Chapter 2 Introduction of SIW and Butler Matrix
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 of the original transmission line. The simple equivalent circuit for the stub phase shifter with short stub is shown in Fig. 4.2.
Fig. 4.2 The equivalent circuit of the stub phase shifter
A short stub with the characteristic impedance of Z2 and electrical length of θ2 is shunted in the middle of two transmission lines with the characteristic impedance of Z1 and electrical length of θ1 , which is equivalent to a transmission line with the characteristic impedance of Z3 and electrical length of θ3. Then, the ABCD matrix of the transmission line and short stub is used to calculate the characteristic impedance of the transmission line and short stub, and the electrical length of the transmission line and short stub. The above derivation is shown in formula ( 4.1 ).
� cos 𝜃𝜃1 𝑗𝑗𝑍𝑍1𝐵𝐵𝐵𝐵𝐵𝐵𝜃𝜃1
Then simplified the formula ( 4.1 ) as
⎣⎢
From formula( 4.2 ), it can obtain three equation as following:
𝑍𝑍 𝐵𝐵𝐵𝐵𝐵𝐵2𝜃𝜃
𝑍𝑍1𝐵𝐵𝐵𝐵𝐵𝐵2𝜃𝜃1 +𝑍𝑍12sin2𝜃𝜃1 For example, we can choose the target impedance and phase as Z3 = 443Ω and θ3 = 45° and also assume Z2 = 337Ω and θ1 = 28.5° are given. Then, the Z1 = 337Ω and θ2 = 68.5° is obtained by the equation ( 4.8 ) and ( 4.9 ). The ADS software is used to verify this data. And also, we can transfer impedance and phase to corresponding structural parameters in AFSIW to simulate a stub phase shifter in HFSS.
The results are shown in Fig. 4.3.
(a)
(b)
(c)
Fig. 4.3 Use the ADS and HFSS to simulate the stub phase shifter (a) ADS circuit (b) Magnitude (c) Phase
From Fig. 4.3, it’s shown that the S11 of ADS is very deep at the 60 GHz, the S21 of ADS is close to 0 dB and the phase is clearly at −45°. So, following the design flow mentioned above, we can design an ideal stub phase shifter. But the simulation results of the HFSS don’t match the results of the ADS. Thus, it implies that the T-junction effect on the AFSIW might not be ignored in the model of phase shifter. In the next section, we will discuss and analyze the model of AFSIW stub phase shifter.
4.2 Analysis of Stub Phase Shifter in AFSIW
From Fig. 4.3(b) and (c), the mismatch of results implies that T-junction effect should be taken into consideration. In this section, we will try to model T-junction to get a revised equivalent circuit model for phase shifter.
The T-junction in the waveguide is discussed by the waveguide handbook [33].
There are two types of T-junction in the waveguide. One is the E-plane T-junction, another is the H-plane T-junction. In the waveguide, the E-plane T-junction is extended in the height direction (E-plane). The H-plane T-junction is extended in the width direction (H-plane). They are shown in Fig. 4.4.
(a) (b)
Fig. 4.4 (a) E-plane T-junction (b) H-plane T-junction
The E-plane T-junction might encounter some problems in the AFSIW fabrication process. First, the substrate thickness cannot fit the desired height. Second, the fabrication for connecting the AFSIW different layers is difficult. So, the H-plane is the best choice for design. The equivalent circuit of the H-plane T-junction in the waveguide is shown in
Fig. 4.5 The equivalent circuit of H-plane T-junction
From Fig. 4.5, two impedance of jXa are shunted a series of complex circuits in the middle, which is composed of jXb, jXc and jXd.The ABCD matrix of the T-junction is
A =x
There are some limits to these equations. First, the wavelength should be inside the range between the width of the waveguide and twice the width of the waveguide (a <
λ < 2a). Second, if the waveguide is in the range of 0 < 2a/λg< 1, the error will be less than 15 percent. Third, the width of the junction’s three-port should be the same width.
For the short-ended stub phase shifter, the short stub with the characteristic impedance of Z1 and electrical length of θ2 is connected after the impedance of jXd. This circuit is called the middle network here. The equivalent circuit is shown in Fig. 4.6.
