ECE 546 Lecture 03 Waveguides - University of Illinois

ECE 546 – Jose Schutt‐Aine 1

ECE 546 Lecture 03Waveguides

Spring 2022

Jose E. Schutt-AineElectrical & Computer Engineering

University of [email protected]


2 2 22+ + = -2 2 2

E E Ex x x Exx y z

2 2 22

2 2 2+ + = -y y yy

E E EE

x y z

2 2 22

2 2 2+ + = -z z zz

E E E Ex y z

2 2 E E 0

Parallel-Plate Waveguide

Maxwell’s Equations


For a parallel-plate waveguide, the plates are infinite in the y-extent; we need to study the propagation in the z-direction. The following assumptions are made in the wave equation

0, but 0 and 0y x z

Assume Ey only

These two conditions define the TE modes and the wave equation is simplified to read

TE Modes

2 22

2 2+ = -y yy

E EE

x z

(¥)


General solution (forward traveling wave)

( , ) x xz j x j xj zyE x z e Ae Be

( , ) sinzj zy o xE x z E e x

At x = 0, Ey = 0 which leads to A + B = 0. Therefore, A = -B = Eo/2j, where Eo is an arbitrary constant

Phasor Solution

a is the distance separating the two PEC plates


sin 0zj zo xE e a

xma

2 2 2 2z x

22

zma

At x = a, Ey(x, z) = 0

This leads to: xa= m, where m = 1, 2, 3, ...

Moreover, from the differential equation (¥), we get the dispersion relation

which leads to

Dispersion Relation


22 m

a

where m = 1, 2, 3 ... Since propagation is to take place in the zdirection, for the wave to propagate, we must have z

2 > 0, or

This leads to the following guidance condition which willinsure wave propagation

2mf

a

Guidance Condition2

2z

ma


2cmf

a 2= c

c

v af m

The cutoff frequency fc is defined to be at the onset of propagation

Each mode is referred to as the TEm mode. It is obvious thatthere is no TE0 mode and the first TE mode is the TE1 mode.

Cutoff Frequency

The cutoff frequency is the frequency below which the mode associated with the index m will not propagate in the waveguide. Different modes will have different cutoff frequencies.


ˆ ˆ ˆ1 0

0 0x z

y

jE

x y zH

sinzj zzx o xH E e x

coszj zxz o x

jH E e x

we have

From = - jE H

which leads to

Magnetic Field for TE Modes

The magnetic field for TE modes has 2 components


As can be seen, there is no Hy component,therefore, the TE solution has Ey, Hx and Hz only.

From the dispersion relation, it can be shown that the propagation vector components satisfy the relationsz = sin, x = cos where is the angle of incidence of the propagation vector with the normal to the conductor plates.

E & H Fields for TE Modes


2

21pz

z c

cvff

2

21 cg

z

fv cf

2

21

y oTE

x c

EH f

f

and

The effective guide impedance is given by:

The phase and group velocities are given by

Phase and Group Velocities


2 2+ = H H 0

2 2 22

2 2 2+ + = -x x xx

H H H Hx y z

2 2 22

2 2 2+ + = -y y yy

H H HH

x y z

2 2 22

2 2 2+ + = -z z zz

H H H Hx y z

The magnetic field also satisfies the wave equation:

Transverse Magnetic (TM) Modes



0, but 0 and 0y x z

2 22

2 2+ = -y yy

H HH

x z

Assume Hy only These two conditions define the TM modes and the equations are simplified to read

General solution (forward traveling wave)

For TM modes, we assume

( , ) x xz j x j xj zyH x z e Ae Be

TM Modes


= -j H E

ˆ ˆ ˆ1 0

0 0x z

y

jH

x y z

E

This leads to

( , ) x xz j x j xj zzxE x z e Ae Be

( , ) x xz j x j xj zxzE x z e Ae Be

Electric Field for TM Modes

we get

From


( , ) coszj zy o xH x z H e x

( , ) coszj zzx o xE x z H e x

( , ) sinzj zxz o x

jE x z H e x

At x=0, Ez = 0 which leads to A = B = Ho/2 where Ho is an arbitrary constant. This leads to

At x =a, Ez = 0 which leads to

TM Modes Fields

xa = m, where m = 0, 1, 2, 3, ...


2

21x cTM o

y

E fH f

This defines the TM modes which have only Hy, Ex and Ezcomponents.

