Chapter 1 Introduction
1.4 Organization of This Thesis
This thesis is organized as follows. In Chapter 2, the theory of the Butler matrix and SIW are introduced. First, the basic concept of the waveguide is introduced. Second, the concept of SIW is introduced. Then, the theory of the antenna array and the array factor would be derived. The theory of the Butler matrix and the phase of the output port would be derived in the end.
In Chapter 3, the fabrication process of the AFSIW would be introduced. Then, the feeding of the AFSIW would be introduced. In Chapter 4, the stub phase shifter is introduced. First, many kinds of phase shifters would be introduced. And the design formula of the stub phase shifter would be derived. Analyzed by taking the important effect of waveguide junction into consideration. In the end, the AFSIW stub phase shifter is proposed.
Chapter 5 starts with the design of the Butler matrix network. First, the crossover and the coupler would be proposed. Second, the slot antenna array would be designed.
Then, each component would be combined to form the Butler matrix antenna array. And the phase difference of each output port and the simulated performance would be given.
The measurement of the Butler matrix would be given and discussed in the end. Finally, the conclusions of this thesis are given in Chapter 6.
Chapter 2
pter 2 Introduction of SIW and Butler Matrix
The rectangular waveguide is widely used in the microwave and mm-wave communication system, and many applications. Since the disadvantage of the waveguide, such as the size is bulk and non-planar, the new structure is evolved. This new structure is substrate integrated waveguide (SIW). SIW can be used to design the butler matrix network and be a planar integrated circuit. This chapter will first introduce the waveguide and SIW. The array factor of the antenna array will be derived. Then, the Butler matrix will be introduced and analyzed to know the phase difference of each output port. Thus, the combination of the Butler matrix and the antenna array can predict that when the input port input, the main beam will be directed to where.
2.1 Introduction of Waveguide
Rectangular waveguides are often used to transport microwave signals and used for many applications. But the size of the rectangular waveguide is often huger than using microstrip line or strip line. There are two modes to be propagated in a rectangular waveguide. One is TE mode, another is TM mode.
A typical rectangular waveguide is shown in Fig. 2.1. The width is a at the x-axis, and the height is b at the y-axis. Sometimes, the width is larger than height so that 𝑎𝑎 > 𝑏𝑏.
And the waveguide is propagated at the z-axis.
Fig. 2.1 Rectangular waveguide
TE modes
TE modes are characterized by Ez= 0 and Hz≠ 0 . Then Hz(𝑥𝑥, 𝑦𝑦, 𝑧𝑧) = ℎ𝑧𝑧(𝑥𝑥, 𝑦𝑦)𝑒𝑒−𝑗𝑗𝑗𝑗𝑧𝑧 , where propagation constant β = �k2− 𝑘𝑘𝑐𝑐2 , kx2+ 𝑘𝑘𝑦𝑦2 = kc2 and hz(x, y) = (𝐴𝐴 cos 𝑘𝑘𝑥𝑥𝑥𝑥 + 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝑘𝑘𝑥𝑥𝑥𝑥)�𝐶𝐶 cos 𝑘𝑘𝑦𝑦𝑦𝑦 + 𝐷𝐷𝐵𝐵𝐵𝐵𝐵𝐵𝑘𝑘𝑦𝑦𝑦𝑦�.
Since Maxwell’s equations,
∇ × 𝐸𝐸�⃑ = −jωµ𝐻𝐻��⃑ ( 2.1 )
∇ × 𝐻𝐻��⃑ = jωϵ𝐸𝐸�⃑ ( 2.2 ),
can be reduced to
Hy = −𝑗𝑗
and apply the boundary conditions on the electric field that tangential to the waveguide walls: ex(𝑥𝑥, 𝑦𝑦) = 0, 𝑎𝑎𝑎𝑎 𝑦𝑦 = 0 𝑎𝑎𝐵𝐵𝑎𝑎 𝑏𝑏, ey(𝑥𝑥, 𝑦𝑦) = 0, 𝑎𝑎𝑎𝑎 𝑥𝑥 = 0 𝑎𝑎𝐵𝐵𝑎𝑎 𝑎𝑎 [26]. where Amn is the amplitude of the multiplication of A and C, the propagation constant is
The cutoff frequency of each mode is
fCmn = kc
2𝑚𝑚√𝜔𝜔𝜔𝜔 = 1
2𝑚𝑚√𝜔𝜔𝜔𝜔��𝑚𝑚𝑚𝑚 𝑎𝑎 �
2+ �𝐵𝐵𝑚𝑚 𝑏𝑏 �
2 ( 2.15 ).
