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ELEctromagnetic DIAgnostics Lab.DIT Università di TrentoDIT - Università di Trento

Via Sommarive 14, I-38050 Trento ItaliaE-mail: [email protected] mail: [email protected]

Couplers and Hybrid Ringsp y g

Master Master DegreeDegree Electronic and TelecommunicationElectronic and TelecommunicationA.A. A.A. 20122012--20132013

Couplers and Hybrid Rings

• The waveguide directional coupler, Bethe single h l lhole coupler.

• Bethe skewed single hole coupler.• The two holes directional coupler.• The multi-holes couplerThe multi holes coupler.• The 180 Hybrid ring (rat race).

Th d t H b id i• The quadrature Hybrid ring.


Couplers and hybrid rings are four ports devices. Powerincident at port 1 is directed (coupled) with port 2 (called theincident at port 1 is directed (coupled) with port 2 (called thethrougt port) with port three the coupled port, but not withport 4 (called isolated port). Similarly power at port 2 will bep ( p ) y p pcoupled with ports 1 and 4 but not 3. Ports 1 and 4 aredecoupled as ports 2 and 3.

CouplersP1

P2

P4P3


The following slide shows two commonly used symbols for directionalcouplers, and power flow conventions.


The scattering matrix of a four ports network matched at all ports has thefollowing form:

In particular considering a lossless device we can have two differentkinds of scattering matrix the symmetric and the anti symmetrickinds of scattering matrix the symmetric and the anti-symmetricscattering matrix reported below:

symmetric anti-symmetric

Couplers and Hybrid RingsIt should be noticed that the two couplers differs only in the choice of thereference planes and that the amplitude of the two terms α and β are notindependent but they must satisfy the following relation:p y y g

Because for a lossless device the sum of the terms along the rows orcolumns must gives 1 as resultcolumns must gives 1 as result.

Couplers and Hybrid RingsThe following three quantities are generally used to characterize adirectional coupler:

dBPP

CCoupling βlog20log10 1 −===P2

dBSP

PDyDirectivit 3 log20log10 β

−===SP 144

dBSP

IIsolation 141 log20log10 −=== dS

Psolation 14

4

og0og0

Lossless Junction – The T-Junction power splitterCouplers and Hybrid Rings

The coupling factor indicates the fraction of the input power that iscoupled to the output port.p p p

The directivity is a measure of the coupler’s ability to isolate forward andbackward waves.

These quantities are then relates as:These quantities are then relates as:

dBCDI += dBCDI +=

An ideal directional coupler have an infinite directivity and isolation (S14= 0).

Waveguide coupler: the Bethe Coupler, single hole waveguide couplerg g p

The structure of the Bethe coupler is reported in the following, it is madewith two segments of waveguide connected together by means of ag g g ysingle hole.

C id i th b t t ith i id t TE d i t t 1Considering the above structure with an incident TE10 mode into port 1we are able to completely describe the different field components

Waveguide coupler: the Bethe Coupler, single hole waveguide couplerwaveguide coupler

The field components are given by the following :The field components are given by the following :

Where Z10 is the wave impedance of the TE10 mode.


The equivalent polarization currents at the aperture are:The equivalent polarization currents at the aperture are:

Considering x=s, y=b, z=0. And the amplitude of the direct and reflectedwaves we obtain:


Where is the power polarization constant from the firstWhere is the power polarization constant from the firstprevious formula we obtain the following condition useful to estimate theparameter s :

The radius of the aperture could be obtained considering the followingrelations:

Where r0 is the radius of the aperture.0 p


The coupling factor and the directivity of the single hole Bethe waveguidecoupler are given by the following relations:

Waveguide coupler: the Bethe Coupler, skewed single hole waveguide couplersingle hole waveguide coupler

The geometry of the skewed single hole Bethe waveguide coupler is givenbelow:

In this case the aperture is centered at s=a/2 being a the waveguide width.The skewed angle is used to slightly change the transverse magnetic fieldThe skewed angle is used to slightly change the transverse magnetic field.

Example 2: T-Junction power splitter designWaveguide coupler: the Bethe Coupler, skewed single hole waveguide coupler

For this geometry the amplitude of the direct and back-waves are given by

single hole waveguide coupler

the following relations:

If we set A+ =0 we obtain the following condition for the skewed angle:If we set A 10=0 we obtain the following condition for the skewed angle:

Waveguide coupler: the Bethe Coupler, skewed single hole waveguide couplersingle hole waveguide coupler

The coupling factor is given by the following relation:

The geometry of the skewed Bethe hole coupler is often a disadvantage interms of fabrication and application. The single hole Bethe couplers workwell only at the design frequency they are narrowband deviceswell only at the design frequency, they are narrowband devices.