E & H Fields for TM Modes

The electric field for TM modes has 2 components

The effective guide impedance is given by:

xma


This defines the TM modes; each mode is referred to as the TMmmode. It can be seen from that m=0 is a valid choice; it is called the TM0, or transverse electromagnetic or TEM mode. For this mode and,

E & H Fields for TM Modes

THE DISPERSION RELATION, GUIDANCE CONDITION AND CUTOFF EQUATIONS FOR A PARALLEL-PLATE WAVEGUIDE ARE THE SAME FOR TE AND TM MODES.


zj zy oH H e

z zj z j zzx o oE H e H e

0zE

x=0 and z = . There are no x variations of the fields within the waveguide. The TEM mode has a cutoff frequency at DC and is always present in the waveguide.

TEM Mode

The TEM mode is the fundamental mode on a parallel-plate waveguide

The propagation characteristics of the TEM mode do not vary with frequency


1 Re2

P E H *Time-Average Poynting Vector

* *1 ˆ ˆ ˆRe2 y x zE H H P y x z

2 221 ˆ ˆRe sin cos sin

2o o

z x x x x

E Ex j x x

P z x

22ˆ sin

2o

z x

Ex

P z

Power for TE Modes

TE modes


*1 ˆ ˆ ˆRe2 x z yE E H P x z y

2 221 ˆ ˆRe cos sin cos

2o o

z x x x x

H Hx j x x

P z x

22ˆ cos

2o

z x

Hx

P z

TM modes

The total time-average power is found by integrating <P>over the area of interest.

Power for TM Modes


2 2 22+ + = -2 2 2

E E Ex x x Exx y z

2 2 22

2 2 2+ + = -y y yy

E E EE

x y z

2 2 22

2 2 2+ + = -z z zz

E E E Ex y z

2 2 E E 0

Waveguide



For a waveguide with arbitrary cross section as shown in the above figure, we assume a plane wave solution and as a first trial, we set Ez = 0. This defines the TE modes.

TE Modes

Fromt

HE , we have

y xzz y x

E HE j E j Hy z t

yx zz x y

HE E j E j Hz x t

y yx xzz

E EE EH j Hx y t x y

(1)

(2)

(3)


TE Modes

j H E

ˆ ˆ ˆ

x y z

jx y z

H H H

x y z

EFrom , we get

yz zx z y x

HH Hj E j H j Ey z y

x z zy z x y

H H Hj E j H j Ez x x

0y xH Hx y

We want to express all quantities in terms of Hz.

(4)

(5)

(6)


TE Modesz x

yEH

2 xzz x

EH j j Ey

2 2z

xz

HjEy

z yx

EH

2z y z

y

E Hj j Ex

2 2z

yz

HjEx

From (2), we have

Solving for Ex

From (1)

in (5)

so that

in (4)


TE Modes

2 2z z

xz

j HHx

2 2z z

yz

j HHy

2 22 2

2 2z z

z zH H Hx y

Ez = 0

Combining solutions for Ex and Ey into (3) gives

(¥)


Rectangular Waveguide

2 22 2

2 2z z

z zH H Hx y

If the cross section of the waveguide is a rectangle, we have a rectangular waveguide and the boundary conditions are such that the tangential electric field is zero on all the PEC walls.

(¥)


y yx xz j y j yj x j xj zzH e Ae Be Ce De

2 2y yx xz j y j yj x j xj zx

yz

E e Ae Be Ce De

2 2y yx xz j y j yy j x j xj z

xz

E e Ae Be Ce De

The general solution for TE modes with Ez=0 is obtained from (¥)

At y=0, Ex=0 which leads to C=D

At x=0, Ey=0 which leads to A=B

TE Modes


TE Modes

The general solution for TE modes with Ez=0 is

cos coszj zz o x yH H e x y

2 2 sin coszj zxy o x y

z

jE H e x y

2 2 cos sinzy j zx o x y

z

jE H e x y

At x=a, Ey=0 which leads to

At y=b, Ex=0 which leads to

xma

ynb

(§)


2 2 2 2z x z

The dispersion relation is obtained by placing (§) in (¥)

The guidance condition is

Dispersion Relation

2 22 2z

m na b

2 22

zm na b

2 22 m n

a b

(23)

(24)

(25)

(26)


Guidance Condition

2 212c

m nfa b

or f > fc where fc is the cutoff frequency of the TEmn modegiven by the relation

The TEmn mode will not propagate unless f is greater than fc.

Obviously, different modes will have different cutoff frequencies.


