DESIGN OF BROADBAND MICROSRTIP BALUN
IN THE 6 GIGAHERTZ WIRELESS BAND
A Project
Presented to the faculty of the Department of Electrical and Electronic Engineering
California State University, Sacramento
Submitted in partial satisfaction of
the requirements for the degree of
MASTER OF SCIENCE
in
Electrical and Electronic Engineering
by
Christopher Anthony Gabris
FALL
2013
© 2013
Christopher Anthony Gabris
ALL RIGHTS RESERVED
ii
DESIGN OF BROADBAND MICROSRTIP BALUN
IN THE 6 GIGAHERTZ WIRELESS BAND
A Project
by
Christopher Anthony Gabris
Approved by:
__________________________________, Committee Chair
Preetham B. Kumar, Ph.D.
__________________________________, Second Reader
Russ. Tatro, M.S.
____________________________
Date
iii
Student: Christopher Anthony Gabris
I certify that this student has met the requirements for format contained in the University
format manual, and that this project is suitable for shelving in the Library and credit is to
be awarded for the project.
__________________________, Graduate Coordinator
Preetham B. Kumar, Ph.D.
Department of Electrical and Electronic Engineering
iv
___________________
Date
Abstract
of
DESIGN OF BROADBAND MICROSRTIP BALUN
IN THE 6 GIGAHERTZ WIRELESS BAND
by
Christopher Anthony Gabris
This thesis investigates wideband balun power dividers (WBPD) and wideband
balun couplers (WBC) implemented on microstrip. Current wideband microstrip circuits
are rather large due to their dependence on quarter wavelengths. Microstrip circuits
become especially large when they operate at frequencies under 10 GHz due to their
quarter wavelength requirement. This has created a demand for wideband microstrip
circuits that operate under 10 GHz and are smaller than those currently available. It is
therefore desired to implement a WBPD or WBC on microstrip in a smaller form, where
the circuit is under a quarter wavelength long.
Agilent’s Advanced Design System (ADS) software, version 2012.08, was used for
this thesis to verify theory and simulate designs.
The objective is to create a WBPD or WBC that is small in size and operates under
10 GHz over a large bandwidth, ideally, the bandwidth would be over 10 GHz. To keep
manufacturing costs low, it is desired to implement a WBPD or WBC on microstrip
without vias or other circuit elements. This thesis determines what is necessary to
implement a WBPD and WBC on microstrip. A WBPD requires a minimum of two
v
resistors and two microstip gaps to achieve a balanced output, and that the maximum gain
between any two ports is -4.77dB. A WBC requires a dielectric with a large relative
permittivity and thickness. A WBC also requires the even and odd mode characteristic
impedances to be extreme opposites; one is very large and the other is very small.
_______________________, Committee Chair
Preetham B. Kumar, Ph.D.
_______________________
Date
vi
TABLE OF CONTENTS
Page
List of Figures ........................................................................................................... viii
Chapter
1. INTRODUCTION ……………………………………………………...……….. 1
2. PASSIVE MICROSTRIP CIRCUITS .................................................................... 3
2.1 Transmission Lines .......................................................................................... 3
2.2 Open Stub Matching Circuits .......................................................................... 4
2.3 Transmission Line Phase ..................................................................................7
2.4 Microstrip Gaps ............................................................................................. 12
3. POWER DIVIDERS ............................................................................................. 14
3.1 Continuous Power Dividers ........................................................................... 14
3.2 Non-Continuous Power Dividers ................................................................... 15
3.3 Balun Power Divider VSWR ..........................................................................18
4. BALUN COUPLERS ........................................................................................... 19
4.1 Balun Couplers ...............................................................................................19
4.2 Wideband Coupler Design ..............................................................................28
5. CONCLUSIONS................................................................................................... 37
References ................................................................................................................... 39
vii
LIST OF FIGURES
Figures
Page
1. Figure 1.1 Block diagram of 3-port power divider or coupler ……………………. 1
2. Figure 2.1 Open stub matching circuit………… ... .……………………………….. 4
3. Figure 2.2 Impedance of a open stub matching circuit …………………………….6
4. Figure 2.3 Voltage on a transmission line……….… ... …………………………….7
5. Figure 2.4 Output phase of a transmission line…….………………………………. 9
6. Figure 2.5 Block diagram of a small terminated circuit ………………………….10
7. Figure 2.6 Output phase of a small terminated circuit ……………………………11
8. Figure 2.7 Output phase of a microstrip gap …….………………………………. 13
9. Figure 3.1 Output phase of a power divider
…….………………………………. 14
10. Figure 3.2 Block diagram of a balun power divider …………………………….15
11. Figure 3.3 Output phase of a balun power divider.………………………………. 17
12. Figure 4.1 Layout of balun coupler …………….………………………………. 19
13. Figure 4.2 Even and odd mode excitation of a balun coupler……………………. 20
14. Figure 4.3 Balun coupler with open load………….………………………………22
15. Figure 4.4 Typical bandwidth of a coupler……….……………………………….24
16. Figure 4.5 Lower cutoff frequency of a coupler….………………………………. 27
17. Figure 4.6 Coupler on microstrip ……………….………………………………. 28
18. Figure 4.7 Contour diagrams of equation 4.23 ….……………………………….29
19. Figure 4.8 2D-field solver for a coupler ………….………………………………. 30
viii
20. Figure 4.9 2D-field solver for a transmission line .………………………………. 32
21. Figure 4.10 Wideband balun coupler design simulation…………………………. 33
22. Figure 4.11 Wideband balun coupler design simulation…………………………. 35
23. Figure 4.12 Tabulated results at coupling frequency…… ………………………. 36
ix
1
Chapter 1
INTRODUCTION
A power divider or coupler is a circuit that takes power from a single input and
divides it among multiple outputs. This thesis will only focus on power dividers and
couplers with one input and two outputs. Figure 1.1 shows the block diagram for a 3-port
power divider or coupler.