Because a is larger than b. So that TE10 mode is the dominant mode, where m = 1, n = 0. The field of TE10 mode is shown in Fig. 2.2. The solid line is the electric field line. The dashed line is the magnetic field line.
Fig. 2.2 TE10 mode
TM modes
𝑒𝑒𝑧𝑧(𝑥𝑥, 𝑦𝑦)𝑒𝑒−𝑗𝑗𝑗𝑗𝑧𝑧 , where propagation constant β = �k2− 𝑘𝑘𝑐𝑐2 , kx2+ 𝑘𝑘𝑦𝑦2 = kc2 and ez(x, y) = (𝐴𝐴 cos 𝑘𝑘𝑥𝑥𝑥𝑥 + 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝑘𝑘𝑥𝑥𝑥𝑥)�𝐶𝐶 cos 𝑘𝑘𝑦𝑦𝑦𝑦 + 𝐷𝐷𝐵𝐵𝐵𝐵𝐵𝐵𝑘𝑘𝑦𝑦𝑦𝑦�.
From equation ( 2.3 )-( 2.6 ) and apply the boundary conditions on the electric field that tangential to the waveguide walls: ez(𝑥𝑥, 𝑦𝑦) = 0, 𝑎𝑎𝑎𝑎 𝑥𝑥 = 0 𝑎𝑎𝐵𝐵𝑎𝑎 𝑎𝑎 , ez(𝑥𝑥, 𝑦𝑦) = where Bmn is the amplitude of the multiplication of B and D, the propagation constant is
The cutoff frequency of each mode is
fCmn = kc
2𝑚𝑚√𝜔𝜔𝜔𝜔 = 1
2𝑚𝑚√𝜔𝜔𝜔𝜔��𝑚𝑚𝑚𝑚 𝑎𝑎 �
2+ �𝐵𝐵𝑚𝑚 𝑏𝑏 �
2 ( 2.24 ).
Because Ez includes m and n. If m or n is equal to zero, the Ez will be equal to zero. So that TM11 mode is the lowest order mode, where m = 1, n = 1.The field of TM11 mode is shown in Fig. 2.3. The solid line is the electric field line. The dashed line is the magnetic field line.
2.2 Introduction of SIW
Because the traditional rectangular waveguide is bulky, non-planar, and more restrictive in manufacturing, it is difficult to be the mm-wave planar integrated circuits [27]. The substrate integrated waveguide (SIW) is a structure using a printed circuit board (PCB) and metallic via holes on both sides, it’s shown in Fig. 2.4. SIW structure is like a rectangular waveguide so that SIW can preserve the advantage of traditional rectangular waveguides, such as the high Q factor and high-power capacity [28]. And also, the advantage of the SIW is a small size and the planar circuit, which can improve the disadvantages of the traditional rectangular waveguide.
Since SIW is made up of metallic via holes, there are many gaps. From these gaps, it can be judged that SIW may produce the leakage problem or wave attenuation. In addition, the substrate of the SIW is lossy dielectric, so some losses were produced by the substrate.
Fig. 2.4 SIW structure (Top view)
The equivalent width of the SIW (weff) is [28]
𝑤𝑤𝑒𝑒𝑒𝑒𝑒𝑒 = 𝑤𝑤 − 1.08 ∙𝑎𝑎2
𝐵𝐵 + 0.1 ∙ 𝑎𝑎2
𝑤𝑤 ( 2.25 ),
where w is the width of the SIW, s is the pitch of the metallic via holes and d is the diameter of the metallic via holes, it’s shown in Fig. 2.4.
