

++++

//

/

/

//

,

/

/

cossin( / )

/

cos (sinh cosh )

(sinh cosh ) cos cos

(sinh cosh ) ,

22

2

2

22

1

2

2

2 2

2==== n

Aa m W a

n n d a n d an

o r

====++++

4 2

2

sin( / )

( ) [sinh( / ) cosh( / )

ππππ

ππππ εεεε ππππ εεεε ππππ



� the voltage of the microstrip wrt the

ground plane is

� the total charge on the strip is

V E x y dy A n d ay nn odd

d==== −−−− ==== ==== ∑∑∑∑∫∫∫∫====

∞∞∞∞( , ) sinh /

,

01

0 ππππ

dx WWW−−−−∫∫∫∫ ====/

/2

2



� the static capacitance per unit length is

� this is the expression for

CQ

V

W

a m W a n d a

n n d a n d ao rn odd

==== ====

++++∑∑∑∑

====

∞∞∞∞ 4 2

21

sin( / ) sinh( / )

( ) [sinh( / ) cosh( / )],

ππππ ππππ

ππππ εεεε ππππ εεεε ππππ(64)

εεεεr ≠≠≠≠ 1

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� the effective dielectric is defined as

� , where is obtained from

Eq. (64) with

� the characteristic impedance is given by

εεεεeC

Co==== Co

εεεεr ==== 1

Zv C cC

op

e==== ====1 εεεε

The Transverse Resonance Techniques

� the transverse resonance technique employs a transmission line model of the transverse cross section of the guide

� right at cutoff, the propagation constant is equal to zero, therefore, wave cannot propagate in the z direction


� it forms standing waves in the transverse plane of the guide

� the sum of the input impedance at any point looking to either side of the transmission line model in the transverse plane must be equal to zero at resonance


� consider a grounded slab and its

equivalent transmission line model

x

zε

r

d

to infinity

Za, kxa

Zd,kxd


� the characteristic impedance in each of the air and dielectric regions is given by

� and

� since the transmission line above the dielectric is of infinite extent, the input impedance looking upward at x=d is simply given by

Zk

ka

xa o

o====

ηηηηZ

k

k

k

kd

xd d

d

xd o

r o

==== ====ηηηη ηηηη

εεεε

Za


� the impedance looking downward is the impedance of a short circuit at x=0 transfers

to x=d

� Subtituting , we have

� Therefore,

Z ZZ jZ l

Z jZ lin o

L o

o L====

++++

++++

tan

tan

ββββ

ββββ

Z Z Z k l dL o d xd==== ==== ==== ====0, , ,ββββ

Z jZ din d==== tanββββ

k

kjk

kk dxa o

o

xd o

r oxd

ηηηη ηηηη

εεεε++++ ====tan 0

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15


� Note that , therefore, we have

� From phase matching,

� which leads to

� Eqs. (65) and (66) are identical to that

of Eq. (38) and (39)

k jhxa ==== −−−−

εεεεr xd xdh k k d==== −−−− −−−− −−−−tan ( )65

k kyo yd====

εεεεr o xd o xa ok k k k k h2 2 2 2 2 2

66−−−− ==== −−−− ==== ++++ −−−− −−−− −−−− ( )

Wave Velocities and Dispersion

� a plane wave propagates in a medium at the speed of light

� Phase velocity, , is the speed at which a constant phase point travels

� for a TEM wave, the phase velocity equals tothe speed of light

� if the phase velocity and the attenuation of atransmission line are independent of frequency,a signal propagates down the line will not be

distorted

1 / µεµεµεµε

vp ==== ωωωω ββββ/


� if the signal contains a band of

frequencies, each frequency will travelat a different phase velocity in a non-

TEM line, the signal will be distorted

� this effect is called the dispersion effect


� if the dispersion is not too severe, a

group velocity describing the speed ofthe signal can be defined

� let us consider a transmission with a

transfer function of

Z Ae Z ej z j( ) | ( )|ωωωω ωωωωββββ ψψψψ==== ====−−−− −−−−


� if we denote the Fourier transform of a time-

domain signal f(t) by F(ω), the output signal

at the other end of the line is given by

� if A is a constant and ψ = aω, the output will

be

f t F Z e doj t

( ) ( )| ( )|( )==== ∫∫∫∫

−−−−

−−−−∞∞∞∞

∞∞∞∞1

2ππππωωωω ωωωω ωωωω

ωωωω ψψψψ

f t A F e d Af t aoj t a

( ) ( ) ( )( )==== ∫∫∫∫ ==== −−−−

−−−−

−−−−∞∞∞∞

∞∞∞∞1

2ππππωωωω ωωωω

ωωωω


� this expression states that the output

signal is A times the input signal with adelay of a

� now consider an amplitude modulated carrier wave of frequency ωωωωo

s t f t t f t eoj to( ) ( ) cos Re{ ( ) }==== ====ωωωωωωωω

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� the Fourier transform of

is given by

� note that the Fourier transform of s(t) is

equal to

f t ej to( )ωωωω

S F o( ) ( )ωωωω ωωωω ωωωω==== −−−−

1

2{ ( ) ( )}F Fo oωωωω ωωωω ωωωω ωωωω++++ ++++ −−−−


� The output signal , is given by

� for a dispersive transmission line, the

propagation constant β depends onfrequency, here A is assume to be

constant (weakly depend on ω)

s to ( )

s t AF e do oj t z( ) Re ( ) ( )==== ∫∫∫∫ −−−−

−−−−∞∞∞∞

∞∞∞∞−−−−1

2ππππωωωω ωωωω ωωωωωωωω ββββ


� if the maximum frequency component of the signal is much less than the carrier

frequencies, β can be linearized using a Taylor series expansion

� note that the higher terms are ignored as the higher order derivatives goes to zero faster

than the growth of the higher power of

ββββ ωωωω ββββ ωωωωββββ

ωωωωωωωω ωωωωωωωω ωωωω( ) ( ) | ( ) . . .==== ++++ −−−− ++++====o o

d

d o

( )ωωωω ωωωω−−−− o


� with the approximation of

ββββ ωωωω ββββ ωωωω ββββ ωωωω ωωωω ωωωω ββββ ββββ ωωωω( ) ( ) '( )( ) ' &&&≈≈≈≈ ++++ −−−− ==== ++++o o o o o

s t A F e doj t z zo o( ) Re{ (&&&) }( ' &&& )==== ∫∫∫∫

−−−−∞∞∞∞

∞∞∞∞−−−− −−−−1


ωωωω ββββ ββββ ωωωω

s t A e F e d

s t A f t z e

oj t z j t z

o oj t z

o o o

o o

( ) Re{ (&&&) &&&}

( ) Re{ ( ' ) }

( ) &&&( ' )

( )

==== ∫∫∫∫

==== −−−−

−−−−

−−−−∞∞∞∞

∞∞∞∞−−−−

−−−−

1


ββββ

ωωωω ββββ ωωωω ββββ

ωωωω ββββ

s t Af t z t zo o o o( ) ( ' ) cos( ) ( )==== −−−− −−−− −−−− −−−− −−−−ββββ ωωωω ββββ 67


� Eq. (67) states that the output signal is

the time-shift of the input signalenvelope

� the group velocity is therefore defined

asv

d

dg

oo

==== ==== ====1

ββββ

ωωωω

ββββωωωω ωωωω

'|

Thank You!!

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