TM Mode

The transverse magnetic modes for a general waveguide are obtainedby assuming Hz =0. By duality with the TE modes, we have

2 22 2

2 2z z

z zE E Ex y

y yx xz j y j yj x j xj zzE e Ae Be Ce De


TM Mode

xma

ynb

The boundary conditions are

At y=b, Ez=0 which leads to

At x=0, Ez=0 which leads to A=-B

At x=a, Ez=0 which leads to

At y=0, Ez=0 which leads to C=-D


NOTE: THE DISPERSION RELATION, GUIDANCE CONDITION AND CUTOFF EQUATIONS FOR A RECTANGULAR WAVEGUIDE ARE THE SAME FOR TE AND TM MODES.

so that the generating equation for the TMmn modes is

TM and TE Modes

sin sinzj zz o x yE E e x y

For additional information on the field equations see Rao (6th Edition), page 607, Table 9.1.


TE and TM Modes

There is no TE00 mode

There are no TMm0 or TM0n modes

The first TE mode is the TE10 mode

The first TM mode is the TM11 mode


Impedance of a Waveguide

y xgTE

x y z

E EH H

2 22 1

4cm nfa b

2

21gTE

cff

For a TE mode, we define the transverse impedance as

From the relationship for z and using

we get

where is th intrinsic impedance


Impedance of a Waveguide

2

21 cgTM

ff

Analogously, for TM modes, it can be shown that


Power Flow in a Waveguide

221 ˆRe sin

2 2o zE x

a

*P E× H z

22

0 0sin

2a b o zE xPower dxdy

a

10

2 2

4 4o oz

gTE

E Eab abPower

TE10 ModeThe time-average Poynting vector for the TE10 mode in arectangular waveguide is given by

The time-average power flow in a waveguide is proportional to its cross-section area.


2 22 2

2 2z z

z zH H Hx y

2 22 2

2 2z z

z zE E Ex y

2

2 22 2

2 2 2

1 1 0

tr z

z z zz

E

E E E Er r r r

For a waveguide with arbitrary cross section, it is known that

TE Modes

TM Modes

We first assume TM modes in cylindrical coordinates:

Circular Waveguide ‐ Fields

See Reference [6].

(1)

(2)

zj


,zE r f r g

22 2

2

1r d df d gr h rf dr dr g d

2 2 2h

Solution will be in the form

Which after substitution gives

where

For equality in (3) to hold, both sides must be equal to the same constant say n2 where n is an integer in view of the azimuthal symmetry since the fields must be periodic in .

Circular Waveguide – TM Modes

(3)


22

2 0d g n gd

2 22

2 2

1 0d f df nh fdr r dr r

1 2cos sing C n C n

3 4n nf r C J hr C Y hr

Solution of (4) is of the form

(5) is Bessel’s equation and has solution

(4)

(5)

Jn and Yn are the nth order Bessel functions of the first and second kinds respectively


(6)

(7)


Bessel Functions of the First Kind

2

0

1 / 2! 1

r n r

nr

xJ x

r n r

1 !n n


3 1 2, cos sinz nE r C J hr C n C n

, cosz n nE r C J hr n

, 0zE r for r a

0nJ ha

Yn has singularity at 0 and must consequently be discarded C4 = 0. The general solution then becomes

Since the origin for is arbitrary, the expression can be written as:

where Cn is a constant. The boundary condition Etan = 0 requires that

Solution exists for only discrete values of h such that



nl

nlTM

tha

0 1 21 2.405 3.832 5.1362 5.520 7.016 8.4173 8.654 13.323 11.620

hamust be a root of the nth order Bessel function. If we assume that tnl is the lth root of Jn, we can define a set of eigenvalues hnl for the TM modes so that:

Each choice of n and lspecifies a particular solution or mode

ln

lth root of Jn(.)=0

n is related to the number of circumferential variations and l describes the number of radial variations of the field.



1/222

nl

nlTM

ta

2cTMnl

nl

at 2

nlcTMnl

tfa

The propagation constant of the nlth propagating TM mode is:

The propagation occurs for < cTMnl or f > fcTMnl where the cutoff frequency and wavelength can be found from = 0 as:

The other field components can be obtained from Ez

cos nlj znlz n n

tE C J r n ea



Circular Waveguide – TE ModesThe solutions for the TE modes can be found in a similar manner except that we solve for Hz(r,) to get:

, cosz n nH r C J hr n

To apply the boundary condition Etan = 0, we require

zHr

to be 0 at r = a

For this, we need the zeros of Jn’(u) given by snl. The propagation constant, cutoff frequency and wavelength have the same expressions as in the TM case with tnl snl.

