Wilkinson Power Divider (1)

1

Chapter 7: The Wilkinson Power Divider

1) Review of Directional Coupler 2) Review of T Junction 3) Three Port Network 4) The Wilkinson Power Divider 5) Even-Odd Mode Analysis 6) Unequal Power Division 7) Wilkinson Power Dividers Example 8) Wilkinson Power Divider Summary

2

Directional Coupler

Directional Coupler

Circuit Diagram for Even and Odd Mode Analysis.

( ))tan()tan(

ljZZljZZZlZ

ooe

oeooe

ein β

ββ++

=

( ))tan()tan(

ljZZljZZZlZ

ooo

ooooo

oin β

ββ++

=

Even mode input impedance.

Odd mode input impedance.

3

Directional Coupler

oooe

oooe

ZZZZC

+−

=

( )( )ljC

ljCVVβ

βtan1

tan23+−

=

( ) ( )ljlCCVV

ββ sincos11

2

2

2+−

−=

04 =V

Coupling Factor:

Voltage at port 3:

Voltage at port 2:

Voltage at port 4:

CCZZ ooe −

+=

11

CCZZ ooo +

−=

11

4

T Junction

Matching requirement for the T junction input: 132

111ZZZ

=+

1) The Lossless T- Junction suffers from the problem of not being matched at all ports.

2) It does not have any isolation between output ports.

5

T Junction

ooo

inoo

ooo

in ZZZZZZZZZZ =+=⇒

+

++=

32

3333

Since the network is symmetric from all three ports, the output ports are also matched. Therefore, S11=S22=S33=0

The resistive divider can be matched at all ports. Even though it is not lossless, isolation is still not achieved.

6

Three Port Network

[ ]

=

333231

232221

131211

SSSSSSSSS

S

• A lossless three port network can be matched at all three ports, but any matched lossless network must be nonreciprocal. • Alternatively, a lossless and reciprocal three port network can be physically matched at only two ports. • A Lossy three port network can be matched at all ports with isolation between ports.

[ ]

=

00

0

3231

2321

1312

SSSSSS

S

[S] matrix of a three port network.

[S] matrix of a lossless and nonreciprocal three port network.

[ ]

=

332313

2312

1312

00

SSSSSSS

S

[S] matrix of a lossless and nonreciprocal three port network.

7

The Wilkinson Power Divider/ Combiner

Four-way divider using 3 Wilkinson dividers in Microstrip form.

8

The Wilkinson Power Divider/ Combiner

• The Wilkinson power divider can be matched at all ports with isolation between the output ports. • The Wilkinson can be made using Microstrip or Stripline. • The Wilkinson power divider can be made to give arbitrary power division. • We will study the equal split (3 dB) case first.

Zo

Zo

Zo

Port 1

Port 3

Port 2

2Zo

oZ2

λ/4

oZ2

9

The Wilkinson Power Divider/ Combiner

2

2

λ/4

2

1

1 1

1

V1

V3

V2 Vg2

Vg3

Normalized Wilkinson Power Divider/ Combiner

• We will analyze this circuit by reducing it to two simpler circuits driven by Symmetric (Even) and Anti-Symmetric (Odd) sources at the output ports.

10

2

2

λ/4

2

1

1 1

1 V1

V3

V2 Vg2

Vg3

• If voltages Vg2 and Vg3 are equal and are in phase, we have even excitation, and magnetic wall (open circuit) can be placed between the two arms • Alternatively, when Vg2= - Vg3 = 2 V we have odd excitation, electric wall (short circuit) can be placed in the plane of the symmetry

The Wilkinson Power Divider/ Combiner

11

Plan: • Consider odd and even excitations • Compute voltages at different ports • Compute reflection coefficient at ports • Create S-matrix

We will consider odd and even excitation of ports 2 & 3. We know already that any other excitation can be built from that. Then we will excite port 1 - this is even with respect to ports 2 and 3.

The Wilkinson Power Divider/ Combiner

2

2

λ/4

2

1

1 1

1 V1

V3

V2 Vg2

Vg3

12

Even Mode Analysis

2

λ/4

1 1

V1

V2

Vg2

For the even mode analysis, Vg2 = Vg3 = 2V and no current flows through the isolation resistor or through the connection at port 1. Hence we can treat the axis of symmetry as an open circuit and draw the circuit as shown below.

We can now examine the voltages at ports 1 and 2, (V1 and V2) and determine the optimum value for Z.