Fig. 4.6 The equivalent circuit of H-plane T-junction with short stub
The ABCD matrix of the middle network is equal to
�𝐴𝐴𝑚𝑚𝑖𝑖𝜋𝜋𝜋𝜋𝑚𝑚𝑒𝑒 𝐵𝐵𝑚𝑚𝑖𝑖𝜋𝜋𝜋𝜋𝑚𝑚𝑒𝑒 middle network are extended to form a stub phase shifter with the characteristic impedance of Z3 and the electrical length of θ3. And this circuit is called the unfinished (UF) network here. The ABCD matrix of the UF network can be obtained by
�𝐴𝐴𝑈𝑈𝐼𝐼 𝐵𝐵𝑈𝑈𝐼𝐼
As shown in ( 4.20 ), the stub phase shifter equivalents to the waveguide with the characteristic impedance of Z3 and the electrical length of θ3. Two sides of the overall short-ended stub phase shifter should be designed with the characteristic impedance of Z3 to match the UF network. As shown in the region AA’- BB’ and EE’- FF’ in Fig. 4.7, two sides of the UF network are connected to the waveguide with the characteristic impedance of Z3 and the electrical length of θ4. So, the equivalent circuit of the overall short-ended stub phase shifter is completed. It’s shown in Fig. 4.7.
Fig. 4.7 The equivalent circuit of the overall short-ended stub phase shifter
The ABCD matrix of the overall short-ended stub phase shifter is given by
�𝐴𝐴𝑗𝑗𝑜𝑜𝑒𝑒𝑟𝑟𝑎𝑎𝑚𝑚𝑚𝑚 𝐵𝐵𝑗𝑗𝑜𝑜𝑒𝑒𝑟𝑟𝑎𝑎𝑚𝑚𝑚𝑚
As shown in ( 4.21 ), the overall short-ended stub phase shifter equivalents to the waveguide with the characteristic impedance of Z3 and the electrical length of θ5.
4.2.1 Design Procedure
As shown in Fig. 4.7 and formula ( 4.21 ), except the parameters Z3 and θ5 are
determine the width of the T-junction in waveguide (a1) at first to obtain the parameters of the junction (𝑗𝑗𝑋𝑋𝑎𝑎, 𝑗𝑗𝑋𝑋𝑏𝑏, 𝑗𝑗𝑋𝑋𝑐𝑐, 𝑗𝑗𝑋𝑋𝜋𝜋) and the characteristic impedance (Z1). Second, the θ1, θ2 and θ4 are calculated in MATLAB. The detailed procedure is listed as follows:
Step 1: The characteristic impedance of the waveguide is related to the width of the waveguide. When the fundamental mode of the waveguide is the TE10 mode, the characteristic impedance of the waveguide is equal to [34]
wherwhere a is the width of the waveguide, and b is the height of the waveguide.
As shown in formula ( 4.11 ) - ( 4.18 ), the parameters 𝑗𝑗𝑋𝑋𝑎𝑎, 𝑗𝑗𝑋𝑋𝑏𝑏, 𝑗𝑗𝑋𝑋𝑐𝑐, 𝑗𝑗𝑋𝑋𝜋𝜋 are related to the width of the T-junction in waveguide (a1) with the characteristic impedance of Z1.
Thus, if the width of the T-junction in waveguide (a1 ) is determined, only parameters θ1, θ2 and θ4 are unknown.
Step 2: Since the physical length of the phase shifter is same to the physical length of the crossover, θ4 can be simplified as
where 𝑙𝑙𝑐𝑐𝑟𝑟𝑗𝑗𝑐𝑐𝑐𝑐 is the physical length of the crossover, 𝜆𝜆𝑔𝑔,𝑎𝑎𝑖𝑖 is the guided wavelength of the waveguide with the characteristic impedance of Z𝑖𝑖, where 𝐵𝐵 = 1,3.
The parameters θ1 ,θ2 and θ4 are calculated by the formula ( 4.11 ) - ( 4.23 ) in MATLAB. Thus, the overall short-ended stub phase shifter is designed.
For example, given the height of the waveguide b is 0.744mm and the width of the waveguide a3 is 2.8mm, the characteristic impedance of the waveguide Z3 = 445Ω at
f = 60 Hz can be calculated by formula ( 4.22 ). And we set the desired phase (θ5) shift from the phase shifter to be 45 degree. Then we can start to analyze the UF network part.
First, we assume the width of the T-junction in waveguide (a1) is 3mm. Second,
θ1 = 70° and θ2 = 40° are calculated by the formula ( 4.11 ) - ( 4.23 ) in MATLAB.
The desired stub phase shifter is proposed. This example is simulated and verified by HFSS. The results of the MATLAB and the HFSS are shown in Fig. 4.8.
In Fig. 4.8, the result of HFSS is slightly mismatched with the result of MATLAB.
In Fig. 4.8, the result of HFSS is slightly mismatched with the result of MATLAB.