Download Task A. Transmission Line and Discontinuities

TASK A. TRANSMISSION LINE AND DISCONTINUITIES
Task A. Transmission Line and Discontinuities............................................................. 1
A.I. TEM Transmission Line ....................................................................................... 2
A.I.1. Circuit Representation of a Uniform Transmission Line ........................... 2
A.I.2. Time Harmonic Voltage and Current Waves in a Transmission Line ...... 4
A.I.3 Transmission Line Termination ..................................................................... 6
A.II. Physical Transmission Lines ............................................................................... 8
A.II.1. TEM Transmission Lines ............................................................................. 8
Properties of TEM Lines: ..................................................................................... 8
Commonly Used TEM Lines: ............................................................................ 10
a) Coaxial Line: ............................................................................................... 10
b) Planar Transmission Lines:....................................................................... 14
Stripline: ...................................................................................................... 14
Microstrip: ................................................................................................... 16
A.II.2. Non-TEM Transmission Lines .................................................................. 20
Rectangular Waveguide: .................................................................................... 20
Circular Waveguide: .......................................................................................... 24
A.II.3 Characteristic Impedance in Waveguide ................................................... 26
A.II.3.1 Rectangular Waveguide ........................................................................... 26
A.III Transmission Line Discontinuities . ................................................................. 31
A.III.1 Two-Port Discontinuities. .......................................................................... 31
1
A.I. TEM Transmission Line
A.I.1. Circuit Representation of a Uniform Transmission Line
Circuit Representation of a Uniform Transmission Line is shown in Figure A.I.1.1
[1,Rizzi], where
R = Series Resistance per unit length of line (ohm/m)
G = Shunt conductance per unit length of line (mho/m)
L = Series inductance per unit length of line (ohm/m)
C = Shunt capacitance per unit length of line (mho/m)
Lz v+v
+
Rz
v
+
i
i+i
Gz
Cz
+z
direction
-
-
Figure A.I.1.1 Circuit Representation of a Uniform Transmission Line.
Applying the Kirchhoffs's voltage law and the Kirchhoffs's voltage law, result in the
following system of equations for the voltage and current (z  0)

v
v
Ri L
z
t
A.I.1.1
2

I
dV
 GV  C
z
dt
A.I.1.2
By solving the above system of equations a second order partial differential
equation for voltage (Equation A.I.1.3) and a second order differential equation for
voltage (Equation A.I.1.4) are obtained
 2v
v
2 v

LC
 RC  GL 
 RGv
2
2
t
z
t
2i
2i
i

LC
 RC  GL   RGi
2
2
t
z
t
A.I.1.3
A.I.1.4
For a lossless line (G=0 and R=0) Equations A.I.1.1 through A.I.1.4 reduces to Equations
A.I.1.5 through A.I.1.8.
v
i
L
z
t
(A.I.1.5)
f 
Z TM   1  c  2
f 
(A.I.1.6)
 2v
2 v

LC
 z2
 t2
(A.I.1.7)
2 i
2i

LC
 z2
 t2
(A.I.1.8)

Equations A.I.1.7 and A.I.1.8 are the wave equations for homogeneous, lossless,
and linear medium.The general solution for such wave equations is
y(t,z)= f +(z - vt) + f -(z + vt)
(A.I.1.9)
where y() is the physical quantity of interest, e.g., voltage (v) or current (i), and f () are
arbitrary functions representing the progression of the quantity in time and distance, and
up is the wave velocity. Here
3
up 
1
(A.I.1.10)
LC
A.I.2. Time Harmonic Voltage and Current Waves in a Transmission Line
When the change of a physical quantity with the time is as a sinusoidal wave with
the angular frequency , it is possible to apply the phasor forms of that quantity. In this
case, partial differential Equations A.I.1.1 through A.1.1.4 become the ordinary
differential Equations A.I.2.1 through A.I.2.4 respectively:

dV
 (R  jL) I  ZI
dz
(A.I.2.1)