Waveguide coupler: the two holes directional couplerThe geometry of the two holes directional coupler is reported below itconsists of two waveguide segments soldered together and two circularholes placed at a distance of quarter wavelength each otherholes placed at a distance of quarter wavelength each other.

Waveguide coupler: the two holes directional couplerA similar geometry could be designed also with planar technology(microstrip). The behavior of the device is the following, the two smallapertures spaced quarter wavelength each other couple the two waveguideapertures spaced quarter wavelength each other couple the two waveguidesections. A wave enter at port 1 is mostly transmitted at port two, but a littleamount of power is coupled trough the two apertures.

The multi-holes coupler

This is a narrowband device, because the structure is composed by twosections of rectangular waveguides moreover the two circular holes areplaced at quarter wavelength each other. In order to obtain a widebandp q gdevice we can introduce different couples of holes.

Hybrid ring: the quadrature (90°) hybrid ringQ d t h b id i 3 dB di ti l l ith 90° hQuadrature hybrid rings are 3 dB directional couplers with a 90° phasedifference at the output ports. Usually the hybrid rings are often made inmicrostrip or stripline form. As reported in the following figure:

This type of hybrids are also known as branch line couplers Other 3 dBThis type of hybrids are also known as branch line couplers. Other 3 dBcouplers, such as coupled line couplers or Lange couplers can also beused as quadrature couplers, we discuss these other devices later.

Lossy Junction – The Three resistors power splitterHybrid ring: the quadrature (90°) hybrid ring

The hybrid ring is a symmetrical device and it can be analyzedconsidering the even odd analysis. It is worth noticed that any port canbe used as input or output piort. We first starts with a circuital schema ofthe normalized branch line (normalization obtained considering thecharacteristic impedance Z0 of the transmission lines connected with thep 0device.).

Normalized circuital schema of the 90° ring


The following schema report the even decomposition:

For the even part we introduce two sources generators with the sameFor the even part we introduce two sources generators with the sameamplitude and the same phase. In this way we can split the device intotwo different circuits.

Hybrid ring: the quadrature (90°) hybrid ring

For the even analysis the lateral transmission line can be considered asytwo open stubs. In this way port s1 and 4 are connected with quarterwavelength line connected with open stubs and matched load.


The following schema report the odd decomposition:

For the odd analysis we consider two sources generators with the sameFor the odd analysis we consider two sources generators with the sameamplitude and a phase shift od 180°. Also in this case we can split thedevice into two different circuits.


The following relations summarize the amplitude of the reflected wave atthe device ports considering the even/odd mode analysis.


The analysis of the devices since it is a chain of two ports devices thebest way to analyzed it is using the ABCD matrix in particular we observeth tthat :

First the estimation of the reflection and transmission coefficients of theeven mode have to be considered. This can be done with the abovechain matrix. In particular the admittance of the shunt open circuit stub (λ/8 length ) is Y=jtan(βl)=jλ/8 length ) is Y jtan(βl) j


The ABCD parameters can be used to convert the ABCD matric into Sparameters which are equivalent to the reflection and transmission

ffi i t Thcoefficients. Thus:


Similarly for the odd mode we obtain a chain matrix that consider theconcatenation of the short circuit stub. The ABCD matrix for the odd

d i i b lmode is given below:

The reflection and transmission coefficients can be obtained converting th ABCD i t th S t i i d t bt i th fl ti dthe ABCD into the S matrix in order to obtain the reflection and transmission coefficents.


Considering the transmission and reflection coefficients for the even andodd modes we obtain the following:

01 =B As can be observed the two portsB and B the amplitudes of the0

2

1

−=jB

B B1 and B4 the amplitudes of theelectromagnetic wave at the twoports is 0 this means that the two

t t h d C id i

122

B

ports are matched- Consideringthe amplitude at ports 2 and 3 weobserve that at port B2 we obtain

023 −=

B

B half power with a 90° phase shiftwhile on port 3 we have halfpower but with a phase shift of04 =B p p180 degrees.