13

Even Mode Analysis

22

1

V3 Vg3 2

λ/4

2

1

1

1

Vg2=2V

V2=V

2

λ/4

1 1

V1

V2

Vg2

Quarter wave transformer

V2

Match!

V1=?

L

oin Z

ZZ2

=

22 =CircuitOpen

14

0

)()( xjxj eeVxV ββ Γ+= −+

)1(1 Γ+= +VV V1)(2 =Γ−= + jjVV

111

V j VΓ +=

Γ −

Vg3 2

λ/4

2

1 V1

x

V2

2222

+−

=Γ

jeejj

−==−−

24πλβ

x=0 x=-λ/4

Even Mode Analysis

15

11

1 −Γ+Γ

= jV2222

+−

=Γ

V2V22

4)22(22

2222V1

2222

12222

V1 jjjjV −=−=+−−++−

=−

+−

++−

=

V21 jV −=

Even Mode Analysis

We had: and

16

Odd Mode Analysis

2

2

λ/4

2

1

1 1

1 V1

V3

V2 Vg2

Vg3

For the odd mode analysis Vg2 = -Vg3 = 2 V. So there is a voltage null along the axis of symmetry of the circuit. We can assume a virtual ground at that point and redraw the circuit. Again we are interested in the behavior at ports 1 and 2 and in determining the optimum value for Z.

17

22 1

1

V3 Vg3

Vg2

Vg2=2V V2=V

V1=0

Vg2 2 λ/4

1 1

V1

V2

2

1

1

Match!

Odd Mode Analysis

CircuitShortCircuitShort =2

Quarter wave transformer CircuitOpenZ

ZZZ o

L

oin ===

0

22

1

1 1

λ/4

11=CircuitOpen

18

Excitation in port 1, when 2 & 3 are matched.

1

2

λ/4

1

1 1

1

V1

V3

V2

1

1

open

V2

Match!

Input Impedance at Port 1

The final question is what is the input impedance at port 1, when ports 2 and 3 are terminated in their matched loads?

112

12

22

==inZ

Z in

Therefore, the input to the divider is matched as long as the output ports are matched.

19

1

2

λ/4 1

1 1

1

V3

V2

Matched, S11=0

2 λ/4

1

V2 Vg2=2V

even

Port 2 matched for even excitation

2 λ/4

1 1

V2

2

odd Port 2 matched for odd excitation

Port 2 matched for any excitation. By symmetry, port 3 is also matched

S22 and S33=0

• Ports 2 and 3 are separated either by electric or magnetic wall. No power goes between (ISOLATION). S23=S32=0 • Circuit is reciprocal -- matrix symmetric

Wilkinson Divider S Parameters

20

Wilkinson Divider S Parameters

2

2

λ/4

2

1

1 1

1 V1

V3

V2

Vg2

We know now that ports are matched. Thus:

02

112

31 ==+

−

++

=VVV

VS

21

V11V0V2

22

11

2

112 jj

VVVV

VVS oe

oe

−=+

+−=

++

==

−

−

−−

=

002

002

220

j

j

jj

S

21

Wilkinson Power Divider Example

Design an equal split Wilkinson power divider for a 50 Ω system impedance at frequency f o.

Zo

Zo

Zo

Port 1

Port 3

Port 2

2Zo

oZ2

λ/4

oZ2

Ω== 7.702 oZZ

Ω== 1002 oZR

The quarter wave transmission lines in the divider should have a characteristic impedance of:

The shunt resistor has a value of:

22

Unequal Power Division

It is possible to design the Wilkinson for an unequal dividing ratio. This can be done by different choices of impedances and isolation resistor as shown below. Note: Lines 2 and 3 are still λ / 4 long and that ports 2 and 3 are no longer matched to Z o.

For a given power ratio K between ports 2 and 3: We have the following formulas to design the width of the arms.

2

32

PPK =

3

2

31

KKZZ oo

+= ( )2

32

2 1 KKZZKZ ooo +==

+=

KKZR o

1

Note: These result will reduce to the equal-split case for K=1.

23

The Wilkinson Divider Summary • The Wilkinson Divider can also be generalized as an N-Way Divider or Combiner. • This circuit can be matched at all ports. • The isolation between ports can be achieved. • The Wilkinson divider can also be made with stepped multiple sections for increased bandwidth.

Photograph of a four-way power divider network using three microstrip Wilkinson power divider.