A balun power divider or coupler has a special type of output. Balun is short for
balanced and unbalanced. A circuit is considered balanced when the voltages at the two
output ports are equal in magnitude and have a 180° phase difference. Therefore, the
block diagram in figure 1.1 would be considered balanced when the voltage at port 2 is
equal to the negative voltage at port 3, when:
𝑉𝑝𝑜𝑟𝑡 2 = −𝑉𝑝𝑜𝑟𝑡 3
(1.1)
Figure 1.1 Block diagram of a 3-port
power divider or coupler.
Newer communication systems, such as ultra wideband systems, have created an
increased demand for wideband passive circuits that are small and inexpensive to
2
manufacture. In an attempt to create a small, inexpensive wideband balun power divider
(WBPD) or wideband balun coupler (WBC), this thesis investigates implementing both
of these circuits on microstip.
To summarize the objectives of this thesis;
i)
Implement a WBPD or WBC on microstrip.
ii)
The microstrip lines should be small; less than a quarter wavelength long.
iii)
The WBPD or WBC should have a large bandwidth, greater than 10 GHz.
iv)
The WBPD or WBC should operate at frequencies less than 10 GHz.
v)
The WBPD or WBC should not contain any vias or circuit elements.
vi)
The VSWR of the WBPD or WBC should be small.
3
Chapter 2
PASSIVE MICROSTRIP CIRCUITS
2.1 Transmission Lines.
The impedance of a transmission line is given by
𝑍𝑖𝑛 = 𝑍𝑜
𝑍𝑙 + 𝑗𝑍𝑜 𝑡
𝑍𝑜 + 𝑗𝑍𝑙 𝑡
(2.1)
Where 𝑍𝑜 is the characteristic impedance of the transmission line
𝑍𝑙 is the load on the transmission line
𝑡 is the tangent of the electrical length 𝑡 = tan(𝛽𝑙)
𝛽𝑙 is the electrical length
𝑙 is the length of the transmission line.
If the length of a transmission line is small, less than an eighth of a wavelength long, then
we can approximate the tangent of the electrical length as:
𝑡𝑎𝑛(𝛽𝑙) ≈ 𝛽𝑙
(2.2)
𝑡𝑎𝑛(𝛽𝑙) can be approximated as βl with an error less than or equal to 20 percent when
βl is less than 45 degrees. Equation 2.2 becomes more accurate for frequencies where βl
is less than 45 degrees. The electrical length of the transmission line will be less than 45
degrees when the length of the transmission line satisfies:
𝑙<
𝑣⃗
8 ∗ 𝑓𝑜
(2.3)
4
Where 𝑣⃗ is the velocity of the wave
𝑓𝑜 is the frequency when βl = 45°
As long as equation 2.3 is satisfied, then equation 2.2 is valid for frequencies below 𝑓𝑜 .
2.2 Open Stub Matching Circuits
Figure 2.1 shows a typical open stub matching circuit with no load. The circuit in
figure 2.1 is called an un-terminated circuit because there is no signal path to ground.
The input impedance to the circuit in figure 2.1 is given by:
𝑍𝑖𝑛 = 𝑗
𝑍2 (𝑍2 𝑡1 𝑡2 − 𝑍1 )
𝑍1 𝑡2 + 𝑍2 𝑡1
(2.4)
where 𝑡1 = tan(𝛽𝑙1 ) and 𝑡2 = tan(𝛽𝑙2 )
Figure 2.1 Open stub matching circuit.
5
If the lengths L1 and L2 in figure 2.1 satisfy equation 2.3, then the electrical lengths of
the lines are less than 45°, and we can make the approximations:
𝑡1 𝑡2 ≈ 0
𝑡1 ≈ 𝑡2
and equation 2.4 reduces to:
𝑍𝑖𝑛 = −
𝑗 𝑍1 𝑍2
𝑡 (𝑍1 + 𝑍2 )
(2.5)
𝑓𝑜𝑟 𝛽𝑙1 & 𝛽𝑙2 < 45°
Equation 2.5 shows that the circuit in figure 2.1 has an input impedance that is purely
imaginary. In fact, equation 2.5 is valid for any un-terminated circuit. From equation
2.5, the input impedance for any open stub circuit can be written as:
𝑍𝑖𝑛 = −
Where 𝑘1 and
𝑗𝑘1
tan(𝑘2 𝑓)
(2.6)
𝑘2 are positive constants
𝑓 is the frequency
Equation 2.6 was derived under the assumption that βl was under 45°, but
surprisingly, equation 2.6 is valid for any un-terminated microstrip circuit up to 90°.
Figure 2.2 shows three un-terminated open stub networks. The magnitude of the input
impedance for each circuit is plotted with equation 2.6 as a comparison. The overall
electrical length of each circuit is kept under 90° at 5 GHz.
