Transcript
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.
top related