DC-1

DIRECTIONAL COUPLERS

I. Transmission-line couplers (proximity couplers) in square/rectangular coaxial technology

Couplers are used whenever power transfer from a primary guide to a secondary guide is required. If coupling takes place in a preferred direction, then the component is referred to as directional coupler.Three basic coupler configurations are employed in current waveguide technology:

II. Aperture couplers

H-plane (narrow wall) E-plane

(broad wall)

III. Branch-guide couplers

Rectangular waveguide (E-plane) Square/rectangular coaxial waveguide

DC-2

from which the four-port scattering coefficients are obtained

The transmission and reflection coefficients are given by

In order to permit synthesis techniques known from transformers or filters to be applied, the four-port coupler must theoretically be reduced to a two-port network. This is achieved by assuming symmetry and introducing even-mode and odd-mode excitation

For a lossless coupling section, the normalized ABCD matrix reads

,, ,

, , 0

, , 0, ,

,

cos sin

sin cos

e oe o e o

e o e o

e o e oe o e o

e o

ZL j L

A B ZC D Zj L L

Z

1. Basic ConceptsA. General Single-Section Coupler

,

,

0

even/odd-mode impedances

even/odd-mode propagation constants

impedance of connected lineslength of coupling section

e o

e o

Z

ZL

,, , , ,

, , , ,,

, , , ,

2e o

e o e o e o e o

e o e o e o e oe o

e o e o e o e o

TA D B C

A D B C

A D B C

11 21

41 31

1 12 21 12 2

e o e o

e o e o

S S T T

S S T T

All other S parameters follow from the symmetry condition.

DC-3

Equalling real and imaginary parts and neglecting the trivial solution for zero coupling (Z0e = Z0o ), the conditions for a contra-directional coupler are those of TEM and quasi-TEM transmission-line couplers.

The phase of the coupled port leads that of the through port by 90 degrees, independent of the coupling factor

In contra-directional couplers, a certain amount of power incident at port 1 is coupled to port 4. Ideally, port 3 is isolated (S31 =0) which leades to Te = To or

B. Contra-Directional Couplers (TEM couplers)

0 0 0 0

0 0 0 02cos sin 2cos sine o

e e o oe o

Z Z Z ZL j L L j LZ Z Z Z

20 0 0e o e oZ Z Z

Note that this condition also leads to e = -o and, therefore, S41 is maximum and S11 =0 independent of frequency.

With the midband voltage coupling factor

0 0

0 0

e o

e o

Z ZcZ Z

the through and coupled ports are

2

21 2

41 2

1

1 cos sin

sin

1 cos sin

cSc L j L

jc LS

c L j L

41221

sin

1

c LS jS c

The section length L is determined by which leads to

2 241S c

2 4L

for maximum power transfer from port 1 to port 4.

DC-4

A co-directional coupler transfers power from port 1 to port 3 while, ideally, port 4 is isolated. The latter condition requires that e =o which also leads to a maximum for S11 . Therefore, co-directional couplers are not ideally matched.The two conditions S41 =0 and S31 0 lead towhich are usually satisfied in waveguide couplers.

C. Co-Directional Couplers (waveguide couplers)

0 0e o e oZ Z

The S parameters of the through and coupled ports are

or in terms of power relations

21

31

1 exp exp21 exp exp2

e o

e o

S j L j L

S j L j L

2 221

2 231

cos2

sin2

e o

e o

LS

LS

Of specific interest is the case where the argument is /2, as the entire power is coupled to port 3. The related section length is

0dBe o

L

Accordingly, for other coupling values, e.g., for 3dB

32

dBe o

L

The possibility of realizing tight coupling is one of the advantages of waveguide couplers.

The phase difference of 90 degrees is maintained independent of the coupling value.

31

21tan

2e oS LjS

DC-5

A small-aperture coupler is essentially a co- directional waveguide coupler, but coupling is localized and confined to the apertures. Small- aperture couplers are characterized by equivalent circuits.If a normalized series impedance jxL = jXL /Z0, is assumed, then the normalized even- and odd- mode ABCD matrices are

D. Small-Aperture Couplers

11 41

21

31

2 12

2 1

2 1

L

L

L

L

L

L

jxS Sjx

jxSjx

jxSjx

Even-mode odd-mode equivalent circuits

Since the imaginary parts are usually small, the phase difference between the two output ports is 90 degrees. Good input return loss and isolation is obtained simultaneously.These conditions are usually maintained for multi- aperture arrangements. If a sufficient number of holes is used, the power can be made to alternate between the two output ports (c.f. Waveguide coupler).