As soon as, the gap of metallic via holes s is too large, the large radiation loss will be produced. The return loss of the SIW transition will be affected by the diameter of the metallic via holes (d). There are two useful criteria to establish upper bounds for s and d [29].
d ≤ λg
5 ( 2.26 )
where λg is the guided wavelength and the width of waveguide equal 𝑤𝑤𝑒𝑒𝑒𝑒𝑒𝑒 . So that when the fundamental mode of SIW is TE10 mode, the guided wavelength of the SIW is equal to
λg10 = 2𝑚𝑚
��𝜔𝜔𝑟𝑟𝜔𝜔2
𝑐𝑐2 �2− � 𝑚𝑚𝑤𝑤𝑒𝑒𝑒𝑒𝑒𝑒�2
( 2.28 ).
If the first higher mode of SIW is TEm0 mode, the limit of the via diameter equation ( 2.26 ) can be simplified from equation ( 2.15 ) and equation ( 2.28 ), which is given by
d ≤ 2weff
5√𝑚𝑚2− 1 ( 2.29 ).
2.3 Introduction of antenna array
An antenna array is composed of N single antennas, which are separated at the same distance. The antenna array can achieve higher directivity and narrower beam than the single element. The antenna array also can control the direction of the beam by the phase difference. The antenna array is regarded as many isotropic point sources be separated the same distance. Then the radiation pattern can be calculated by the principle of pattern multiplication [30],
Radiation pattern of antenna array
= Array factor × Radiation pattern of individual antenna ( 2.30 ).
Fig. 2.5 shows the antenna array with n elements. Assume all individual antennas are isotropic point sources with the same distance. Then the electric field of the antenna array can be calculated as
E = 𝐸𝐸0+ 𝐸𝐸0ej𝜓𝜓+ 𝐸𝐸0ej2𝜓𝜓+ ⋯ + 𝐸𝐸0ej(n−2)𝜓𝜓+ 𝐸𝐸0ej(n−1)𝜓𝜓 ( 2.31 ),
where 𝜓𝜓 =2𝑚𝑚𝜋𝜋𝜆𝜆 𝑐𝑐𝑐𝑐𝐵𝐵 (90° − 𝜃𝜃) + 𝑏𝑏 , d is the distance between the point sources, and b is the phase difference. Then equation ( 2.31 ) can be simplified as
E = E01 − 𝑒𝑒𝑗𝑗𝑛𝑛𝜓𝜓
1 − 𝑒𝑒𝑗𝑗𝜓𝜓 = E0𝑒𝑒𝑗𝑗𝑛𝑛𝜓𝜓2
𝑒𝑒𝑗𝑗𝜓𝜓2 �𝑒𝑒𝑗𝑗𝑛𝑛𝜓𝜓2 − 𝑒𝑒−𝑗𝑗𝑛𝑛𝜓𝜓2
𝑒𝑒𝑗𝑗𝜓𝜓2 − 𝑒𝑒−𝑗𝑗𝜓𝜓2 � = E0𝑒𝑒𝑗𝑗𝑛𝑛−12 𝜓𝜓sin 𝐵𝐵𝜓𝜓2
sin 𝜓𝜓2 ( 2.32 ).
The maximum of the electric field occurs on 𝜓𝜓 = 0. The array factor is the normalized total electric field. Thus, the equation of the array factor (AF) of the antenna array can be obtained by
This analysis is assumed that the amplitude and the phase difference of all point sources are the same.
Fig. 2.5 Antenna array with N elements
2.4 Introduction of Butler Matrix
Butler matrix is a beamforming network for the multibeam antenna array. Butler matrix is a low loss network. The block diagram of the Butler matrix is shown in Fig. 2.6.
Butler matrix is composed of four 3-dB couplers, two crossovers, two 45° phase shifters, and two 0° phase shifters.