I
 (G  jc) V  Y V
z
(A.I..2.2)
d2 V
 (R  jL) (G  jC) V  ZYV
 z2
(A.I.2.3)
d 2I
 (R  jL) (G  jC) V  ZYV
d z2
(A.I..2.4)
with Z = R+ jL and Y = G + jC. The solution for the voltage phasor of the Equation
A.I.2.3 is
V= V0  e   z  V
=
ZY 

e   z  V  V 
(A.I.2.5)
R  jLG  jC)
(A.I.2.6)
0
where = +j is the propagation constant of the line,  is the attenuation constant
(Np/m), and  is the phase constant (rad/m) of the line. V = V 0 e   z are the voltage
4
waves in  z (forward and reverse) directions. From A.I.2.1 and A.I.2.2 it is easy to show
that
I= I 0 e   z _ I 0 e   z  I   I 
(A.I.2.7)
where I+0 = V+0/Z0 and I-0 = V0/Z0. The quantity
Z0 =
V V
 
I
I
(A.I.2.8)
is a line parameter called the line characteristic impedance and is related to the various
lumped element components of Figure A.I.1.1 through the following relation:
Z0=
Z
R  jL

Y
G  JC
(A.I.2.9)
The reciprocal of Z0, Y0=1/Z0, is called the line characteristic admittance.
5
A.I.3 Transmission Line Termination
Figure A.I.3.1 shows a uniform transmission line terminated to a load Zl located at z=0.
l
Z
G
EG
ZL
z = -l
z=0
Figure A.I.3.1 A terminated uniform transmission line
Recalling Equations A.I.2.5, A.I.2.7 and   V knowing that ZL=V (0)/I (0), it can
f
be easily shown that
L = I-/I+ = V-/V+= (Zl-Z0)/(Zl+Z0)
(A.I.3.1)
Parameter L is called the load reflection coefficient. It is also possible to define the
reflection coefficient  and the input impedance Zin in an arbitrary point z. Equations
A.I.2.5, A.I.2.7 and A.I.3.1 yield
(z)= I-(z)/I+(z)
=V-(z)/V+(z) = (Zin(z)-Z0)/(Zin(z)+Z0) = L exp(2jz)
and
6
(A.I.3.2)
Zin=
1 Le2  z
1  L e2  z
(A.I.3.3)
or
Zin= Z 0
Z L  Z 0 tanh  z 
Z 0  Z L tanh  z 
(A.I.3.4)
Example A.I.2.1. The voltage and current distribution inside a uniform lossless
transmission line terminated to a load
7
A.II. Physical Transmission Lines
For analytical basis see Propagation of E&M Waves in Uniform Cylindrical Waveguides.
A.II.1. TEM Transmission Lines
Properties of TEM Lines:
1)
The number of TEM modes is N-1 where N is number of conductors.
Example:
Microstrip line
N=2
1 TEM mode
Coaxial Cable
N=2
1 TEM mode
Waveguide
N=2
1 TEM mode
Coupled Mircostrip
N=3
2 TEM modes
Parallel Plate
Even mode of exatation
Odd mode of exatation
2)
3)
Electric field and magnetic field are normal to the direction of propagation; i.e.
Et  Ht  kz
no axial field
Er; H; kz
for coaxial.
Ex; Hy; kz
for parallel plate waveguide
Ratio electric field over magnetic field is always a constant everywhere in space.
 is characteristic impedance of wave.
8


Et 

 a z   t
   ; E t  η H t  â z and H t 
η
Ht
4)


 2t E  0   2t H
 t2 
where  2t   2 
2
i.e. for the Cartesian coordinates
z 2


  1 2
2 1  



and
for
cylindrical
coordinates
it
is


r
t
x 2 y 2
r  r   r  r 2 φ 2


These relationships implies that E and H are calculated using the Laplace’s Equation,
hence it behaves quasi -static.
5)
A unique value for voltage and current is found for TEM lines. [Note: A path
dependent voltage and current is found for metallic waveguides.]
 
V   E d l = Unique
 
I   H  d l = Unique
6)
Knowing V and I, then the chrematistic impendence of a line is defined as:
Z0 =
L
C
7)
L / length
C / length
V
=
I
length
= Inductance per unit length of line
length
= Capacitance per unit of line
Equal values of phase and group velocities that are related to L and C.
[Note: Vp =
Vp = Vg =
8)

d
ωo ].
; Vg =
dk
k
1
L / length  C / length
Equation partition of electric and magnetic energy:
9
2
2
WE
W
1
1
=
  E d ; H =
  H d
2
2
z
z
W  2W  2W  W  W
 2 
 E

t
1

 1
Pf = Pz =
Re 
d = Re 
  η Ht
2
 
 2 


9)
2
dσ
.