1 2 1 0,

0 1 0 1e e o oL

e e o o

A B A Bj xC D C D

Following p. DC-2, the S parameters are obtained as

DC-6

2. Design of TEM-Line CouplersThe TEM-line coupler can be designed from a transformer prototype. However, since the coupler is contra-directional, the prototype transformer’s reflection coefficient (coupling factor) must be nearly constant over the band of operation. The ideal coupler will be ideally matched.

For a specified coupling C in dB,

The related transformer ratio of the prototype is

The normalized bandwidth of the coupler, wc , is related to that of the transformer by wq = 2wc . The maximum transformer VSWR for the contra- directional coupler is

2010CdBc

11

csc

max 1 1s c s

With these input parameters, the prototype’s VSWR’s, vn , are calculated. The related impedances are

and determine the individual coupling factors as1 0with 1n n nz v z z

2

211

nn

n

zcz

For a given feed-line impedance Z0 , the individual even- and odd-mode impedances are obtained as

0 01 1,1 1

n nen on

n n

c cZ Zc c

and the length of every section is a quarter- wavelength.

Note: Closed-form expressions or numerical techniques must be used to determine the cross- section dimensions for given Z0e and Z0o .

Cross sections of a square-coaxial coupler

Feed line Coupling section

DC-7

Example: 10 dB coupler, 3.6 - 4.1 GHz, square coaxial design

TEM design and analysis (solid lines); FEM analysis including housing discontinuities (symbols).

For stronger coupling or for improved performance, optimize and, if necessary, include compensating elements in field solver.

Note: This design method is valid only for TEM couplers with an odd number of sections. For even numbers, see Levy, Trans MTT, July 1963.

- NOT used in square coaxial technology

- branch-guide couplers preferred

DC-8

For multiple apertures separated by waveguide sections, cascade ABCD matrix of first aperture and normalized ABCD matrix of empty waveguide; cascade ABCD matrix of second aperture, etc.Obtain S parameters from p. DC-2.

3. Analysis and Design of Aperture Couplers

Coupling apertures in narrow and broad walls

Equivalent circuit representations of even and odd modes.

g are 0 are the guided and free-space wavelengths, respectively. Z0 and Y0 are the wave immittances of the fundamental TE10 mode. Pmx,z and Pey are magnetic and electric polarizability functions which depend on frequency and aperture dimensions.

The normalized ABCD matrices are

2

0

23

0

Series inductance for coupling by (inductive):4 sin

Shunt susceptance for coupling by (inductive):

cos

Shunt susceptance for coupling by E (capaci

x

x mxx

g

z

g mzxz

y

HX P xxZ ab a

HPB xb

Y aa b

22

0 0

tive):

4siny g ey

yB P xbY aab

1 20 1

1 0

2 1

e e x

e e

o o

o o y z

A B j xC D

A BC D j b b

even

odd

DC-9

which strictly hold only for very small holes (a/t<5) but yield good results also for larger apertures.

A. Circular Apertures

, 3, , ,

,

tan2

exp

2

cm em e m e m e

cm e

ff

P a A tff

The first term accounts for the aperture resonance (cutoff) frequencies of the TE11 or TM01 modes

11 01

1.841 2.405,2 2cm cTE ce cTM

c cf f f fa a

m,e are the attenuation constants

2 2, ,

2m e cm ef f

c

and Am,e are correction factors

1.0064 0.0819 , 1.0103 0.0579m ea aA At t

The general form of the polarizability function for circular apertures of radius a and thickness t is

Analysis Example: 6 dual-hole aperture coupler [Matthaei, Young, Jones]

DC-10

With these approximations, the correction factors Am,e can be made constant: , 3

, , , ,

,

tan2

(0) exp

2

cm em e m e m e m e

cm e

ff

P R a A tff

The first term accounts for the aperture resonance (cutoff) frequencies of the TE10 or TM11 modes

10 11 2 21 1,

2 2cm cTE ce cTMc cf f f fa a b

2 2, ,

2m e cm ef f

c

Rm,e (0) are dimensionless quantities associated with the field distributions of the TE10 or TM11 modes. It is assumed that the tangential magnetic field is parallel to the a dimension.