The output port of the butler matrix will be satisfied with two conditions. One is that the amplitude of each output port will be the same. Another is that when the same input port is input, the phase difference between adjacent output ports are the same.
Fig. 2.6 Block of a 4x4 butler matrix
In the ideal case, the insertion loss of the coupler is 3 dB. The insertion loss of the crossover and the phase shifter are 0 dB. Thus, the total loss of each output port of the butler matrix is 6 dB. On the coupler, the phase difference between the coupled port and
the through port is 90 degrees. And the phase difference between the crossover and the 0° or 45° phase shifter is 0 degrees and 45 degrees, respectively. Assumed that couplers and crossovers are symmetric structure. The clearer descriptions are transformed into formulas as following:
The phase of each output port of the Butler matrix and phase difference of the Butler matrix are computed at the following. And at the following, the PD is represented the phase difference of the Butler matrix.
Input port at port 1
∠S51= ∠SBA+ ∠SEB+ ∠SFE+ ∠SIF ( 2.39 )
∠S61 = ∠SDA+ ∠SOD+ ∠SPO+ ∠SJP ( 2.40 )
∠S71 = ∠SBA+ ∠SEB+ ∠SHE+ ∠SSH ( 2.41 )
∠S81= ∠SDA + ∠SOD+ ∠SRO+ ∠STR ( 2.42 )
PD1 = ∠S61− ∠S51
PD5 = ∠S72− ∠S62
When the input ports are port 3 and port 4, the calculation method is the same as the input ports are port 2 and port 1. So that when the input ports are port 1, port 2, port 3, and port 4, the phase difference is −45°, 135°, −135°, and 45°, respectively. The phase of the output port and the phase difference can be summarized in Table 2.1.
Output
Table 2.1 Phase of output ports of the Butler matrix
Then, the antenna array is fed by the Butler matrix. From the equation ( 2.33 ), the array factor of the antenna array with 4 elements can be expressed as
AF = 1
where 𝜃𝜃 is the angle of the main beam, b is the phase difference. Thus, the different phase difference can obtain the different direction of the main beam.
Assumed the distance (d) equal half wavelength (λ/2) to substitute into the equation ( 2.53 ) - ( 2.55 ). And the phase difference (b) is substituted by the phase difference of each output port of the Butler matrix. The radiation pattern of the butler matrix is shown in Fig. 2.7. The radiation pattern is the cut section on the spherical coordinate system when ϕ = 0 . However, the angle of Fig. 2.7 is the polar angle (θ) on the spherical coordinate system. From Fig. 2.7, it’s shown that when input port is port 1, port 2, port3, and port 4, the direction of the main beam is 14.5°, −48.6°, 48.6°, and −14.5° on the spherical coordinate, respectively. So that when phase difference is ±45° , the direction of the main beam is ∓14.5°. The direction of the main beam is mirrored at 0°.
Fig. 2.7 The radiation pattern of the butler matrix when ϕ = 0 (x-z) cut on the spherical coordinate system.
Chapter 3
pter 3 Novel Structure of AFSIW
AFSIW on PCB is a novel structure. So, the fabrication process and stacking of AFSIW on PCB will be introduced in this chapter. And the feeding method of AFSIW
also will be introduced in this chapter.
3.1 AFSIW Fabrication
In this thesis, we use the AFSIW structure, which is composed of three cores stuck by the prepreg (PP) and copper paste. The description of each layer of AFSIW is shown in Fig. 3.1. The CORE1 is a low loss material, whose Dk = 3 and Df = 0.0019 at 10 GHz. The properties of the material of the CORE2 and CORE3 is Dk = 4.1 and Df = 0.016 at 10 GHz.
Fig. 3.1 The description of each layer of AFSIW
The fabrication process of the AFSIW as following:
Step1: CORE2, M3, and M4 are dug with the desired width and length of the AFSIW. It’s shown in Fig. 3.2(a).
Step2: CORE2 and CORE3 are stuck by the PP2. It’s shown in Fig. 3.2(b).