6
Figure 2.2 Impedance of three open stub circuits with overall lengths less than 45°
at 5 GHz. The input impedance for each circuit is plotted along with the
approximation given by 2.6. The constants 𝑘1 and
through curve fitting.
𝑘2 were determined
The circuit specifications in figure 2.2, such as the characteristic impedances, are not
important. Figure 2.2 is used to show that any un-terminated microstrip circuit will have
an input impedance given by equation 2.6, for βl<90°.
7
2.3 Transmission Line Phase
In figure 2.3, the voltage at the load of the transmission line, is given by:
𝑉𝑙 = 𝑉𝑖 𝑒 −𝑗𝛽𝑙
1 + 𝛤𝑙
1 + 𝛤𝑙 𝑒 −𝑗2𝛽𝑙
(2.7)
Where 𝑉𝑖 is the input voltage
𝛤𝑙 is the reflection coefficient at the load
Figure 2.3 voltages on a transmission line.
If the transmission line is terminated into a purely real load, the reactance of the
reflection coefficient at the load will be zero, and 𝛤𝑙 will be purely real. If 𝛤𝑙 is real,
equation 2.7 gives the phase of the output voltage, as:
𝛩𝑙 = 𝛩𝑖 − 𝑡𝑎𝑛−1 (𝑡
𝛤𝑙 − 1
)
𝛤𝑙 + 1
𝑓𝑜𝑟 𝑟𝑒𝑎𝑙 𝛤𝑙
(2.8)
Where 𝛩𝑙 is the phase of the output voltage
𝛩𝑖 is the phase of the input voltage
𝑡 is the tangent of the electrical length 𝑡 = tan(𝛽𝑙)
8
Assuming 𝑓𝑜 is the frequency when βl = 45°, and 𝛤𝑙 is real, equation 2.8 can be
approximated by:
𝛩𝑙 = 𝛩𝑖 +
𝜋 𝛤𝑙
1
( −
)𝑓
4𝑓𝑜 2 𝛤𝑙 + 1
𝑓𝑜 =
𝑓𝑜𝑟 𝑓 < 𝑓𝑜
(2.9)
→
𝑣
8𝑙
→ is the velocity of the wave
𝑣
𝑙 is the length of the transmission line
𝑓 is the frequency
Equation 2.9 shows that any transmission line terminated into a purely real load, will
have approximately, an output phase that is linear with frequency for βl<45°.
Figure 2.4 shows three microstrip lines with load reflection coefficients equal to
-0.5, 0, and 0.5 for βl=45° at 5 GHz. The approximate phase given by 2.9 is also plotted
as a comparison.
In fact, any microstrip circuit laid out in a square area less than
(
→
𝑣
8𝑓𝑜
2
) , and
terminated into a purely real load, will have an output phase that is approximately linear
→
𝑣
for 𝑓 < 𝑓𝑜 . Figure 2.5 shows a box with a square area equal to (
8𝑓
𝑜
2
)
, and any
9
Zo = 25 Ohms
Figure 2.4 Output phase for three microstrip lines with load reflection coefficients equal
to -0.5, 0, and 0.5 with 𝑓𝑜 = 5 GHz. The phase of the output voltage is shown by the red
curves, and the approximation given by equation 2.6 is shown in blue. Note: when the
load reflection coefficient is 0, the output phase and equation 2.6 are the same.
10
continuous passive microstrip circuit that can fit in the area will have an output phase that
is approximately linear.
Figure 2.5 Block diagram of a small terminated circuit.
The area for any microstrip circuit for which the
phase of the output voltage will be approximately linear for 𝑓 < 𝑓𝑜 .
Any microstrip circuit that can fit in the box in figure 2.5 will have an output phase that
can be approximated by:
𝛩𝑜𝑢𝑡 = 𝛩𝑖𝑛 − 𝑚𝑓
(2.10)
Where 𝛩𝑖𝑛 is the phase of the input voltage
𝑚 is some constant
𝑓 is frequency
Figure 2.6 shows how equation 2.10 compares to the actual output phase of three circuits.
11
Figure 2.6 Output phase for three microstrip circuits that are terminated into real loads.
→
2
The overall area of each circuit is less than (8𝑓𝑣 ) . The output phase of each circuit is
𝑜
plotted with the approximate phase given by equation 2.10.
12
2.4 Microstrip Gaps
Section 2.3 showed that the output phase of any continuous microstrip circuit
starts at zero degrees, and decreases linearly with frequency up to the point where the
electrical length equals 45°. This means the output phase of all small circuits will have
plots like those in figure 2.6, except with different slopes. Another way to change the
output phase is to introduce a gap. When a gap is introduced into a microstrip circuit, it
is no longer considered a continuous circuit. Section 2.3 covered the output phase for
continuous circuits; this section covers the phase of non-continuous circuits.
Figure 2.7 shows three random microstrip gaps and the phase of the output
voltage. From figure 2.7, we can see that the output phase of a gap can be approximated
as a linear function in the form:
𝛩𝑜𝑢𝑡 = 𝛩𝑖𝑛 − 𝑚𝑓 +
𝜋
2
(2.11)
Comparing equations 2.10 and 2.11, we see that a gap produces the same output phase as
a continuous line, only shifted up 90°.
13
Figure 2.7 The output phase of
three microstrip gaps.