Hybrid ring: the quadrature (90°) hybrid ringThe scattering matrix of this device is reported in the following:

⎤⎡ −− 010 j

[ ] ⎥⎥⎤

⎢⎢⎡

−− 1001 jj

S[ ]⎥⎥⎥

⎢⎢⎢

−−=

0012 jj

S

⎥⎦

⎢⎣ −− 010 j

j

⎦⎣

Hybrid ring: the quadrature (90°) hybrid ringAn example of application of Hybrid ring is to obtain the the circularpolarization in patch antennas. In particular the following schema isusually consider:usually consider:

0Z0 / 2Z0Z

λg/4

feed

0Zg

50 Ohm load

0

λg/450 Ohm load

30

Even/Odd analysis of the Wilkinson power splitterHybrid ring: the quadrature (90°) hybrid ringThe Hybrid ring is a narrowband device since it is composed by fourquarter wavelenght transmission lines. For this reason techniques toenhance the bandwidth of such kind of device have been developed Inenhance the bandwidth of such kind of device have been developed. Inparticular multi arms hybrid ring can be considered to improve thebandwidth of the device.

Even/Odd analysis of the Wilkinson power splitterHybrid ring: the multi arms quadrature (90°) hybrid ring

First arm

Second armSMA coax connectors

Hybrid ring: the multi arms quadrature (90°) hybrid ringIn the following the design formula for the multi arms branch line arereported in particular a two arms branch line has been considered.

0Z 0Z0Z

2Z

Port 1 Port 2

λg/402Z

0)12( Z+0)12( Z+λg/4

0Z Z

gPort 4 Port 3

0Z0Z

Wilkinson power splitter with lumped elementsHybrid ring: the single/multi arms quadrature (90°) hybrid ring with lumped elements

Also the quadrature hybrid ring could be reduced in size consideringlumped elements. In particular the four quarterwave transmission linesp p qcould be simulated with a Pi-Greca networks realized with two capacitorsand one inductor:

21

ZfCeq π

= 0

2 fZLeq π

=002 Zfπ 02 fπ

This technique could be used also for low frequencies and high power.

Hybrid ring the 180° hybrid ring (rat-race)

The 180° hybrid junction is a four ports network with a 180° phase shiftbetween the two output ports. The schema of the 180° hybrid is reportedp p y pbelow:

A signal applied at port 1 will be equally split into two components with a180° phase difference at port 2 and 3 and port 4 wuill be isolated When180 phase difference at port 2 and 3, and port 4 wuill be isolated. Whenthe Hybryd operated as a combiner, with input signals applied at ports 2and 3, the sum of the inputs will be formed at port 1, while the differencet t 4 P t 1 d 4 f d t th d diff tat port 4. Port 1 and 4 are referred to as the sum and difference ports

respectively.


The scattering matrix of an ideal hybrid 180° junction is reported in thefollowing:g

A signal applied at port 1 will be equally split into two components with ag pp p q y p p180° phase difference at port 2 and 3, and port 4 wuill be isolated. Whenthe Hybryd operated as a combiner, with input signals applied at ports 2and 3, the sum of the inputs will be formed at port 1, while the differenceand 3, the sum of the inputs will be formed at port 1, while the differenceat port 4. Port 1 and 4 are referred to as the sum and difference portsrespectively.


The hybrid ring can be fabricated in several forms, waveguide microstrip.Since the geometrical structure is symmetrical we can use the even oddg yanalysis to analyze the behavior of this device. The following photoshows the geometry of a rat-race fabricated with microstrip.


The schema of the rat-race is reported below:

Now the even and odd analysis of the rat-race will be used to analyze the devicebehavior Let us first consider a unit amplitude wave incident at port 1 (the sumbehavior. Let us first consider a unit amplitude wave incident at port 1 (the sumport). At the ring junctions this wave will be divided into two components. Thesetwo components arrive in phase at ports 2 and 3, and 1port 4 is isolated. If thesignal is applied at port 4 it will be splitted at ports 2 and 3 with a phasesignal is applied at port 4 it will be splitted at ports 2 and 3 with a phasedifference of 180°. Using the even odd analysis we can decompose this case intoa superposition of two simpler circuits considering the symmetry of the structure.

Even odd analysis of the (rat-race)

For the even mode two identical power sources (same phases and amplitudes)have to be considered at ports 1 and 3, and considering the horizontal symmetrywe obtain the following schema:

is like an open circuit

Even odd analysis of the (rat-race)The horizontal line of symmetry in the middle is like an open circuit so we obtaina quarter wavelength line connected with two open stubs.


For the odd mode two power sources with a 180° phase shift have to beconsidered at ports 1 and 3, and still considering the horizontal symmetry weobtain the following schema:

is like a short circuit

Even odd analysis of the (rat-race)The horizontal line of symmetry in the middle is like a short circuit so we obtain aquarter wavelength line connected with two short circuited stubs.