Step3: The copper is plated around the dug M3 to PP2. The bottom of the AFSIW is formed. It’s shown in Fig. 3.2(c).
Step4: CORE1 and CORE2 are stuck by the PP1. It’s shown in Fig. 3.2(d)
Step5: A row of the through-hole via on the two sides of the air channel to form the metal wall on the PP1 layer. The complete AFSIW is formed. It’s shown in Fig. 3.2(e).
(a)
(b)
(c)
(d)
(e)
Fig. 3.2 The fabrication process of the AFSIW (a) step1 (b) step2 (c) step3 (d) step4 (e) step5
The front view of the AFSIW is shown in Fig. 3.3. It’s shown that the air channel of the AFSIW is formed by M2 to M5, the dug CORE2, and some PP1. The structure of AFSIW is like a T-shaped waveguide, which is the slash section in Fig. 3.3. Due to the fabrication process, there is a prepreg layer embedded in the top of the T-shaped structure (only on two sides, shown by the dot part in Fig. 3.3 slash section). We can still regard it as a traditional rectangular waveguide, as long as the prepreg sizes on two sides are not too large.
But compared with the waveguide equation derived in Chapter 2.1, there will be some slight errors on the properties of the AFSIW, such as the cutoff frequency, guided wavelength or propagation constant. And also, there will be a slight error on transmission loss. Those errors are caused by the T-shaped structure and the embedding prepreg. Since the width of the AFSIW is larger than the height of the AFSIW. The fundamental mode propagating in the AFSIW is TE10 mode.
The geometrical dimensions of the AFSIW is shown in Fig. 3.3 and Table 3.1. In Table 3.1, s is the pitch of the through-hole via.
Fig. 3.3 The front view and geometrical dimensions of the AFSIW
Geometrical dimensions
Table 3.1 The geometrical dimension of the AFSIW
However, when the Butler matrix is fabricated, some places are surrounded by air channels from three sides or all around. At this time, the above fabrication process might encounter some fabrication process problems which makes the AFSIW collapse. So, the other method must be used.
The modified method is that the PP1 is changed to the copper paste on step 4 and step 5 is removed because the copper paste forms the metal wall on the PP1 layer. The front view of the modified AFSIW is shown in the Fig. 3.4. The modified AFSIW is almost equal to the waveguide. So, the properties of the AFSIW, such as the cutoff frequency, guided wavelength or propagation constant, can use the waveguide equation derived on Chapter 2.1. Since the width of the AFSIW is larger than the height of the AFSIW. The fundamental mode propagating in the AFSIW is TE10 mode. The geometrical dimensions of the AFSIW is shown in Fig. 3.4 and Table 3.2.
Fig. 3.4 The front view and geometrical dimensions of the modified AFSIW
Geometrical dimensions h1, h5 = 5 mil m1, m6 = 48 um h2, h4 = 3 mil m2, m5 = 35 um h3 = 20 mil m3, m4 =42 um a = 2.8 mm @60GHz b = 0.744 mm
Table 3.2 The geometrical dimensions of the modified AFSIW
The simulation results and measurements of the AFSIW and modified AFSIW on HFSS software are shown in Fig. 3.5. In Fig. 3.5(a), it’s shown that when the width of the AFSIW is 2.8mm, the cutoff frequency is equal to 52.3 GHz, which is not equal to the equation (2.15). But, when the width of the modified AFSIW is 2.8mm, the cutoff frequency is equal to 53.5 GHz, which matches the equation (2.15). The loss of the AFSIW and the modified AFSIW are 0.1 dB/cm and 0.064 dB/cm, respectively. These results verify that modified AFSIW doesn’t include the prepreg and is similar to the waveguide, so loss is very low and the cutoff frequency the same as the waveguide.
In Fig. 3.5(b), the cutoff frequency of AFSIW and modified AFSIW are about 51.9
(a)
(b)
Fig. 3.5 The simulation and measurement of the AFSIW on HFSS (a) Simulation (b) Measurement
3.2 AFSIW Feeding
There are several methods for feeding the SIW, such as the transition between microstrip line to SIW with tapered microstrip feeding, by probe feeding or by slot coupling [31].The slot coupling for feeding AFSIW is used in this thesis, which refers to [31] and [32] to design AFSIW feeding.