0.8

0.16695 0.0161 , 0.5(0)

0.1939 0.061 , 0.5m

b ba aR

b ba a

The polarizability function for a rectangular aperture of width a, height b and thickness t is

B. Rectangular Apertures

1.01095, 0.45m eA A

2,

(0) 0.0925 0.0209 ,,

e

b b aaR x x xa a bb

Analysis Example: H-plane two-hole 20dB coupler

Equivalent circuits Mode matching

DC-11

1

1

1 2 2arctan arctan2

n nn

nn n

b vv

b b

Coupler bandwidth: 1 2

1 22 g g

cg g

w

2 1 sinarcsin( ) ,1 sin

N

c sN

To determine the number of transformer sections, i.e. coupling apertures, the coupling factor c is related to the transformer ratio s by 2n z x yb b x b

The design procedure requires the normalized lumped elements to be known. They are calculated from the VSWR’s of a transformer prototype.

Specify width a, height b, thickness t, frequency of operaton (f1 to f2 ), voltage coupling factor c and input reflection coefficient r.

C. Design Procedure

Transformer bandwidth: 2q cw w

Max input VSWR: max11

rsr

Once the number of transformer sections and related VSWR’s, vn , are computed, the normalized susceptances bn and associated electrical lengths n are calculated from

A search algorithm is now required to determine the dimensions of the normalized susceptances at midband wavelength g0

Note: Only one parameter is varied during this search algorithm:- the radius an of a circular aperture- the width an of a square aperture- the width an of a rectangular aperture of given

height.

DC-12

Example: 15-dB, 7-slot narrow-wall coupler with aperture heights identical to the waveguide height (40 dB return loss).

Dashed lines: Equivalent circuit analysisSolid lines: Mode-matching analysis

Dashed lines: Equivalent circuit analysisSolid lines: Mode-matching analysis

Note: Agreement is good for weak coupling but deteriorates for tight coupling.

Example: 3-dB, 10-slot narrow-wall coupler with aperture heights identical to the waveguide height (40 dB return loss).

DC-13

Example: 10-dB and 3-dB Ku-band designs with square apertures (40 dB return loss).

10-dB design

3-dB design

Note: Again agreement is good for weak coupling but deteriorates for tight coupling. The spikes in the performance of the 3-dB design are due to higher-order mode propagation. This must be included in a final optimization using field-theory-based codes. Higher-order mode propagation may limit the achievable bandwidth of the coupler.

DC-14

Example: 10-dB and 3-dB Ku-band designs with circular apertures (40 dB return loss).

10-dB design

3-dB design

Conclusion: The design procedure works well for weak coupling (>10 dB). For tight coupling, it yields good initial designs for further optimization.

Radii had to be reduced, because the design produced overlapping apertures.

DC-15

4. Branch-Guide Couplers

According to even- and odd-mode analysis, the symmetry plane is placed half way between the upper and lower impedance (admittance) row. Therefore, jXn and jBn are the immittances seen into the n-th branch with an open or short at /4.

Since the branches are spaced /2 or g0 /4,

1 1

1 1

Impedance notation: ,

Admittance notation: ,

n nn n

n n

n nn n

n n

X Zu wZ ZB Yu w

Y Y

All impedances and admittances are normalized to those of the connected waveguides. Therefore,

Branch-guide coupler notation and equivalent junction circuits for impedances and admittances.

1 2n n

Design:The design follows the VSWR‘s calculated from the transformer prototype.

Specify width a, height b, frequency of operaton (f1 to f2 ), voltage coupling factor c and input reflection coefficient r. Then the prototype parameters are:

Coupler bandwidth: 1 2

1 22 g g

cg g

w

2

0.393 4

0.028 0.15 0.462, 1 3.5

0.474 0.12 , 3.5s

s s s

e s

cq

ww

max max max1 1, , 1 11 1c c

c rs s s c sc r

Transformer bandwidth: with

DC-16

For these input parameters, the prototype‘s junction VSWR’s vn and the number N of transformer sections are computed. The number of branches in the coupler is N+1. The junction reflection coefficients n are obtained from 1 1n n n

Computation of un and wn

N odd (even number N+1 of branches):