Even odd analysis of the (rat-race)The amplitude of the signals at the input output ports is given by the followingrelations:

We first calculate the chain matrix for the upper or lower section of the devicethen we will transform it into the scattering matrix in order to obtain thetransmission and the reflection coefficients for the even and odd analysis.

Even odd analysis of the (rat-race)The ABCD matrices of the upper lower device section are reported in thefollowing:

After the conversion we obtain:

Even odd analysis of the (rat-race)Now consider a unit amplitude wave incident at port 4 (the differenceport). The two waves arrive at ports 2 and 3 with a net phase differenceof 180° between these portsof 180 between these ports.As in the previous case we first consider the even source

Even odd analysis of the (rat-race)Now consider we must consider the odd source s obtaining:


The amplitude of the scattered waves could be obtained considering the reflection and transmission coefficients:reflection and transmission coefficients:


This is the typical narrowband behavior of a rat race.

Wilkinson power splitter with lumped elementsQuestion: It is possible to simulate a rat-race with lumped elements ?

In this way we can obtain a more compact layout. Unfortunately it is notpossible because only four quarter wave transmission lines could be

lumped elements ?

possible because only four quarter-wave transmission lines could besimulated with a Pi-Greca networks realized with two capacitors and oneinductor:

21

ZfCeq π

= 0

2 fZLeq π

=002 Zfπ 02 fπ

Coupled Line Directional Couplers

when two unshielded transmission lines are closet th b l d b t th litogether, power can be coupled between the lines

Wh

S C12S

d

C12

C11 C22d


• C11 and C22 are the self capacitance in theabsence of the other line

• C12 is the mutual capacitance between the twoplines in the absence of the ground plane


• for the even mode, the electric field has evensymmetry and the field lines of one transmission linesymmetry and the field lines of one transmission linerepel those of the other line, therefore, C12 iseffectively open-circuitedeffectively open circuited

C C Ce = =11 22C C Ce 11 22


• the characteristic impedance for the even modeis:is:

Z LC vCoe

e e= =

1C vCe e

• for the odd mode, the electric field have an oddsymmetry about the symmetry plane and avoltage null exists between the two stripconductors

• this is effectively putting a ground plane betweenthe conductors


• the effective capacitance between either stripconductor and ground isconductor and ground is

C C C C Co = + = +11 2 12 22 2 12

• the characteristic impedance for the odd mode is

C C C C Co + +11 2 12 22 2 12

p

Z L= =

1ZC vCoo

o o= =

• the transmission lines are assumed TEM lines,this is true for stripline but only approximately truefor microstrip line


• a single-section coupled line coupler is shown belowbelow

33 4

isolatedcoupled isolatedcoupled

1 2

i t throughinput through

Coupled Line Directional Couplers 3 4Zo V3 V4

I1 I2I3 I4Zoe, Zoo

1 2θV1 V2V

• the input impedance at Port 1 of the coupler is given by g y

Z V V Vin

e o= =+1 1 1Z

I I Iine o+1 1 1


• the input impedance for the even and odd modes are given bymodes are given by

Z jZ+ tanθZ Z Z jZZ jZin

eoe

o oeoe o

=++

tantan

,θθZ jZoe o+ tanθ

Z Z Z jZino

ooo oo=+ tanθZ Z

Z jZin oooo o+ tanθ


• by voltage division, we have

V V Z V V Zino

ine

1 1= =V VZ Z

V VZ Z

oino

oe

ine

o1 1=

+=

+, ,

V VI V

Z ZI V

Z Zo

ino

oe

ine

o1 1=

+=

+, ,

in o in o


• the input impedance is given by

Z Z Z Z Z Z Zi

ino

ine

o ine

ino

o=+ + +( ) ( )Z

Z Z Zin

ine

ine

o=

+ + 2

Z Z Zo e( )2 2Z Z Z Z

Z Z Zo

ino

ine

oe e= +

−

+ +

( )2

2

2

Z Z Zin in o+ + 2


• note that the input impedance should benote that the input impedance should bematched to Zo, we have to choose

and so that this condition is satisfiedZino Zin

eand so that this condition is satisfiedZin Zin


• Let , the even and odd d h t i ti i d b

Z Z Zo oe oo=mode characteristic impedances become

Z ZZ j ZZ j Zin

eoe

oo oe=++

tantan

,θθZ j Zoe oo+ tanθ

Z ZZ j Z

ino

oooe oo=

+ tanθZ Z

Z j Zin oooo oe+ tanθ


• It can be shown that Z Z Z Z Zine

ino

oe oo o= = 2

• which leads to Z Zin o=

• Port 1 is matched, due to symmetry, all other ports are matchedports are matched