14
Chapter 3
POWER DIVIDERS
3.1 Continuous Power Dividers
Since any small continuous microstrip circuit will have an output phase given by
equation 2.10, we can write the output phase as a linear function of frequency for any
small power divider. Figure 3.1 shows a 3-port continuous power divider and the phase
at the two output ports.
Figure 3.1 Output phase for the block diagram of a small microstrip power
divider and the phase of the output voltages. 𝑚12 is the phase slope from
port 1 to port 2. 𝑚13 is the phase slope from port 1 to port 3.
For the power divider in figure 3.1 to have balanced outputs, the difference in
phase of the two output ports should be equal ±180°. This is written as:
𝛩2 − 𝛩3 = ±𝜋
Plugging in for
(3.1)
𝛩2 and 𝛩3 we get
(𝛩1 − 𝑚12 ∗ 𝑓) − (𝛩1 − 𝑚13 ∗ 𝑓) = ±𝜋
(3.2)
15
Solving for f gives
𝑓=±
𝜋
= 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝑚13 − 𝑚12
(3.3)
Equation 3.3 is only satisfied at a specific frequency, and therefore, a small continuous
microstrip power divider cannot have a balanced output over a large bandwidth.
3.2 Non-Continuous Power Divider
Section 2.4 showed that introducing a gap in a microstrip circuit gives a 90° phase
shift. Therefore, achieving a 180° phase shift would require a circuit with two 90° gaps,
which add up to 180°. In figure 3.2, a balanced output is achieved by introducing two
gaps between ports 1 and 2. The path between ports 1 and 3 does not have any gaps.
Figure 3.2 Block diagram of a balun power divider using two 90° gaps.
If we look at figure 3.2, moving from port 1 to node a crosses a gap which causes a 90°
phase shift. Then, the phase at node a is given by equation 2.11:
𝛩𝑎 = 𝛩1 − 𝑚1𝑎 𝑓 + 90°
(3.4)
16
Moving from node a to port 2 crosses another gap causing another 90° phase shift.
Therefore, the phase at port 2 is also given by equation 2.11:
𝛩2 = 𝛩𝑎 − 𝑚𝑎1 𝑓 + 90°
= 𝛩1 − 𝑓(𝑚1𝑎 + 𝑚𝑎2 )
+ 180°
(3.4)
Moving from port 1 to node b crosses a continuous circuit which causes a linear phase
shift given by equation 2.10:
𝛩𝑏 = 𝛩1 − 𝑚1𝑏 𝑓
(3.5)
Moving from node b to port 3 crosses another continuous circuit which is another linear
phase shift. Therefore, the phase at port 3 is also given by equation 2.10:
𝛩3 = 𝛩𝑏 − 𝑚𝑏3 𝑓
= 𝛩1 − 𝑓(𝑚1𝑏 + 𝑚𝑏3 )
(3.4)
For the output to be balanced:
𝛩2 − 𝛩3 = 180°
Plugging in for 𝛩2 and 𝛩3
𝛩2 − 𝛩3 =
= 𝛩1 − 𝑓 (𝑚1𝑎 + 𝑚𝑎2 ) + 180° − (𝛩1 − 𝑓(𝑚1𝑏 + 𝑚𝑏3 ))
= 𝑓(𝑚1𝑏 + 𝑚𝑏3 − 𝑚1𝑎 − 𝑚𝑎2 ) + 180°
If we can set
(3.4)
𝑚1𝑏 + 𝑚𝑏3 − 𝑚1𝑎 − 𝑚𝑎2 = 0 in equation 3.4, then
𝛩2 − 𝛩3 = 180° for all frequencies, and the output of the power divider is balanced.
Figure 3.3 shows an example of a balun power divider consisting of two microstrip gaps.
17
Figure 3.3 Output phase of a balun power divider for 𝜀𝑟 = 3.38
3.3 Balun Power Divider VSWR
The block diagram in figure 3.2 shows that two resistors, 𝑅𝑎 and 𝑅𝑏 , are required
to achieve a balanced output. Even if 𝑅𝑎 and 𝑅𝑏 are really large, the maximum input
18
impedance looking into port 1 is
𝑍𝑜
2
, which means the lowest reflection coefficient
looking into port 1 is:
𝑍𝑜
− 𝑍𝑜
1
2
𝛤1 =
=−
𝑍𝑜
3
+ 𝑍𝑜
2
(3.5)
And the minimum VSWR is:
𝑉𝑆𝑊𝑅𝑚𝑖𝑛
1
1 + ‖− ‖
3 =2
=
1
1 − ‖− ‖
3
(3.6)
And the maximum power gain is:
𝑆21 = 𝑆31 = −3.52 𝑑𝐵
(3.7)
19
Chapter 4
BALUN COUPLERS
4.1 Balun Couplers
Couplers require three lines for balanced outputs; an input line, and two output
lines that are in opposite directions. Figure 4.1 shows two configurations for couplers
with balanced outputs. In figure 4.1, port 1 is the input, and ports 3 and 4 are the outputs,
port 2 is the isolated port. Notice how ports 3 and 4 are in opposite directions; this is
required to achieve a balanced output.
(a)
Figure 4.1 Layouts for a balun coupler
(b)
We will now apply even and odd mode analysis to coupler (a) in figure 4.1. The
two balanced ports are terminated into the impedance Zo, and the input is run by a
generator of 3Vo and internal impedance Zo. The isolated port enters a non-terminated
network. Figure 4.2 shows the even and odd mode excitations for coupler (a).
