coupled line - even and odd mode
TRANSCRIPT
Chapter 7: Directional Coupler
1) Coupled Line 2) Even and Odd Mode3) Coupled Line Even Mode4) Coupled Line Odd Mode5) Even and Odd Mode Analysis6) Coupled Line Directional Couplers7) Even and Odd Mode DC Voltage Analysis8) Directional Coupler Example
Coupled Line - Even and Odd Mode
3 4
1 2
input through
coupled isolated
A very common type of power divider used in Microwave circuits is the coupled line. In the structure the coupling between the ports is due to the interaction of the electromagnetic fields along transmission lines which have been placed in close proximity.
Coupled Line - Even and Odd Mode
Coupled lines
V V V V
E wall - Odd modeH wall- Even mode
Electric wallMagnetic wall
Even and Odd Mode AnalysisOne method of analyzing multi-port transmission line circuits such as coupled line is through even and odd mode analysis. In this case, the circuit input voltage is split into two, an even or symmetric mode and an odd or anti-symmetric mode.The total response of the circuit can then be described as the superposition of the two separate responses (Even and Odd).
21: VVModeEven
21: VVModeOdd
Coupled Line - Even and Odd Mode
C11 and C22 represent the capacitor to ground for one strip (without the presence of the other). C12 is the capacitance between the two lines without any ground plane. If the strips are identical, then C11= C22.
Coupled Line – Even Mode
In the even mode, a virtual open circuit occurs along the axis of symmetry, which removes the C12 from the circuit. The resulting capacitance to the ground is the even mode capacitance Ce= C11= C22. If the two strips are identical, the even mode impedance Zoe can be calculated as:
ee
e
eoe vCC
LC
C
LZ
1
Coupled Line – Odd Mode
In the odd mode, a virtual ground forms along the axis of symmetry, which may be considered as a ground plane through the middle of C12, resulting in the capacitance as it shown above. The resulting capacitance for each conductor to the ground in the odd mode is Co: Co= C11+ 2C12. The Zoo can be calculated as:
oo
o
ooo vCC
LC
C
LZ
1
Coupled Lines – Even and Odd Modes
Zoe is the characteristic impedance of one of the transmission lines under even mode operation and Zoo is the characteristic impedance of one of the lines under the odd mode excitation.
The design relations for the dimensions of the transmission line for given values of Zoe and Zoo have been tabulated by Pozar, with a complete solution for the microstrip lines. But only for r = 10.
These results are required for the design of coupled line directional couplers.
Coupled Lines – Even and Odd Modes
Even and Odd mode characteristic impedance design data for coupled microstrip lines on a substrate with r = 10.
Coupled Line Directional Couplers
By using the even and odd mode analysis we can show that two coupled lines can be used as a power divider with predictable coupling ratio and isolation.
Coupled Line Directional Couplers
For the analysis we will use the schematic as shown in the above figure, with the excitation at port 1 and all other ports terminated in the characteristic impedance.
Coupled Line Directional Couplers- Even and Odd Modes
= +
Using the superposition the excitation at port 1 can be treated as the sum of the even and odd mode excitations.
Even Mode
Odd Mode
Coupled Line Directional Couplers- Even Mode
For the Even Mode from the symmetry we can see that:
ee II 31 ee II 42
ee VV 31 ee VV 42
Coupled Line Directional Couplers- Odd Mode
For the Odd Mode from the symmetry we can see that:
oo II 31 oo II 42
oo VV 31 oo VV 42
Even and Odd Mode Characteristic Impedances
The input impedance at port 1 can be written as:
oe
oein II
VV
I
VZ
11
11
1
1
We can calculate the even and odd input impedances for the line terminated in Zo using:
)tan(
)tan(
ljZZ
ljZZZlZ
Lo
oLoin
We can define Zine and Zin
o as:
)tan(
)tan(
ljZZ
ljZZZlZ
ooe
oeooe
ein
)tan(
)tan(
ljZZ
ljZZZlZ
ooo
ooooo
oin
Even mode input impedance. Odd mode input impedance.
Note: The line looks line a transmission line of characteristic impedance Zoe and Zoo terminated in a load impedance Zo.
Even and Odd Modes - DC Voltage Analysis
Now we can calculate the input voltage and current as a function of the even and odd modes input impedances:
ooin
oino
ZZ
ZVV
1
oein
eine
ZZ
ZVV
1
oein
e
ZZ
VI
1
ooin
o
ZZ
VI
1
Even mode Odd mode
Even and Odd Modes - DC Voltage Analysis
Substitution of these values in this equation yields:
oe
oein II
VV
I
VZ
11
11
1
1
o
ein
oin
oein
oin
oin ZZZ
ZZZZZ
2
2 2
Now if we set:
Then the input impedance at port 1 is matched. By symmetry all of the other ports are matched as well.What about the other ports voltages and currents?Can we find V3, V4 or V2?
oinoein
oin ZZZZZ 2
Even and Odd Modes - DC Voltage Analysis
From the even and odd mode voltage and current relations we can write:
o
oin
oin
oein
einoeoe
ZZ
Z
ZZ
ZVVVVVV 11333
V3 can also be written as a function of the input voltage and even and odd mode impedances:
lZZjZ
lZZjVV
oooeo
oooe
tan2
tan3
Even and Odd Modes - DC Voltage Analysis
lZZjZ
lZZjVV
oooeo
oooe
tan2
tan3
Therefore, we could derive the voltage at port 3 as a function of input voltage and even and odd mode characteristic impedances. Now if we define the Coupling Factor C as:
Then we have:
By similar reasoning:
oooe
oooe
ZZ
ZZC
ljC
ljCVV
tan1
tan23
ljlC
CVV
sincos1
12
2
2
and 04 V
Directional Coupler Performance
If l = /4, then 223 1 CjV
VandC
V
V
The final step is to relate the coupling factor to the even and odd mode impedances to determine the dimensions of the coupler.
C
CZZ ooe
1
1
C
CZZ ooo
1
1and
This allows us to use the even and odd mode characteristic impedance plots to determine the width and separation of the lines for a given coupling coefficient.
Directional Coupler ExampleDesign a 5 dB microstrip directional coupler for an r of 10
(Zo=200).
The required coupling is –5 dB,
So 5623.010 20
5
C
49.11910
87.377
1
1
C
CZZ
r
ooe
48.3310
87.105
1
1
C
CZZ
r
ooo
From the figure 1, we have:
3.0/05.0/ dWdS
S=Separation W=Width of Microstrip lines d=Dielectric thickness