ˆ 0ztr z

Hn H at r ar

We must have


1/222

nl

nlTE

sa

The propagation constant of the nlth propagating TE mode is:

0 1 21 3.832 1.841 3.0542 7.016 5.331 6.7063 10.173 8.536 9.969

lth root of Jn‘(.)=0

ln From the tables, it can

be seen that the lowest cutoff frequency is the TE11 mode.

cos nlj znlz n n

sH C J r n ea

and for TE modes,

Circular Waveguide – TE Modes


Circular Waveguide – TE & TM Modes

See Reference [6].


TE11 Mode in Circular Waveguide

EHSee Reference [1].


EH

TE11

TM11

Modes in Circular Waveguide

See Reference [1].


11

1.84122cTE

cfa

Example: Circular Waveguide Design

Design an air‐filled circular waveguide such that only the dominant mode will propagate over a bandwidth of 10 GHz.

Solution: the cutoff frequency of the TE11 mode is the lower bound of the bandwidth.

The next mode is the TM01 with cutoff frequency:

01

2.40492cTM

cfa


Example: Circular Waveguide Design

01 11

2.4049 1.8412 102cTM cTE

cBW f f GHza

The BW is the difference between these two frequencies

From which we find a = 0.269 cm

11 1132.7 42.76cTE cTMf GHz and f GHz

So that


Coaxial Waveguide

• Most common two-conductor transmission system• Dielectric filling in most microwave applications is

polyethylene or Teflon


Coaxial Waveguide – TEM Mode

• Two-conductor system Dominant mode is TEM• Tangential E-field and normal H field must be 0 in

conductor surfaces

0 and 0 at ,rE H r a b



ˆˆ , and ,rE rE r z H H r z

o or r

Hj E j H r j E r

z

1 10 0o

oH HH H r

r r r r

TEM solution can exist only with

with no dependence because of azimuthal symmetry

we get

Where propagation in z direction is assumed.



ˆ j zoH er

H ˆ j zoHr er

E

We get

where Ho is a constant. No cutoff condition for TEM mode.

ln / j zoV z H b a e

2 j zoI z H e

ln( / )2ob aZ

The voltage between the two conductors is given by

The current in the inner conductor is given by

The characteristic impedance Zo is thus given by


3 4, cosoz n nE r C J hr C Y hr n

' '3 4, coso

z n nH r C J hr C Y hr n

, 0 for TM modeszE r

0 for TE modeszHr

Coaxial Waveguide – TE and TM ModesTE and TM modes may also exist in addition to TEM. In a coaxial line, they are generally undesirable.

For TM modes, we have:

For TE modes, we have:

With boundary conditions at r =a, b of


for TM modesn n n nJ ha Y hb J hb Y ha

' ' ' ' for TE modesn n n nJ ha Y hb J hb Y ha

Coaxial Waveguide – TE and TM Modes

These conditions lead to

Solutions of these transcendental equations determine the eigenvalues of h for given a, b. As in the circular waveguide case, the modes for coaxial waveguide are denoted TEnl and TMnl.


The mode with the lowest cutoff frequency is the TE11mode for which the eigenvalue h is approximated as:

2ha b

11 11

2 1andc ca b fh a b

The cutoff frequency and cutoff wavelength are given by




TM01

See Reference [3].


[1]. C. S. Lee, S. W. Lee, and S. L. Chuang, "Plot of modal field distribution in rectangular and circular waveguides", IEEE Trans. Microwave Theory and Techniques, 33(3), pp. 271-274, March 1985.

[2]. J. H. Bryant, "Coaxial transmission lines, related two-conductor transmission lines, connectors, and components: A U.S. historical perspective", IEEE Trans. Microwave Theory and Techniques, 32(9), pp. 970-983, September 1984.

[3]. H. A. Atwater, "Introduction to Microwave Theory", p. 76, McGraw-Hill, New York, 1962.

[4]. N. Marcuvitz, "Waveguide Handbook", IEEE Press, Piscataway, New Jersey, 1986.

[5]. S. Ramo, J. R. Whinnery, and T. Van Duzer, "Fields and Waves in Communication Electronics", John Wiley & Sons, New York, 1994.

[6]. U. S. Inan and A. S. Inan, "Electromagnetice Waves", Prentice Hall, 2000.

References

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ECE 546 Lecture 03 Waveguides - University of Illinois