20
Figure 4.2 Even and odd mode excitation for coupler (a)
for a generator of 3Vo and internal impedance of Zo.
Even and odd mode analysis yields:
𝑍𝑖𝑛
𝑒 𝑜
2(𝑍𝑖𝑛
𝑍𝑖𝑛 − 𝑍𝑜2 )
= 𝑍𝑜 + 𝑒
𝑜
𝑍𝑖𝑛 + 𝑍𝑖𝑛
+ 2𝑍𝑜
(4.1)
𝑒
Where 𝑍𝑖𝑛 is the input impedance for even mode excitation given by:
𝑒
𝑍𝑖𝑛
= 𝑍𝑜𝑒
𝑍𝑙 + 𝑗𝑍𝑜𝑒 𝑡
𝑍𝑜𝑒 + 𝑗𝑍𝑙 𝑡
(4.2)
𝑜
and 𝑍𝑖𝑛 is the input impedance for odd mode excitation given by:
𝑜
𝑍𝑖𝑛
= 𝑍𝑜𝑜
𝑍𝑙 + 𝑗𝑍𝑜𝑜 𝑡
𝑍𝑜𝑜 + 𝑗𝑍𝑙 𝑡
(4.3)
Where 𝑍𝑜𝑒 and 𝑍𝑜𝑜 are the characteristic impedances for the even and odd mode
21
excitations.
𝑡 is the tangent of the electrical length from port 1 to port 2
Since port 2 is an open network (un-terminated), the load impedance,
𝑍𝑙 = −
𝑗𝑘1
tan(𝑘2 f)
(4.4)
Plugging equation 4.4 into equations 4.2 and 4.3 gives:
𝑒
𝑍𝑖𝑛
= 𝑗𝑍𝑜𝑒
𝑍𝑜𝑒 𝑡1 𝑡2 − 𝑘1
𝑍𝑜𝑒 𝑡2 + 𝑘1 𝑡1
(4.5)
𝑜
𝑍𝑖𝑛
= 𝑗𝑍𝑜𝑜
𝑍𝑜𝑜 𝑡1 𝑡2 − 𝑘1
𝑍𝑜𝑜 𝑡2 + 𝑘1 𝑡1
(4.6)
Where 𝑡1 is the tangent of the electrical length of the coupler
𝑡1 = tan(𝛽1 𝑙1 )
𝑡2 is the tangent of the electrical length of the load
𝑡2 = tan(𝑘2 𝑓)
From equation 4.1, Zin = Zo when:
𝑒 𝑜
𝑍𝑖𝑛
𝑍𝑖𝑛 − 𝑍𝑜2 = 0
Substituting equations 4.5 and 4.6 for
𝑒 𝑜
𝑍𝑖𝑛
𝑍𝑖𝑛 − 𝑍𝑜2 =
(4.7)
𝑒 𝑜
𝑍𝑖𝑛
𝑍𝑖𝑛 we get that:
𝑍𝑙 , is given by:
22
−𝑍𝑜𝑒 𝑍𝑜𝑜
(𝑍𝑜𝑒 𝑡1 𝑡2 − 𝑘1 ) (𝑍𝑜𝑜 𝑡1 𝑡2 − 𝑘1 )
− 𝑍𝑜2
(𝑍𝑜𝑒 𝑡2 + 𝑘1 𝑡1 ) (𝑍𝑜𝑜 𝑡2 + 𝑘1 𝑡1 )
(4.8)
To find the frequency for which the right side of equation 4.8 is smallest, we make the
approximations:
𝑡1 = tan(𝛽1 𝑙1 ) ≈ 𝑘𝑐 𝑓
𝑡2 = tan(𝑘2 𝑓) ≈ 𝑘2 𝑓
Where 𝑘𝑐
and
𝜀𝑓𝑓
𝑐
=
𝑘2 =
2𝜋√𝜀𝑓𝑓 𝑙1
𝑐
2𝜋√𝜀𝑓𝑓,𝑙𝑜𝑎𝑑 𝑙𝑙𝑜𝑎𝑑
𝑐
(4.9)
(4.10)
(4.11)
(4.12)
is the effective permittivity of the coupler
is the speed of light
𝑙1 is the length of the coupler, and is shown in figure 4.3
Figure 4.3 Balun coupler with open load circuit.
Plugging 4.9 and 4.10 into the right side of equation 4.8 we get:
𝑒 𝑜
𝑍𝑖𝑛
𝑍𝑖𝑛 − 𝑍𝑜2
23
= −𝑍𝑜𝑒 𝑍𝑜𝑜
2
2
(𝑍𝑜𝑒 𝑘𝑐 𝑘2 𝑓 − 𝑘1 ) (𝑍𝑜𝑜 𝑘𝑐 𝑘2 𝑓 − 𝑘1 )
(𝑍𝑜𝑜 𝑘2 + 𝑘1 𝑘𝑐 )
2
𝑓 (𝑍𝑜𝑒 𝑘2 + 𝑘1 𝑘𝑐)
2
− 𝑍𝑜 (4.13)
To find the minimum value of 4.13, we take the derivative with respect to frequency, and
set it equal to zero; this will give us the coupling frequency at:
𝑓𝑐 = (
𝑘12
𝑍𝑜𝑒 𝑍𝑜𝑜 𝑘𝑐2 𝑘22
1
4
)
(4.14)
If we plug in equations 4.11 and 4.12 into equation 4.14, we get that the coupling
frequency will be inversely proportional to the root of 𝑙1 and 𝑙2 ; this can be written as:
𝑓𝑐 =
𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
√𝑙1 𝑙2
(4.15)
Ideally, the coupling frequency would be independent of length. However, from equation
4.15, shortening the length of a coupler will increase the coupling frequency. This means
we cannot simultaneously decrease the length and coupling frequency. The only other
way to keep the length small and the coupling frequency low is to couple at a higher
frequency, which keeps the coupler short, and then expand the bandwidth into the lower
frequency range.