Power transmitted in z is the total energy per unit length multiplied by the group
velocity
Pz =
W
 Vg
z
If TEM line is lossy these equations are modified; however in practice, these results are
very good approximation for the low loss TEM Lines.
Commonly Used TEM Lines:
a) Coaxial Line:
For analytical basis see Propagation of E&M Waves in Uniform Cylindrical Waveguides.
A homogeneously filled, low loss coaxial cable (Figure A.II.1.1) is an example of
a physical TEM transmission line The dielectric with relative permittivity r fills the
region between inner and outer conductors (diameters 2a and 2b).
10
2b
2a
z
r
Front
Side
Figure A.II.1.1. A homogeneously filled, low loss coaxial cable.
As the term TEM (i.e., Transverse Electric and Magnetic) represents, for a dominant
mode, there are no electric and magnetic components in z direction inside the line. This
mode is able to propagate from zero frequency. If the line is low loss, the following
equations are always valid for the phase velocity, wavelength, and propagation constant
with enough accuracy
V
C
r

C

f r

8 m
 C  3 10

s


(A.II.1.1)
0
A.II.1.2
r

2
r 
C

(A.II.1.3)
For a coaxial cable the characteristic impedance is given by
Z0=
b
ln  
r a 
60
(A.II.1.4)
11
The electric and magnetic fields of a coaxial cable in cylindrical coordinate system are
given by:

V0 a r  jz
E
e
b r
ln  
a

H
(A.II.1.5)
 r V0 a   jz
e
b r
120 ln  
a
(A.II.1.6)
where, V0 is a constant and a and ar are unit vectors in cylindrical coordinates.
A coaxial line can operate single mode up to a certain point that the first higher order
non-TEM mode starts propagating. This mode is known as TE11 and an approximate
frequency that it starts propagating is
fc 
C
 a  b   r
(A.II.1.7)
For instance, a 50  teflon filled line (r = 2.05) and 2a = 0.91 mm and 2b = 2.98, will
support TE11 mode from about 35.0 GHz.
Example A.II.1.1 Field distribution inside a uniform coaxial line for the dominant (TEM)
and the second order (TE11) modes
The attenuation (NP/m) of a coaxial line is obtained from the following equation
  c  d 
R m r
1 1 
  
 b   a b  0
240 ln  
a
 r tan 
(A.II.1.8)
where c and d are the conductor loss and dielectric loss respectively, and
12
Rm 

is the surface resistance of the conductor.
26
Example A.II.1.2 The loss of a coaxial cable
Example A.II.1.3  diagram of the TEM and the TE11 modes of a coaxial cable
13
b) Planar Transmission Lines:
Several different forms of waveguiding structures made from parallel metal strips
on a dielectric substrate have found use in microwave and millimeter-wave circuits as
well as in high –speed digital circuits. In this section, we will examine in some detail
three types, called stripline, microstrip, and coplanar waveguide. Emphasis will be on the
lowest-order mode, which is a TEM wave in a homogeneusly filled stripline and a quasiTEM wave in microstrip and coplanar waveguide.
Stripline:
The stripline consists of a conducting strip lying between, and parallel to, two
wide conducting planes, as shown in Figure A.II.1.2.a The region between the strip and
the planes is filled with a uniform dielectric. Such a structure, with a uniform dielectric
and more than one conductor, can support a TEM wave. If the strip width w is much
greater than the spacing d and the two-plane lines connected in parallel. More precise
results are found from the capacitance per unit length. For a TEM wave, the phase
1 / 2
velocity is     
, and from the transmission-line formalism, it is given by
   LC 1 / 2 . Then the characteristic impedance is
Zo 

L
LC


C
C
C
(A.II.1.9)
Thus, for systems with uniform  and  the characteristic impedance can be found from
the capacitance, which can be determined in a number of ways, as we saw in Chapters 1
and 7. An approximate expression for the charateristic impedance of the stripline,
assuming a zero-thichness strip, has been found by conformal transformation to be
14
Zo 


