zo and ze concept.pdf

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Whites, EE 481 Lecture 28 Page 1 of 11

2012 Keith W. Whites

Lecture 28: Coupled Line andLange Directional Couplers.

These are the final two directional couplers we will consider.

They are closely related and based on two TLs that interact with

each other, but are not physically connected.

Coupled Line Directional Coupler

When two TLs are brought near each other, as shown in the

figure below (Fig. 7.26), it is possible for power to be coupled

from one TL to the other.

This can be a serious problem on PCBs where lands are close

together and carry signals changing rapidly with time. EMC

engineers face this situation in high speed digital circuits and in

multiconductor TLs.

For coupled line directional couplers, this coupling between TLs

is a useful phenomenonand is the physical principle upon which

the couplers are based.


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Consider the geometry shown in Fig 7.27:

r

+++++

+ ++++

++

- -- - -- - -- - --

1 2

When voltages are applied, charge distributions will be induced

on all of the conductors. The voltages and total charges are

related to each other through capacitance coefficients Cij:

V1

V2

1 2C12

C11 C22

By definition (Q CV ): 1 11 1 12 2Q C V C V

2 21 1 22 2Q C V C V

where 11C capacitance of conductor 1 with conductor 2present but

grounded.

22C capacitance of conductor 2 with conductor 1present but

grounded.

12C mutual capacitance between conductors 1 and 2. (If the

construction materials are reciprocal, then 21 12C C .)

By computing only these capacitances and the quasi-TEM mode

wave speed, well be able to analyze these coupled line

problems. Why? Assuming TEM modes, then


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1

p

L LCZ

C C v C (1)

Notice that Ldoesnt appear here. Hence we only needp

v and

C, as conjectured. This is a widely used approach in all TL

problems, not just microstrip or coupled lines.

Even-Odd Mode Characteristic Impedances

To simplify the problem analysis, well assume the two strips

are identical and located on top of a dielectric. Because of the

symmetry, we can use an even-odd mode solution approach.

Even mode. The voltages and currents are the same on both

strips, as shown in Fig 7.28a:

Hence, the electric field has even symmetryabout the plane of

symmetry (POS). This plane is called an H-wall.

Notice that no Efield lines from conductor 1 (2) terminate on

conductor 2 (1). Consequently, the two halves are decoupled,

as we expect. This leads us to the equivalent circuit:


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V1

V2

1 2

C11 C22

We define the effective capacitance to ground for either

conductor in this configuration as

11 22eC C C (7.68),(2)

Then using (1)

0

1e

pe eZ v C (7.69),(3)

which is the characteristic impedance of either TL when both

are operated in the even mode. This is called the even mode

characteristic impedance.

Odd mode. The voltages and currents are opposite on each

line as shown in Fig 7.28(b):

Notice here that the electric field lines areperpendicular to the

POS. Therefore, similar to image theory, we can consider the

POS as an equipotential surface (or an E wall). That is, as a

ground plane where 0V .


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This approach leads to the equivalent capacitance circuit:

V1

V2

1 22C12

C11 C22

2C12

Sum = C12

The effective capacitance to ground of either conductor in this

configuration is then

12 11 11 122 2oC C C C C (7.70),(4)so that, from (1)

0

1o

po o

Zv C

(7.71),(5)

This is the odd mode characteristic impedance. It is the

characteristic impedance seen when a voltage wave is

launched on the structure with odd symmetry, as shown inFig. 7.28b.

The computation ofe

C ando

C required in (3) and (5) is a bit

involved. Alternatively, the text presents two examples of

graphical design datafor specific geometries. Fig 7.29 contains

design data for striplines and Fig 7.30 for microstrips with

10r .


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Recall that striplines support pure TEM modes, while

microstrips do not. To use the stripline design curves, scalethe

even and odd mode impedances upward byr , while the

microstrip design curves apply only for 10r .

Even-Odd Mode Solution for Coupled LineDirectional Coupler

A coupled line directional coupler (in stripline or microstrip) is

show in Fig 7.31. It is excited at port 1 and terminated with 0Z

at all ports.


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3 1

e eI I , 4 2e eI I (6)

3 1

e eV V , 4 2e eV V

while for the odd

3 1o oI I , 4 2

o oI I (7)

3 1

o oV V , 4 2o oV V

By definition, 1 1 1in1 1 1

e o

e o

V V VZ

I I I

(7.72),(8)

These TL problems in Fig 7.32 are simple and easy to solve. As

shown in the text, by choosing

0 0 0e oZ Z Z (7.77),(9)

leads to in 0Z Z (7.78),(10)

That is, with (9) then port 1 is matched and given the symmetry

of the structure, allports will then be matched!

As shown in the text,

1V V (11)

2

2 2

1

1 cos sin

CV V

C j

(7.84),(12)

32

tan

1 tan

jCV V

C j

(7.82),(13)

4 0V (total isolation) (7.83),(14)

where l is the electrical length and


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0 0

0 0

e o

e o

Z ZC

Z Z

(7.81),(15)

is the voltage coupling coefficient. (Here, C stands for

coupling, not capacitance.)

Typical plots of2

2V V and2

3V V are shown in Fig 7.33:

Notice from these plots that we can maximize2

3V V and

simultaneously minimize2

2V V when 2, 3 2, so that

4, 3 4,l . In these cases, (11)-(14) become

1V V

2

2 1V jV C (7.86),(16)

3V CV (7.85),(17)

4 0V

Lastly, it can be easily shown that power is conserved, which is

a valuable check of the analysis.

Example N28.1(Similar to text example 7.7). Design a 20-dB,

single section, coupled line directional coupler using stripline.


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Assume a 0.158-cm ground plane separation, 2.56r , 0 50Z

, and a center frequency of 3 GHz.

From (17), 3C V V is the coupling coefficient, which must be

less than onefor a passive device. In this case,

1020 20log C 0.1C (18)

Next, we need to determine 0eZ and 0oZ . From (9) and (15),

0 0

1

1e

CZ Z

C

1.150 55.28

0.9

(7.87a),(19)

and 0 01

1o

CZ Z

C

0.950 45.23

1.1 (7.87b),

These values can be used in Fig 7.29 to determine Wand S for

the stripline dimensions. With 2.56r

, then

0 88.4r eZ and 0 72.4r oZ Using these values in Fig 7.29we find

0.72W

b and 0.34

S

b .

Given that 0.158b cm, then

0.114W cm and 0.054S cm.

This value for S is rather small and may lead to fabrication

difficulties with basic milling machines.

The coupling and directivity for this coupled line directional

coupler computed by a CAD package are:


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Lange Coupler

This device is shown in Fig 7.38:

This is one method for obtaining higher coupling coefficients(up to approximately 2-3 dB or so) than what is possible with

regular coupled line directional couplers.

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