We can determine the bandwidth from equation 4.13. Coupling occurs when
equation 4.13 is smallest, which is at the frequency given by equation 4.14. At this
coupling frequency, equation 4.13 reduces to:
(4.11)|𝑓=𝑓𝑐 =
24
𝑘1
(𝑍 + 𝑍𝑜𝑜 − 2√𝑍𝑜𝑒 𝑍𝑜𝑜 )
𝑘2 𝑜𝑒
− 𝑍𝑜2 (4.16)
𝑘
𝑘
(𝑍𝑜𝑒 + 𝑘𝑐 1 ) (𝑍𝑜𝑜 + 𝑘𝑐 1 )
𝑘2
𝑘2
𝑍𝑜𝑒 𝑍𝑜𝑜 𝑘𝑐
Figure 4.4 shows the bandwidth for a coupler.
Figure 4.4 Typical bandwidth of a coupler
At the lower cutoff frequency f1:
𝑡1 𝑡2 ≈ 0
𝑡1 = tan(𝛽1 𝑙1 ) ≈ 𝑘𝑐 𝑓
𝑡2 = tan(𝑘2 𝑓) ≈ 𝑘2 𝑓
and equations 4.5, 4.6, and 4.13 reduce to:
𝑒
𝑍𝑖𝑛
𝑘1
𝑘2
=
𝑘
𝑓1 (𝑍𝑜𝑒 + 𝑘𝑐 1 )
𝑘2
(4.17)
𝑜
𝑍𝑖𝑛
𝑘1
𝑘2
=
𝑘
𝑓1 (𝑍𝑜𝑜 + 𝑘𝑐 1 )
𝑘2
(4.18)
−𝑗 𝑍𝑜𝑒
−𝑗 𝑍𝑜𝑜
25
𝑒 𝑜
𝑍𝑖𝑛
𝑍𝑖𝑛 − 𝑍𝑜2
𝑘1 2
−𝑍𝑜𝑒 𝑍𝑜𝑜 ( )
𝑘2
=
− 𝑍𝑜2
𝑘
𝑘
𝑓12 (𝑍𝑜𝑒 + 𝑘𝑐 1 ) (𝑍𝑜𝑜 + 𝑘𝑐 1 )
𝑘2
𝑘2
(4.19)
We can make a rough estimate for the lower cutoff frequency, f1, in figure 4.4.
We can say that when the input impedance has increased by some percentage, the gain
has decreased. Let’s say that when the input impedance has increased by 35 percent (a
real loose estimate), the gain is too low. This occurs when equation 4.19 is 35 percent
larger than equation 4.16. Then:
𝑘1 2
−𝑍𝑜𝑒 𝑍𝑜𝑜 ( )
𝑘2
− 𝑍𝑜2
𝑘
𝑘
𝑓12 (𝑍𝑜𝑒 + 𝑘𝑐 1 ) (𝑍𝑜𝑜 + 𝑘𝑐 1 )
𝑘2
𝑘2
𝑘1
(𝑍 + 𝑍𝑜𝑜 − 2√𝑍𝑜𝑒 𝑍𝑜𝑜 )
𝑘2 𝑜𝑒
− 𝑍𝑜2 ) (4.20)
𝑘
𝑘
(𝑍𝑜𝑒 + 𝑘𝑐 1 ) (𝑍𝑜𝑜 + 𝑘𝑐 1 )
𝑘2
𝑘2
𝑍𝑜𝑒 𝑍𝑜𝑜 𝑘𝑐
= 1.35 (
Which gives the lower cutoff frequency as:
𝑓1 =
1.69
√𝑍𝑜2 (𝑍𝑜𝑒 + 𝑘𝑐
𝑘1
𝑍
𝑘2 √𝑍𝑜𝑒 𝑜𝑜
𝑘1
𝑘
𝑘
) (𝑍𝑜𝑜 + 𝑘𝑐 1 ) − 3.857 𝑍𝑜𝑒 𝑍𝑜𝑜 𝑘𝑐 1 (𝑍𝑜𝑒 + 𝑍𝑜𝑜 − 2√𝑍𝑜𝑒 𝑍𝑜𝑜 )
𝑘2
𝑘2
𝑘2
26
(4.21)
Assuming the coupling frequency is halfway between the lower and upper cutoff
frequencies, the bandwidth will be given by:
𝐵𝑊 = 2(𝑓𝑐 − 𝑓1 )
Solving for
𝑘1
𝑘2
(4.22)
in equation 4.12, and plugging it into equations 4.19 we get:
𝑓1 =
1.69 𝑘𝑐 𝑓𝑐2 𝑍𝑜𝑒 𝑍𝑜𝑜
√𝑥 − 𝑦
(4.23)
Where 𝑥 = 𝑍𝑜2 (𝑍𝑜𝑒 + 𝑘𝑐2 𝑓𝑐2 √𝑍𝑜𝑒 𝑍𝑜𝑜 )(𝑍𝑜𝑜 + 𝑘𝑐2 𝑓𝑐2 √𝑍𝑜𝑒 𝑍𝑜𝑜 )
𝑦 = 3.857( 𝑍𝑜𝑒 𝑍𝑜𝑜 )3/2 𝑘𝑐2 𝑓𝑐2 (𝑍𝑜𝑒 + 𝑍𝑜𝑜 − 2√𝑍𝑜𝑒 𝑍𝑜𝑜 )
If we choose a coupling frequency, equation 4.23 gives the lower cutoff frequency that is
dependent only on Zoe, Zoo, and the length and effective permittivity of the coupler. The
even and odd mode characteristic impedances are positive constants that range from 0 to
200, and therefore, we can plot equation 4.23 for 0<Zoe<200 and 0<Zoo<200. Figure 4.5
shows four plots of equation 4.23. Each plot has a characteristic impedance of 50 Ω, a
coupling frequency of 6 GHz, and a coupler length of 200 mils.