K k 
4 K 1 k2

(A.II.1.10)
(b)
(a)
Figure A.II.1.2 (a) Stripline, (b) Microstrip
where  

, k is given by


  w 
k  cosh 

 4d 

1
(A.II.1.11)
and K(k) is the complete elliptic integral of the first kind (see Ex. 4.7a). A convenient
approximate expression for Z o , accurate over the range
Zo 

w
 0.56 , is
2d
(A.II.1.12)

  w 
8 ln 2 exp 

 4d  

Approximate techniques used to obtain corrections to Z o for t  0 are discussed
by Hoffmann. The velocity of propagation does not depend on thickness and is 1/ 2
as for all TEM waves. Both velocity of propagation and characteristic impedance,
neglecting loss, are independent of frequency and may be used up to the cutoff frequency
of modes between the ground plates, Eq 8.3(14).
15
An approximate expression for attenuation resulting from conductor surface resistivity
Rs is
 w
 8d  
 ln   

R
2d
 t   nepers / m
c  s 
2 d  ln 2   w 

4d 

which is valid if w > 4d and t <
(A.II.1.13)
d
. Approximations for other dimensions are given in
5
footnote 4. Attenuation from lossy dielectrics is exactly as in equation 8.5(2).
Microstrip:
Probably the most widely used thin-strip line is one formed with the strip lying on
top of insulator with a conductive backing. A different dielectric (usually air) is above the
insulator and strip (Figure A.II.1.2.b). In such an arrangement, there cannot be a true
TEM wave. The reason is that such a wave, as shown in connection with equation 8.3(1),
requires that  2  k 2  0 . The propagation constant  is a single quantity for the wave, so
 2  k 12 and  2  k 22 cannot both be zero if k 1  k 2 Exact solution of this problem is
complicated by the finite width of the strip and the two different dielectrics. As
mentioned above for the stripline, the lowest-order approximation for the micostrip is a
section of parallel-plane guide neglecting fringing. A much better, though still
approximate, approach is to consider that the lowest-order wave is approximately a TEM
wave so the distribution of fields in the transverse plane is nearly the same as that for
static fields. This so-called quasi static approach employs calculations made with static
fields to determine the transmission-line parameters for propagation of the lowest-order
16
mode, even though it is not a pure TEM wave. The approximation is very useful in
practical applications but has some limitations that we will mention below.
A common approach to obtaining simple, accurate expression is to first find the
characteristic impedance Zoo if an electrode structure identical to the one of interest but
with the strip electrode having zero thickness and the dielectric being free space
everywhere. This problem can be solved exactly by the method of conformal mapping,
but the expressions are very complex and more useful results are the various
approximations to the exact expressions. A particularly useful expression is
0.172 
w
 w
oo  377   1.98 

d 
 d

1
(A.II.1.14)
w
which is accurate to <0.3% for all   >0.06. Then to get the characteristic impedance
d
of the actual line, it is corrected by the so-called effective dielectric constant eff, which if
filling the entire space, would give the same capacitance as that of the actual structure.
Since the inductance is unaffected by the presence of a dielectric, correction for the
capacitance gives the correction for the characteristic impedance.
The Static approximation to the effective relative dielectric constant is also found
by conformal mapping methods; one of several useful approximations is
 eff