• the voltage at Port 3 is given by:

V V V V V V Z Ze o e o

ine

ino

3 3 3 1 1= + = − = −⎡⎢

⎤⎥

Z Z Z ZZ jZ Z jZ

e o e oine

o ino

o

o oe o oo

3 3 3 1 1+ +⎣

⎢⎢ ⎦

⎥⎥

+ +tan tanθ θV Z jZZ j Z Z

Z jZZ j Z Z

o oeo oe oo

o ooo oe oo

3 2 2=

++ +

−+

+ +tan

( ) tantan

( ) tanθ

θθ

θ

j Z Z( ) tanθV j Z ZZ j Z Z

oe ooo oe oo

3 2=

−+ +( ) tan

( ) tanθ

θ


• we can now define a coupling factor C so that

Z ZC Z ZZ Z

oe oooe oo

=−+

• The voltage at 3 is given by

V V jC

C j3 21=

− +

tan

tan

θ

θj


• At Port 4, we have

V V V V Ve o e o4 4 4 2 2 0= + = − =

• At Port 2, we have

V V V V Ce o2 2 2

2

21

= + =−

C je o2 2 2 21− +cos sinθ θ


• when q is small, virtually all the power will bed li d t P t 2 l d t P t 3delivered to Port 2. none coupled to Port 3,Port 4 is always isolated

• for q = p/2, the coupler is l/4 long and V V C3 / =q p , p gand

3V V j C2

21/ = − −


• the results satisfy power conservation, therei 90 h hift b t th t t tis a 90o phase shift between the two outputport voltages which can be used as a

d t h b idquadrature hybrid


• if the characteristic impedance and thecoupling coefficient are specific we use thecoupling coefficient are specific, we use thedesign formulas to obtain the even and oddmode characteristic impedancemode characteristic impedance

Z Z C Z Z Coe o oo o=

+=

−1 1,Z ZC

Z ZCoe o oo o− +1 1

,


• we have assumed that the even and oddmodes have the same propagation velocitiesmodes have the same propagation velocitieswhich is not valid at higher frequencies formicrostrip linesmicrostrip lines

• the coupling of a single-section coupled linel i li it d i b d idth d t thcoupler is limited in bandwidth due to the

quarter-wave length requirement, we cani th l ’ fimprove the coupler’s performance vsfrequency by using multisections


• for weak coupling, i.e., C << 1, q = 90o we have

V jC jC jC j3 −tan tan iθ θθ θV

VjC

C j

jCj

jC e j31 21 1=

− +≈

+=

tan

tan

tantan

sinθ

θ

θθ

θ θ

VV

C

C je j2

1

2

21

1=

−≈ −

iθ θ

θV C j1 21− +cos sinθ θ


• using these results, we can cascade the multi-sections so thatsections so that

V jC e V jC e V ej j j3 1 1 2 1

2= +− − −( sin ) ( sin )θ θθ θ θ

+Λ

V jC e V jC e V e3 1 1 2 1+( sin ) ( sin )θ θ

jC e Vj1+ −( sin )θ θ+Λ jC e VN

j1+ ( sin )θL


• assuming that the coupler is symmetric soassuming that the coupler is symmetric so that , etc., for an odd number of segments, we havewe have

V jV e C N C NjN3 1 1 22 1 3= − + − +−sin [ cos( ) cos( )θ θ θθV jV e C N C N3 1 1 22 1 3= − + − +sin [ cos( ) cos( )θ θ θ

N1 1+• +ΛAt at center frequency, we can define a

N12

12

++( )]

q y,coupling factor

C V3C VVo =

=

31 2θ π /


Example of coupled line directional coupler

The Lange Directional Couplers Generally in the coupled line directional coupler it is very difficult reach a coupling factor of 3 dB or 6 dB. To increase the coupling between edge coupled line wires are used tothe coupling between edge-coupled line, wires are used to connect several lines so that fringing fields at both edges of a line contribute to the coupling .p g

The most practical implementation of this idea is the langeThe most practical implementation of this idea is the lange coupler.

This coupler is quite expensive, and it can hand only low power.

The Lange Directional Couplers

Example of lange directional coupler:

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