2

2 2 2 22

12 2

1. Set 1 / 2 and4

2. For 1, solve for and to simultaneously satisfy

12 1, 1 2 0tan 21

213. Calculate arctan ,2 21

4. Set : 1 and repea

n

n n n

n n nn n n n n

nn

nn n n

n n

n N

w u w

u wu w w u w

uu w

n n

t from step 2 until 0.n

N even (odd number N+1 branches):

1

1. Set 1 and 12

12. Compute

1 2arctan ,2 2

3. Set : 1 and repeat procedure for odd from step 2 until 0.

n

n nn

n n n nn

Nn w

u vv

un n N

n

DC-17

For Z0 and Y0 given, and all un and wn known, the individual immittances of the branch guide coupler can be determined by successive denormalization. In E-plane waveguide technology, the impedance of the fundamental mode is directly proportional to the height. In other technologies, e.g., square coax, the immittances must be calculated using numerical techniques or closed-form expressions. The center-to-center spacing between branches as well as between the upper and lower waveguide trace is g0 /4.

Analysis Example: 6-dB, five-branch E-plane branch-guide coupler with two-step impedance transformers at all ports [Matthaei, Young, Jones]

Dashed and solid lines: Mode matchingSymbols: Measurements

DC-18

Design Example: 4 dB

0.15 dB coupling, 28 dB return loss, 10.9 – 12.8 GHz, WR75 waveguide.

Repeated synthesis and mode-matching analysis: specs NOT met.

Synthesis, mode-matching optimization, measurements: specs MET.

Same example, but 8.34 dB

0.15 dB coupling.

Repeated synthesis and mode-matching analysis: specs MET.

DC-19

Design Example: 3 dB coupling in square coaxial technology. Initial design by lumped-element approach; re-optimized using TLM analysis. Comparison with measurements

DC-20

Periodic Six-Port Branch-Guide Coupler:

The periodic branch-guide coupler has identical impedances in the through-sections. Under such conditions, a lumped-element synthesis procedure can be derived [Kühn, Schmiedel, Waugh, EuMC 1986].

Mode-matching analysis of synthesized / optimized

dimensions

Note: Although the synthesized structure, in this specific case, meets all specifications, the component can still be improved by optimization.

DC-21

Asymmetric Periodic Six-Port Branch-Guide Coupler:

[Esteban, Rebollar, IEEE Trans MTT, Oct. 1991].

Initial design using synthesis for Periodic Six-Port Branch-Guide Coupler.

Mode-matching analysis and optimization to obtain asymmetric configuration.

S11

S22

S33

S42

S52

S62

1 4

3

2 5

6

DC-22

Asymmetric Periodic Eight-Port Branch-Guide Coupler:

[Carle, MTT-S Digest, June 1991].

Mode-matching analysis (solid lines); measurements (dashed lines)

S33 S31 S32

DC-23Mode-matching analysis (solid lines); measurements (dashed lines)

S34S35 S36

S37 S38

DC-24

5. E-Plane Couplers With Multi-Mode Coupling Section

At millimeter-wave frequencies, branch-guide couplers in waveguide technology are difficult to fabricate. A single large coupling section was demonstrated to be easier for the manufacturing process and having better performance or smaller size than traditional branch-guide couplers of a similar bandwidth.

7.5 GHz / WR1127.5 GHz / WR112

77 GHz / WR1277 GHz / WR12

26 GHz / WR3426 GHz / WR34The single coupling section operates with three modes:TE10 , TE11 and TM11(Rosenberg, Speldrich, IEEE MTT-S,

2000)

Triple-mode coupling section:

DC-25

Return Loss and Isolation Transmission and Coupling

-50

-25

0

36 38 40Frequency, GHz

s11,

s21

, d

B

-3.5

-3.0

-2.5

36 38 40Frequency, GHz

s31,

s41

, d

B

Example: Measured Response of 3 dB / 38 GHz Multi-Mode CouplerUsable bandwidth: 37 GHz – 39.5 GHz 6.5 percent

DC-26

The single coupling section operates with two modes: TE10 , TEM (Beyer, Rosenberg, IEEE MTT-S, 2005) Dual-mode coupling section:

Rectangular coax

Rectangular coax

Usable bandwidth: 10.5 GHz – 13.7 GHz 26 percent

Transmission and CouplingIsolation and Return Loss