permittivity is increased from 2 to 8.
The effective
27
Figure 4.5 Lower cutoff frequency of a coupler given by equation 4.23 for a
characteristic impedance of 50 Ω, a coupling frequency at 6 GHz, and coupler
length of 200 mils for; (a) 𝜀𝑓𝑓 = 2; (b) 𝜀𝑓𝑓 = 4; (c) 𝜀𝑓𝑓 = 6; (d) 𝜀𝑓𝑓 = 8.
28
4.2 Wideband Coupler Design
For the coupler in figure 4.6, let’s say we have a characteristic impedance of 50Ω,
and a coupling frequency at 6 GHz. From figure 4.5, the minimum lower cutoff
Figure 4.6 Coupler on microstrip.
frequency possible occurs around 1.2 GHz, when Zoe and Zoo are both small and the
effective permittivity is largest (Figure 4.5 plot (d)). The gain of a coupler is largest
when the product of the even and odd mode characteristic impedances equals the
characteristic impedance of the system squared. That is, the gain is maximized when:
𝑍𝑜𝑒 𝑍𝑜𝑜 = 𝑍𝑜2
Therefore, if we design the coupler for maximum bandwidth, the gain will be low, but if
we choose a cutoff frequency a little higher than 1.2 GHz, let’s say at 2 GHz, we have
more freedom to choose values for Zoe and Zoo to keep the gain high. Figure 4.7 shows
the contour diagrams of equation 4.23 with the lower cutoff frequency 𝑓1 = 2 𝐺𝐻𝑧. In
figure 4.7, any point on the curves gives the values of Zoe and Zoo for 𝑓1 = 2 𝐺𝐻𝑧. The
goal is to choose a point on the curve where the product of Zoe and Zoo equals 𝑍𝑜2 .
None of the curves in figure 4.7 have a point where 𝑍𝑜𝑒 𝑍𝑜𝑜 = 𝑍𝑜2 , but we can see that as
the permittivity increases, we get a better chance of getting a point on the curve where
29
Figure 4.7 Contour diagrams of equation 4.21 for a characteristic impedance
of 50 Ω, a coupling frequency at 6 GHz, a coupler length of 200 mils,
and a lower cutoff frequency of 2 GHz for;
(a) 𝜀𝑓𝑓 = 2; (b) 𝜀𝑓𝑓 = 4; (c) 𝜀𝑓𝑓 = 6; (d) 𝜀𝑓𝑓 = 8.
𝑍𝑜𝑒 𝑍𝑜𝑜 = 𝑍𝑜2 , and this occurs when Zoe and Zoo are opposites; one is large and the other
is small. Therefore, we want the permittivity to be large, and we want to choose a large
value of Zoe. We need to keep 𝑍𝑜𝑒 𝑍𝑜𝑜 = 𝑍𝑜2 ; let’s choose Zoe=150Ω, then Zoo will
equal 16Ω. We can now use a 2D field solver to determine the width, spacing, height and
relative permittivity of the dielectric for Zoe=150Ω and Zoo=16Ω. The electrical length
30
is kept under 45° up to the coupling frequency. Figure 4.8 shows the ADS solution for
the width and spacing for Zoe=150Ω and Zoo=16Ω.
Figure 4.8 2D field solver gives the width and
length of a coupler for a desired electrical length and
even and odd mode characteristic impedances.
Figure 4.8 shows that for Zoe=150Ω, Zoo=16Ω, and an average effective permittivity of
8, and a dielectric height of 64 mils, the relative permittivity is 16, and the width and
spacing are 5.2 mils and 0.3 mils respectively. For the electrical length to be 45° at 6
GHz, the length of the coupler is 88.1 mils.