The design method of AFSIW feeding is introduced as follows. The signal of the microstrip line is at M1 and the ground of the microstrip line is at M2. The signal is propagated from M1 to the slot at M2 to couple into the air channel. There is a one-quarter wavelength open stub, which is used to short the microstrip line. The slot whose length is half-wavelength is used to make the power couple into the air channel to complete feeding.
The structure of AFSIW feeding is shown in Fig. 3.6. In Fig. 3.6(c), the air channel is stepped because it makes the power easily couples into the air channel.
(a)
(b)
(c)
Fig. 3.6 The structure of the AFSIW feeding. (a) 3D view (b) Top view (c) Side view
The simulation results and measurement of AFSIW feeding on HFSS software are shown in Fig. 3.7. It’s shown that the bandwidth (defined by S11 lower than -15 dB) is 58.3 to 62.3 GHz on simulation. The minimum loss of the feeding is about -0.37 dB on simulation. The bandwidth of the measured is 58.4 to 61.6 GHz. The measured minimum loss is about 0.4 dB.
Fig. 3.7 The simulation results of AFSIW feeding on HFSS.
Chapter 4
pter 4 AFSIW Stub Phase Shifter
Corresponding to the Butler matrix architecture in Fig. 2.6, the main goal of this chapter is to design a phase shifter which provides a 0/45-degree phase difference compared with the crossover. And in order to reduce the complexity of the path on Butler matrix circuit design and reduce the circuit size, the phase shifter should be designed with the same physical length of the crossover.
There are many methods to design the phase shifter on the SIW structure. However, in the AFSIW fabrication process, some methods will encounter some fabrication problems. Thus, this chapter will introduce the methods to design the phase shifter on the SIW structure and discuss the problems about these methods on the AFSIW structure.
And the reason that the stub phase shifter is chosen in this thesis is explained. The stub phase shifter designed in AFSIW structure is analyzed. Next, the design rule of the proposed phase shifter is introduced.
4.1 Introduction of Stub Phase Shifter
Several typical methods for SIW phase shifter design are shown in Fig. 4.1. In Fig.
4.1, it is shown the top view of the SIW phase shifter.
The first method is a curved phase shifter as shown in Fig. 4.1(a). The curved phase shifter can achieve the requirements of the desired electrical length and the same physical length of the crossover. The disadvantage of this method is that the return loss is relatively poor. The second method can improve this disadvantage. The second method is the straight phase shifter which changes the width of the SIW to achieve the requirement of the desired electrical length, as shown in Fig. 4.1(b). Because the propagation constant is affected by the variation of the width of the SIW, the guided wavelength is also affected.
So, the phase shifter with the desired electrical length and the same physical length of the crossover is proposed. The overall size of the second method is smaller than the first method.
But these methods will encounter some problems in the AFSIW fabrication process.
For the first method, the curved line should be fabricated accurately. As soon as the curvature is wrong, the significant phase error will be provided. For the second method, if the minimum unit of the width of the SIW is too small, the fabrication process cannot be accurate. Thus, in order to avoid the above problems, we choose stub phase shifter for our design, as shown in Fig. 4.1(c).
Stub phase shifter doesn’t require the accurate curvature and too precise unit width in the AFSIW fabrication process. And also, the stub phase shifter can be easily designed with desired electrical length and the same physical length of crossover. So, the stub type is chosen for the AFSIW Butler matrix design at this thesis.
(a) (b)
(c)
Fig. 4.1 Top view of SIW phase shifter (a) Curved phase shifter (b) Straight phase shifter(c) Stub phase shifter
As shown in Fig. 4.1(c), a short-ended or open-ended stub is connected to the middle
As shown in Fig. 4.1(c), a short-ended or open-ended stub is connected to the middle