Lecture 27. Rectangular Metal Waveguides

ECE 303 – Fall 2006 – Farhan Rana – Cornell University

Lecture 27

Rectangular Metal Waveguides

In this lecture you will learn: • Rectangular metal waveguides

•TE and TM guided modes in rectangular metal waveguides

d z x o

µ

ε

( )

r

{

K

K

r

,

3

,

2

,

1

sin

ˆ

⎟

=

⎠

⎞

⎜

⎝

⎛

=

x

e

−

m

d

m

E

y

r

E

_o

π

jkzz m = 1 m = 2 E_y E_y

Parallel Plate Metal Waveguides

TE Modes: 2 2

⎟

⎠

⎞

⎜

⎝

⎛

−

=

d

m

k

_z

ω

µ

_o

ε

π

Dispersion relation: d z o

µ

ε

m = 1 Hy _{m = 2} Hy x TM Modes: Dispersion relation:

( )

r

{

K

K

r

,

3

,

2

,

1

,

0

cos

ˆ

⎟

=

⎠

⎞

⎜

⎝

⎛

=

x

e

−

m

d

m

H

y

r

H

_o

π

jkzz 2 2

⎟

⎠

⎞

⎜

⎝

⎛

−

=

m

k

_z

ω

µ

_o

ε

π

ECE 303 – Fall 2006 – Farhan Rana – Cornell University

Rectangular Metal Waveguides

a b

x y

z

Somewhat like a parallel plate metal waveguide that is closed by metal walls on the remaining two sides

o

µ

ε

Rectangular Metal Waveguides: TE Guided Modes - I

The electric field of the guided wave will satisfy the complex wave equations:

( )

r

j

H

( )

r

E

o

r

r

r

r

ω

µ

−

=

×

∇

( )

r

j

E

( )

r

H

r

r

=

ω

ε

r

r

×

∇

( )

r

E

( )

r

E

o

r

r

r

r

ω

2

µ

ε

2

=

−

∇

( )

r

H

( )

r

H

r

r

ω

2

µ

_o

ε

r

r

2

=

−

∇

We look for solutions of the equation:

( )

( )

( )

r

E

( )

r

E

z

y

x

r

E

r

E

o o

r

r

r

r

r

r

r

r

ε

µ

ω

ε

µ

ω

2 2 2 2 2 2 2 2 2

−

=

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂

+

∂

∂

+

∂

∂

⇒

−

=

∇

b a y x z

( )

[

( ) ( )

( ) ( )

]

jk z o

y

A

x

B

y

x

C

x

D

y

e

z

E

r

E

r

r

=

ˆ

+

ˆ

−

Try separation of variables (with E-field pointing in directions transverse to the direction of propagation)

Where:

( ) ( )

x

C

x

(

k

x

)

(

k

x

)

A

,

∝

sin

x

or

cos

x

( ) ( )

y

D

y

( )

k

y

( )

k

y

B

,

∝

sin

_y

or

cos

_y

ε

µ

ω

_o z y x

k

k

k

2

+

2

+

2

=

2

ECE 303 – Fall 2006 – Farhan Rana – Cornell University

Rectangular Metal Waveguides: TE Guided Modes - II

b a y x z

( )

[

( ) ( )

( ) ( )

]

jk z o

y

A

x

B

y

x

C

x

D

y

e

z

E

r

E

r

r

=

ˆ

+

ˆ

− Boundary condition:

Components of E-field parallel to the metal walls must go to zero at the metal walls:

0

, 0 =

=

= y b y x

E

₀

, 0 =

=

= x a x y

E

This implies:

( )

x

(

k

x

)

A

=

sin

x

( )

y

( )

k

y

D

=

sin

_y

K

K

,

3

,

2

1

,

0

:

where

m

,

a

m

k

_x

=

π

=

K

K

,

3

,

2

1

,

0

:

where

n

,

b

n

k

_y

=

π

=

So we have:

( )

[

(

)

( )

( )

( )

]

jk z y x o

y

k

x

B

y

x

C

x

k

y

e

z

E

r

E

r

r

=

ˆ

sin

+

ˆ

sin

− b a y x z

Rectangular Metal Waveguides: TE Guided Modes - III

Another Boundary Condition:

Components of H-field normal to the metal walls must go to zero at the metal walls:

0

, 0 =

=

= x a x x

H

H

y_y=0,y=b

=

0

Turns out that the above solution form already satisfies the H-field boundary conditions What do we do now?

We use something that we never had to use before in the context of guided waves:

( )

0

.