 1 1  1
1  r
2 
10d
1

w





(A.II.1.15)
which is always between unity and r. Applying Equations (A.II.1.14) and (A.II.1.15) in
o 
 oo
(A.II.1.16)
 eff
17
we obtain the static approximation for the actual characteristic impedance. The results for
a variety of insulator dielectric constants are shown in figure 8.6c. Corrections for
nonzero strip thickness have been published.
1 / 2
Since the phase velocity in a TEM wave is  p   
, the ratio
c
  eff in
vp
the quasi static case As frequency increases, the longitudinal field components become
increasingly important; this can be represented by a frequency-dependent effective
dielectric constant eff(f) to express the variation of phase velocity or phase constant.
Approximate expressions for eff(f) and for attenuation from conductor and dielectric
losses have been obtained from numerical calculations or empirically. Some results
follow:
Example A.II.1.4 Attenuation In a Microstrip Transmission Line
Also see
A.S1 more on microstrip equations and applications
A.S3 Microstrip Discontinuity Compensation
A.S4 Microstrip Applications
18
19
A.II.2. Non-TEM Transmission Lines
Rectangular Waveguide:
For analytical basis see Propagation of E&M Waves in Uniform Cylindrical Waveguides.
Figure A.II.2.1 illustrates a rectangular waveguide. The modes are either T (Hz = 0) or
TM (z = 0). The fields are given in Table A.II.2.1
y
b
z
a
Figure A.II.2.1 A rectangular waveguide.
20
x
Table A.II.2.1 parameter and field inside a rectangular wave guide.
TMmn
Tmn
f 
 z  jk c   c  sin( k x x ) sin k y y 
f 
Hz 
jk c  f c 
  cos( k x x ) cosk y y 
 f 
H x   k y sin( k x x ) cosk y y 
 x   k y cosk x x sin k y y 
H y  k x cosk x x sin k y y 
 y  k x sin k x x cosk y y 
 x  ZTM H y
Hy 
 y   Z TM H x
x
Z T
Hz  0
Hx  
f 
ZTM   1  c  2
f 
z  0
kc 
ZTE
Z T 



y

  k 2c  k 2
K
2
x
 K y2 
kx 


f 
1  c 2
f 
k   
ma
a
ky 


k x2  k y2
Vk c
kc
fc 


2 2 
2  

b
kc is called the cutoff wave number fc is the cutoff frequency and  is the propagation
constant. The other important parameters are the phase velocity
Vp 



V
(A.II.2.1)
f 
1  c  2
f 
and group velocity
21
Vg 
d
f 
 V 1  c  2
d
f 
(A.II.2.2)
V is the plane wave speed in a medium with dielectric constant  .
1

V
(A.II.2.3)
The wavelength inside the waveguide for a mode with cutoff frequency fc is
g 

(A.II.2.4)
f 
1  c  2
f 
and
f 
  k 1  c 
f 
where  
2
(A.II.2.5)
V
is the wavelength and k = /V is the phase constant of the plane wave.
f
Furthermore,

2
g
(A.II.2.6)
Example A.II.2.1 Electric field distribution inside a rectangular waveguide for various
modes (xy plane).
Example A.II.2.2 Magnetic field distribution inside a rectangular waveguide for various
modes (xy plane).
Example A.II.2.3  diagram if the first few modes in X-band rectangular wave guide.
22
Example A.II.2.4 phase and group velocity of TE10 mode in rectangular waveguide.
Example A.II.2.5 TE mode attenuation of an x-band rectangular wave guide.
For waveguide excitation see: excitation of waveguides (from Pozar)
23
Circular Waveguide:
For analytical basis see Propagation of E&M Waves in Uniform Cylindrical Waveguides.
Figure A.II.2.2 illustrates a hollow circular waveguide. The modes are either T (Hz = 0)
or TM (z = 0). The fields are given in Table A.II.2.2. The relations A.II.2.1 through
A.II.2.4 are also valid for the circular waveguide.
y
r

x
a
Figure A.II.2.2 A hollow circular waveguide.
Table A.II.2.2 parameter and field inside a circular waveguide.
TM
TE
u = kc r

x   fc 
  j   Jn u  cos m =  Hz
  f 

Hr =
m J u   sin m  
n
r
u 

H = J ' u  cos m  E
 n


24

r

 H   r (   r YTE )
Z TM
Z TE
TM: kc =
Z TM
p nm
; Jm (p nm) = 0
a
  Hr 
TE: kc =
f 
ZTM   1  c  2
f 
Z T 