Now we need to determine the load circuit. For simplicity, we will use a single
open stub as the load circuit. It too has an electrical length of 45° at the coupling
frequency (6GHz). The load impedance is given by equation 4.4:
31
𝑍𝑙 =
−𝑗𝑘1
tan(𝑘2 𝑓)
If we use a single open stub with an input impedance of
𝑍𝑙 =
−𝑗𝑧
tan(𝛽𝑙𝑙𝑜𝑎𝑑 )
as the load then:
−𝑗𝑘1
−𝑗𝑧
=
tan(𝑘2 𝑓) tan(𝛽𝑙𝑙𝑜𝑎𝑑 )
Where z is the characteristic impedance of the open stub
𝑙𝑙𝑜𝑎𝑑 is the length of the open stub
Comparing the two sides of the equation we can see that, 𝑘1 equals z, which is the
characteristic impedance of the open stub. If the electrical length of the open stub is set
equal to 45° at the coupling frequency, then:
𝑘2 𝑓𝑐 =
𝜋
4
→
𝑘2 = 1.309 ∗ 10−10
𝑘𝑐 is given by equation 4.11:
𝑘𝑐 =
2𝜋√𝜀𝑓𝑓 𝑙1
𝑐
= 1.32 ∗ 10−10 𝑠
By equation 4.14
𝑘1 = 𝑘2 𝑘𝑐 𝑓𝑐2 √𝑍𝑜𝑒 𝑍𝑜𝑜 = 30.5 Ω
Therefore, the characteristic impedance of the open stub is 30.5Ω, and the electrical
length is 45° at 6 GHz. Figure 4.9 shows the ADS solution for the width and length of
the open stub load.
32
Figure 4.9 2D-field solver gives the width and length given for a characteristic
impedance of 30.5Ω, and an electrical length of 45° at 6 GHz.
From figures 4.8 and 4.9 we have all the widths and lengths to simulate a wideband balun
coupler, with a coupling frequency at 6GHz, a relative dielectric permittivity of 16, a
dielectric height of 64 mils, and a characteristic impedance of 50Ω. Figure 4.10 shows
the simulation results for four different layouts.
33
34
Figure 4.10 Wideband balun coupler design simulation for four balun
Layouts. The dielectric has a relative permittivity of 16, and a height of 64 mils.
35
The circuit layout in figure 4.10c produces the best results, so we will use that one for our
design. For better results, we tune the lengths of the coupler. Figure 4.11 shows the
circuit and simulation results after tuning the lengths.
Figure 4.11 Wideband balun coupler design simulation after tuning the
lengths of the coupler in figure 4.10c.
36
Figure 4.12 shows a table of the simulation at the coupling frequency of 6 GHz.
‖𝑆21 ‖
0.690
‖𝑆31 ‖
0.687
𝑝ℎ𝑎𝑠𝑒(𝑆31 ) − 𝑝ℎ𝑎𝑠𝑒(𝑆21 )
175°
VSWR1
1.48
VSWR2
2.63
VSWR3
2.68
Figure 4.12 Tabulated results of wideband design
at the coupling frequency.
37
Chapter 5
CONCLUSIONS
Achieving a balanced output requires a specific layout. A power divider requires
two resistors and two gaps (figure 3.2), and a coupler requires two grounded lines in
opposite directions (figure 4.1). From the specific layouts, we can derive the gains and
bandwidths.
The design for a wideband balun power divider (WBPD) revolves around the
microstrip gaps. Ideally, we want the microstrip gaps to act as shorts and have a linear
phase over the passband. Therefore, the microstrip material used to design a WBPD,
would be chosen so the gaps have the desired gain and phase over the passband. The
continuous lines in a WBPD will always have a linear phase no matter what material is
used, so they do not need to be considered when choosing a microstrip material. The two
resistors required for a WBPD affect the gain and phase of the outputs. Lowering the
resistor values will make the two output phases more balanced, and lower the gains at the
two output ports.
Increasing the resistor values will make the output phases less
balanced, and increase the gains at the two output ports. Therefore, there is a tradeoff
between gain and a balanced output, and the resistor values are chosen to achieve
whichever is desired more.
The design for a wideband balun coupler (WBC) revolves around the even and
odd mode characteristic impedances. We want to choose a microstrip material where the
even and odd mode characteristic impedances are extreme opposites; the even mode is
very large, and the odd mode is very small. And the product of the even and odd mode
38
characteristic impedances equals the system’s characteristic impedance squared. The
electrical length of the coupler will be 45° at the coupling frequency. The load of the
coupler can be easily derived from the equations which are valid for any microstrip
material used, so the load does not need to be considered when choosing a microstirp
material for a WBC.
The most important thing to determine when designing either a WBPD or WBC,
is the microstrip material. When designing a WBPD, the microstip material is chosen so
that the gaps act as shorts with linear phase, and when designing a WBC, the microstrip
material is chosen so the even and odd mode characteristic impedances are extreme
opposites.
39
REFERENCES
[1] Eric Bogatin, “Signal and Power Integrity-SIMPLIFIED” Second edition. Upper
Saddle River, NJ: Prentice Hall, 2010.
[2] David M. Pozar, “Microwave Engineering” Third Edition. Hoboken, NJ: John Wile &
Sons, 2005.
[3] Guillermo Gonzalez, “Microwave Transistor Amplifiers Analysis and Design”
Second Edition. Upper Saddle River, NJ: Prentice Hall, 1997.
[4] Rowan Gilmore, Les Besser, “Practical RF Circuit Design for Modern Wireless
Systems Active Circuits and Systems” Volume II. Norwood, MA: Artech House, Inc,
2003.
[5] Fawwaz T. Ulaby, “Fundamentals of Applied Electromagnetics” Fifth Edition. Upper
Saddle River, NJ: Prentice Hall, 2007.