=

∇

ε

E

r

r

r

Gauss’ Law:

Plugging in the above solution form in Gauss’ Law gives:

(

)

( )

( )

( )

k

y

x

x

C

y

y

B

x

k

x

sin

y

sin

∂

∂

−

=

∂

∂

( )

[

(

)

( )

( )

( )

]

jk z y x o

y

k

x

B

y

x

C

x

k

y

e

z

E

r

E

r

r

=

ˆ

sin

+

ˆ

sin

−

( )

( )

k

y

k

k

y

B

=

x

cos

y

( )

(

k

x

)

k

x

C

=

−

y

cos

_x Gauss’ Law can only be satisfied if:

ε

µ

ω

_o

ECE 303 – Fall 2006 – Farhan Rana – Cornell University b a y x z

Rectangular Metal Waveguides: TE Guided Modes - IV

( )

(

)

( )

(

)

( )

jk z y x y y x x o

k

x

k

y

e

z

k

k

x

y

k

x

k

k

k

y

E

r

E

⎥

−

⎦

⎤

⎢

⎣

⎡

−

=

ˆ

sin

cos

ˆ

cos

sin

r

r

Finally, the solution is:

K

K

,

3

,

2

1

,

0

:

where

m

,

a

m

k

_x

=

π

=

K

K

,

3

,

2

1

,

0

:

where

n

,

b

n

k

_y

=

π

=

Where:

These modes are called TEmnmodes

By convention, the first subscript in “TE_mn”is related to the component of k-vector that is along the longer transverse dimension of the rectangular waveguide 2 2 2 2 2 2 2

⎟

⎠

⎞

⎜

⎝

⎛

−

⎟

⎠

⎞

⎜

⎝

⎛

−

=

⇒

=

+

+

b

n

a

m

k

k

k

k

o z o z y x

π

π

ε

µ

ω

ε

µ

ω

NOTE:The TE₀₀mode does not exist (i.e. it corresponds to field being trivially zero everywhere)

TE Guided Modes – Cut-off Frequency

b a y x z

( )

(

)

( )

(

)

( )

jk z y x y y x x o

k

x

k

y

e

z

k

k

x

y

k

x

k

k

k

y

E

r

E

⎥

−

⎦

⎤

⎢

⎣

⎡

−

=

ˆ

sin

cos

ˆ

cos

sin

r

r

{

m

0 ,

,

1

2

,

3

,

K

K

a

m

k

_x

=

π

=

{

n

0 ,

,

1

2

,

3

,

K

K

b

n

k

_y

=

π

=

2 2 2 2 2 2 2

⎟

⎠

⎞

⎜

⎝

⎛

−

⎟

⎠

⎞

⎜

⎝

⎛

−

=

⇒

=

+

+

b

n

a

m

k

k

k

k

o z o z y x

π

π

ε

µ

ω

ε

µ

ω

The cut-off frequency ωmnfor the TEmnmode is:

ε

µ

π

π

ω

o mn

b

n

a

m

2 2

⎟

⎠

⎞

⎜

⎝

⎛

+

⎟

⎠

⎞

⎜

⎝

⎛

=

If the frequency ω is less than the cut-off frequency then k_zbecomes entirely imaginary and the mode does not propagate (but decays exponentially with distance)

ECE 303 – Fall 2006 – Farhan Rana – Cornell University

TE Guided Modes – Dispersion Relation

b a y x z

( )

(

)

( )

(

)

( )

jk z y x y y x x o

k

x

k

y

e

z

k

k

x

y

k

x

k

k

k

y

E

r

E

⎥

−

⎦

⎤

⎢

⎣

⎡

−

=

ˆ

sin

cos

ˆ

cos

sin

r

r

{

m

0 ,

,

1

2

,

3

,

K

K

a

m

k

_x

=

π

=

{

n

0 ,

,

1

2

,

3

,

K

K

b

n

k

_y

=

π

=

2 2 2 2 2 2 2

⎟

⎠

⎞

⎜

⎝

⎛

−

⎟

⎠

⎞

⎜

⎝

⎛

−

=

⇒

=

+

+

b

n

a

m

k

k

k

k

o z o z y x

π

π

ε

µ

ω

ε

µ

ω

z

k

ω

plane wave dispersion

relation: kz=ω µoε 10

ω

ω

01 TE₁₀mode dispersion relation TE₀₁mode dispersion relation

TE Guided Modes – Field Profile for TE

₁₀

Mode

( )