Z TE
pnm
; J’m (p’ nm ) = 0
a

f 
1  c 2
f 
pnm
n
0
1
2
3
1
4.405
3.832
5.136
6.380
2
5.520
7.016
8.417
9.761
3
8.654
10.173
11.620
13.015
4
11.792
13.324
14.796
m
p’nm
n
0
1
2
3
1
3.832
1.841
3.054
4.201
2
7.016
5.331
6.706
8.015
3
10.173
8.536
9.969
11.346
4
13.324
11.706
13.170
m
25
(    YTE )
Example A.II.2.6 Electric field distribution inside a circular waveguide for various
modes (xy plane).
Example A.II.2.7 Magnetic field distribution inside a circular waveguide for various
modes (xy plane).
Example A.II.2.8 - diagram of the first few modes in an x-band circular waveguide.
Example A.II.2.9 phase and group velocity of TE11 mode in circular waveguide.
Example A.II.2.10 TE11 mode attenuation of an x-band circular waveguide.
A.II.3 Characteristic Impedance in Waveguide
A.II.3.1 Rectangular Waveguide
In order to apply transmission-line theory to waveguides, one must be able
determine the phase constant () or guide wavelength (g) as well as the characteristic
impedance (O) of the particular guide configuration. Equations (A.II.2.4) – (A.II.2.6)
provide the required relationships for  and  g. Characteristic impedance is defined in
Equation (A.I.2.8) as
V
. Where V+ is the voltage between the conductors and I+ is the
I
conduction current in the propagation direction for the traveling wave. For TEM lines o
 
is uniquely defined since the value of V       dl is independent of the integration
26
path. In the waveguide case, V+ is function of the integration path and therefore many
definitions of o are possible.
For rectangular waveguide in the TE10 mode, from Table A.II.2.1
E y  E0
c
 x 
sin   sin( t  z)
g
 a 
E x  H 0
(A.II.3.1)
c
 x 
sin   sin( t  z)
g
 a 
(A.II.3.2)
 x 
E y  H 0 cos  sin( t  z)
 a 
(A.II.3.3)
where c = 2a and E0 = H0ZTE.
The electric field is maximum at x 

a
and its value is  v   o c . Integrating
2
g
directly from y  0 to y  b yields the voltage across the centerline of the waveguide.
Namely,
VcL   o b
c

  o  T Eb c
g
g
(A.II.3.4)
However, this represents only one possible voltage value between the points y  0 , x 
and y  b , x 
a
2

a
. For instance, one could integrate   dl by choosing a path along the
2
conducting walls. In this case VCL  0 . Other paths leading to different values of voltage
are indicated in Figure A.II.3.1. Clearly, voltage and hence o does not have a unique
value in waveguide. One possible definition is the ratio of Vg as defined in Equation
(A.II.3.4) to the longitudinal current Iz, where
27
a
a
I z   K z dx    x dx
o
(A.II.3.5)
o
a
VcL  0
VcL 
VcL  o b
c
g
o  c
b
2 g
a/4
Figure A.II.3.1 Possible integration paths for determining VCL in rectangular waveguide.
Integration along the bottom wall (y = 0) yields
c a


2a
Iz  o
sin x dx   o c

g o
a

g
(A.II.3.6)
Therefore,
o 
VCL  b

T E 
Iz
2a
b
59 a
R
R
f 
1  c 
f 
2
ohms
(A.II.3.7)
where TE is given by Table A.II.2.1. Since VCL and Iz are in phase, o is real for f > fc.
The above expression is known as the voltage-current definition of o. Other commonly
used definions are based upon power flow along the waveguide. These are
28
1. Power-Current Definition:
o 
2. Power-Voltage Definition:

P
I 2z
Vt2
P
3. Modified Power-Voltage Definition:  o 
Vt2
P
Where VCL is given by Equation (A.II.3.4) and Iz by Equation (A.II.3.6). An expression
for power flow in the direction can be developed can be developed with the aid of
Poynting’s theorem. With x and y in phase.


   ab
P       dS     x  y dy dx
S
(A.II.3.8)
oo
and therefore
2
 c  a 2 
2
P   0  T E   b  sin
x dx
 g 
a
  o
(A.II.3.9)
or