(

)

jk z x o

k

x

e

z

E

y

r

E

r

r

=

ˆ

sin

−

⎩

⎨

⎧

=

a

k

_x

π

b a y x z b E y z x π / k_z z π / k_z E H a x y y H b a x

ECE 303 – Fall 2006 – Farhan Rana – Cornell University

Rectangular Metal Waveguides: TM Guided Modes - I

The electric field of the guided wave will satisfy the complex wave equations:

( )

r

j

H

( )

r

E

r

r

=

−

ω

µ

_o

r

r

×

∇

( )

r

j

E

( )

r

H

r

r

=

ω

ε

r

r

×

∇

( )

r

E

( )

r

E

r

r

ω

2

µ

_o

ε

r

r

2

=

−

∇

( )

r

H

( )

r

H

r

r

ω

2

µ

_o

ε

r

r

2

=

−

∇

We look for solutions of the equation:

( )

( )

( )

r

H

( )

r

H

z

y

x

r

H

r

H

o o

r

r

r

r

r

r

r

r

ε

µ

ω

ε

µ

ω

2 2 2 2 2 2 2 2 2

−

=

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂

+

∂

∂

+

∂

∂

⇒

−

=

∇

b a y x z

( )

[

( ) ( )

( ) ( )

]

jk z o

y

A

x

B

y

x

C

x

D

y

e

z

H

r

H

r

r

=

ˆ

+

ˆ

−

Try separation of variables (with H-field pointing in directions transverse to the direction of propagation)

Where:

( ) ( )

x

C

x

(

k

x

)

(

k

x

)

A

,

∝

sin

_x

or

cos

_x

( ) ( )

y

D

y

( )

k

y

( )

k

y

B

,

∝

sin

_y

or

cos

_y

ε

µ

ω

_o z y x

k

k

k

2

+

2

+

2

=

2

Rectangular Metal Waveguides: TM Guided Modes - II

b a y x z Boundary condition:

Components of H-field normal to the metal walls must go to zero at the metal walls:

This implies:

( )

y

( )

k

y

B

=

sin

_y

where

:

n

₀

,

1

,

2

,

3

,

K

K

b

n

k

_y

=

π

=

So we have:

( )

[

( )

( )

(

)

( )

]

jk z x y o

y

A

x

k

y

x

k

x

D

y

e

z

H

r

H

r

r

=

ˆ

sin

+

ˆ

sin

−

0

, 0 =

=

= x a x x

H

H

y_y=0,y=b

=

0

( )

x

(

k

x

)

C

=

sin

_x

where

:

m

0

,

1

,

2

,

3

,

K

K

a

m

k

_x

=

π

=

( )

[

( ) ( )

( ) ( )

]

jk z o

y

A

x

B

y

x

C

x

D

y

e

z

H

r

H

r

r

=

ˆ

+

ˆ

−

ECE 303 – Fall 2006 – Farhan Rana – Cornell University

( )

[

( )

( )

(

)

( )

]

jk z x y o

y

A

x

k

y

x

k

x

D

y

e

z

H

r

H

r

r

=

ˆ

sin

+

ˆ

sin

− b a y x z

Rectangular Metal Waveguides: TM Guided Modes - III

Another Boundary Condition:

Components of E-field parallel to the metal walls must go to zero at the metal walls:

Turns out that the above solution form already satisfies the E-field boundary conditions

What do we do now?

( )

0

.

=

∇

µ

_o

H

r

r

r

Gauss’ Law for H-fields:

Plugging in the above solution form in Gauss’ Law for H-fields gives:

( )

x

( )

k

y

k

D

( )

y

(

k

x

)

A

k

_y

cos

_y

=

−

_x

cos

_x

( )

(

k

x

)

k

k

x

A

=

x

cos

_x

( )

( )

k

y

k

k

y

D

=

−

y

cos

_y Gauss’ Law for H-fields can only be satisfied if:

ε

µ

ω

o

k

=

0

, 0 =

=

= y b y x

E

0

, 0 =

=

= x a x y

E

( )

(

)

( )

(

)

( )

jk z y x y y x x o

k

x

k

y

e

z

k

k

x

y

k

x

k

k

k

y

H

r

H

⎥

−

⎦

⎤

⎢

⎣

⎡

−

=

ˆ

cos

sin

ˆ

sin

cos

r

r

b a y x z

Rectangular Metal Waveguides: TM Guided Modes - IV

Finally, the solution is:

K

K

,

3

,

2

1

,

0

:

where

m

,

a

m

k

_x

=

π

=

K

K

,

3

,

2

1

,

0

:

where

n

,

b

n

k

_y

=

π

=

Where:

These modes are called TMmnmodes

By convention, the first subscript in “TM_mn”is related to the component of k-vector that is along the longer transverse dimension of the rectangular waveguide 2 2 2 2 2 2 2

⎟

⎠

⎞

⎜

⎝

⎛

−

⎟

⎠

⎞

⎜

⎝

⎛

−

=

⇒

=

+

+

b

n

a

m

k

k

k

k

o z o z y x

π

π

ε

µ

ω

ε

µ

ω

NOTE:The TM₀₀, TM_m0TM_0nmodes do not exist (i.e. they correspond to field being trivially zero everywhere)

ECE 303 – Fall 2006 – Farhan Rana – Cornell University

TM Guided Modes – Cut-off Frequency

b a y x z

{

m

1,

2

,

3

,

K

K

a

m

k

_x

=

π

=

{

n

1,

2

,

3

,

K

K

b

n

k

_y

=

π

=

2 2 2 2 2 2 2

⎟

⎠

⎞

⎜

⎝

⎛

−

⎟

⎠

⎞

⎜

⎝

⎛

−

=

⇒

=

+

+

b

n

a

m

k

k

k

k

o z o z y x

π

π

ε

µ

ω

ε

µ

ω

The cut-off frequency ωmnfor the TMmnmode is:

ε

µ

π

π

ω

o mn

b

n

a

m

2 2

⎟

⎠

⎞

⎜

⎝

⎛

+

⎟

⎠

⎞

⎜

⎝

⎛

=

If the frequency ω is less than the cut-off frequency then k_zbecomes entirely imaginary and the mode does not propagate (but decays exponentially with distance)

( )

(

)

( )

(

)

( )

jk z y x y y x x o

k

k

x

k

y

e

z

k

x

y

k

x

k

k

k

y

H

r

H

⎥

−

⎦

⎤

⎢

⎣

⎡

−

=

ˆ

cos

sin

ˆ

sin

cos

r

r

TM Guided Modes – Dispersion Relation

b a y x z

{

m

0 ,

,

1

2

,

3

,

K

K

a

m

k

_x

=

π

=

{

n

0 ,

,

1

2

,

3

,

K

K

b

n

k

_y

=

π

=

2 2 2 2 2 2 2

⎟

⎠

⎞

⎜

⎝

⎛

−

⎟

⎠

⎞

⎜

⎝

⎛

−

=

⇒

=

+

+

b

n

a

m

k

k

k

k

o z o z y x

π

π

ε

µ

ω

ε

µ

ω

z

k

ω

plane wave dispersion

relation: kz=ω µoε 11

ω

ω

₂₁ TM₁₁mode dispersion relation TM21mode dispersion relation

( )

(

)

( )

(

)

( )

jk z y x y y x x o

k

k

x

k

y

e

z

k

x

y

k

x

k

k

k

y

H

r

H

⎥

−

⎦

⎤

⎢

⎣

⎡

−

=

ˆ

cos

sin

ˆ

sin

cos

r

r

ECE 303 – Fall 2006 – Farhan Rana – Cornell University

TM Guided Modes – Field Profile for TM

₁₁

Mode

b a y x z b E y z x π / k_z E H

⎩

⎨

⎧

=

=

b

k

a

k

_x

π

_y

π

( )

(

)

( )

(

)

( )

jk z y x y y x x o

k

k

x

k

y

e

z

k

x

y

k

x

k

k

k

y

H

r

H

⎥

−

⎦

⎤

⎢

⎣

⎡

−

=

ˆ

cos

sin

ˆ

sin

cos

r

r

H y b a x a x z y π / k_z E H

Rectangular Metal Waveguides

Rectangular metal waveguides are commonly used to guide electromagnetic power when dealing with high power levels (radars, satellite and space communications, wireless/mobile base stations, etc)

They are usually made of copper – and the best are gold plated

Integrated versions are used in sub-millimeter wavelength ultrahigh speed electronics (e.g. Gunn oscillators, superconducting THz electronics, nonlinear Schottky diode mixers) operating at frequencies between 300 GHz to 1000 GHz

ECE 303 – Fall 2006 – Farhan Rana – Cornell University

Waveguides to Coax Adapters

Waveguide to Coax Adapter Waveguide to Coax Adapters

Sometimes it is necessary to transfer power between a waveguide and a coax cable

power in power in

power out _{power out}

The pin is placed at the maximum E-field location, and oriented in its direction, so that the E-field produces current in the pin The loop is placed

and oriented so that maximum magnetic flux goes through it and induces current in it (Faraday’s law)