1
p   02  T E ab  c
 g
2





2
(A.II.3.10)
Applying this equation to the three definitions results in the following expressions for o
in rectangular guide.
Power-Current Definition:
 o  465
b R g
ohms
a R 
Power-Voltage Definition:
 o  754
b R g
ohms
a R 
29
(A.II.3.11)
Modified Power-Voltage Definition:  o  377
Example A.II.3.1
30
b R g
ohms
a R 
A.III Transmission Line Discontinuities
See Discontinuities and Modal Analysis (from Pozar)
A.III.1 Two-Port Discontinuities
When a transmission line changes its shape or terminates to a line with different
dimension a discontinuity will be produced. In this case, part of the energy from the
dominant mode will be coupled to the higher order modes. These modes are usually
evanescent in the frequency of operation and can introduce capacitive and / or inductive
effects.
By either necessity or design, microwave networks often consist of transmission
lines with various types of transmission line discontinuities. In some cases discontinuities
are an unavoidable result of mechanical or electrical transitions from one medium to
another (e.g., a junction between two waveguides, or a coax-to-microstrip transition), and
the discontinuity effect is unwanted but may be significant enough to warrant
characterization. In other cases discontinuities may be deliberately introduced into the
circuit to perform a certain electrical function (e.g., reactive diaphragms in waveguide or
stubs in microstrip line for matching or filter circuits). In any event, a transmission line
discontinuity can be represented as an equivalent circuit at some point on the
transmission line. Depending on the type of discontinuity, the equivalent circuit may be a
simple shunt or series element across the line or, in the more general case, a T- or equivalent circuit may be required. The component values of an equivalent circuit depend
on the parameters of the line and the discontinuity, as well as the frequency of operation.
In some cases the equivalent circuit involves a shift in the phase reference planes on the
31
transmission lines. Once the equivalent circuit of a given discontinuity is known, its
effect can be incorporated into the analysis or design of the network using the theory
developed previously in this chapter.
The basic procedure for obtaining equivalent circuits for transmission line
discontinuities; starts with a field theory solution to a canonical discontinuity problem
and develop a circuit model, with component values. This is thus another example of our
objective of replacing complicated field analyses with circuit concepts.
Figures A.III.1.1 and A.III.1.2 show some common transmission line
discontinuities and their equivalent circuits. As shown in Figure A.III.1.1 a-c, thin
metallic diaphragms (or "irises") can be placed in the cross-section of a waveguide to
yield equivalent shunt inductance, capacitance, or a resonant combination. Similar effects
occur with step changes in the height or width of the waveguide, as shown in Figures
A.III.1.1 d,e. Similar discontinuities can also be made in circular waveguide. The best
reference for waveguide discontinuities and their equivalent circuits is The Waveguide
Handbook [7].
Some typical microstrip discontinuities and transitions are shown in Figure
A.III.1.2; similar geometries exist for stripline and other printed transmission lines such
as slotline, covered microstrip, coplanar waveguide, etc. Since printed transmission lines
are newer, relative to waveguide, and much more difficult to analyze, more research
work-is needed to accurately characterize printed transmission line discontinuities; some
approximate results are given in reference [8].
For a two-port discontinuity, these effects can be well represented by a T or 
network (cf Section C.I.3). The study of these effects can be done either by approximate
32
analytical techniques (e.g. Variational technique, see Harington) or numerical techniques
(Finite Element, Finite difference, mode matching, see Gupta or Itoh), and in general are
very complicated.
Following are some examples
illustrating some two-port
discontinuities.
Figure A.III.1.1 Rectangular waveguide discontinuities
33
Figure A.III.1.2 Some common microstrip discontinuities . (a) Open –ended
microstrip.(b) Gap in microstrip. (c) Change in width (d) T-junction (e) Coax-tomicrostrip junction
34
A.S2 Modal Analysis of an H-Plane Step in Rectangular Waveguide
Example A.III.1.1 Capacitive iris inside a rectangular wave guide.
Example A.III.1.2 Inductive iris inside a rectangular wave guide.
Also see
A.S3 Microstrip Discontinuity Compensation
Example D.III.2.1 Aperture in transverse wall of a rectangular waveguide.
Example A.III.1.3 Microstrip step discontinuity
Example A.III.1.4 Microstrip capacitive gap
Example A.III.1.5 Microstrip Mitered Bend
Example A.III.1.6 Microstrip Curved Bend
Example A.III.1.7 Microstrip line loaded with an open stub or a radial stub.
35
A.III.2 Three-Port Discontinuities
Example A.III.2.1 Coaxial T
Example A.III 2.2 Microstrip T
A.III.2 Four-Port Discontinuities
Example A.III.3.1 Microstrip Cross Junction
Example A.III.3.2 Waveguide Magic T
Example D.III.2.2 Aperture in broad wall of a rectangular waveguide.
36