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1. Introduction 1.1 Edge-coupled directional coupler Microstrip directional couplers are extensively used in microwave engineering to monitor forward and backward traveling waves on a microstrip line. For instance a directional coupler placed on the output of a power amplifier can detect if the load is disconnected, which can be further used to shut down the amplifier to prevent and damge. Below is a typical microstrip directional coupler implemented using coupled microstrip lines. when two unshielded transmission lines are close together, power can be coupled between the lines due to the interaction of the electromagnetic fields of each line. Such lines are referred to as coupled transmission lines. The microstrip lines are parallel to each other for this reason this circuit is also described as an edge-coupled circuit. As shown in the graph the separation between the lines is s and the width of the lines in the coupling region is w. We can apply an even-odd mode analysis to a length of coupled line to arrive at the design equations for a single-section coupled line coupler. The four-port network is terminated in the impedance Zo at three of its ports and driven with a voltage generator of 2V and internal impedance Zo at port 1. The maximum coupling from port 1 to port 3 occurs when the coupling length is one quarter-wavelength, i.e. θ= π/2. It should be noted that port 4 always has zero output, irrespective of the electrical length of the coupling region. In practical circuits, a major cause of the poor isolation may be unequal even and odd mode phase velocities. The maximum coupling to port 2 occurs at the frequency that gives quarter wave coupling length. This will be the mid-band frequency. Because of this property, these couplers are also known as quarter wavelength couples. At this maximum coupling frequency, the through-line voltage V2 is 90 degree out of phase with the coupled line voltage V3, i.e. this coupler may be described as a quadrature coupler. The coupled voltage V3 is in phase with V1 and thus V 2 lags V1 by 90 degree. 1.2 broadband directional coupler As the coupling of a single-section coupled line coupler is limited in bandwidth due to the quarter wavelength requirement, to increase the bandwidth, multiple sections are used in couplers. Because the phase characteristics are usually better, multisection coupled line couplers are generally made with an odd number of sections. We assume that N is odd and each section is quarter wavelength at the center frequency, where N is the total number of sections in the coupler. Multisection couplers of this form can achieve decade bandwidths. But coupling levels must be low. Because of the longer electrical length, it is more critical to have equal even and odd mode phase velocities than it is for the single section coupler. Mismatched phase velocities will degrade the coupler directivity, as will junction discontinuities, load mismatches and fabrication tolerances. 2. Design method 2.1 Design of coupled-line 10 dB directional coupler, 11.5 -12.5 GHz To calculate the values of S and W, first we need to know the even and odd mode impedance of the coupled lines ( Z 0e and Z 0O ). While by leaving complicated mathematic aside, the two can be obtained from the following equations. Z 0 Z 0 e Z 0O C Z 0e Z 0o Z 0e Z 0o so Z 0e Z 0 1 C 1 C Z 0O Z 0 1 C 1 C Z 0 is the characteristic impedance of our design, or 50 Ω, specifically. C is the coupling coefficient, whose unit is usually in dB c xdB 10 X / 20 Based on the calculated even and odd mode impedance Z 0e and Z 0O , the ratios of and S h W can be read from figure 1. Since h, height of the substrate is already known, h S W can be found. Figure 1 To find the coupling length L, we start by finding the ¼ wavelength of central frequency and then divide it by the root square of effective dielectric constant εeff, namely l 1 1 0 ……………………(2.X) 4 4 e f f o c f ( c is the velocity of light in free space ) To find εeff eff eff ( e ) eff ( o ) 2 εeff for the even and odd mode, namely eff (e ) & eff (o ) , can be read from graph below Coupling coefficient c 2.2 Broadband 10 dB directional coupler, 6- 18GHz Multisection coupled line couplers are generally made with an odd number of coupled line sections. If the multisection coupler is symmetrical, then C1=CN, C2=CN-1 and so on. We have the Fourier series form 1 V3 jV1 sin e1 jN [C1 cos( N 1) C 2 cos( N 3) ... C M ………(b) 2 At the center frequency, the voltage coupling factor C0 V3 V1 / 2 In our design, we use three sections, with section 1 and 3 being symmetrical. For a three-section (N=3) coupler, we require dn C ( ) d n / 2 0 , for n=1, 2. From (b), C V3 1 2 sin C1 cos 2 C 2 V1 2 C1 sin 3 (C 2 C1 ) sin ………………………… (3) To find second derivative of (b) dC 3C1 cos 3 (C 2 C1 ) cos / 2 0 d d 2C 9C1 s i n3 (C 2 C1 ) s i n / 2 0 ……………………(4) d 2 Based on equation (3) and (4), with = π/2, relation between C1 and C2can be found C C2 2C1 10C1 C2 0 Because C1 =C3, coupling factor of each section is known. Then dimension of S W & L for each section is calculated the same with narrow band design. 3. Calculation 3.1 Coupled-line 10 dB directional coupler, 11.5 -12.5 GHz, on a substrate with a 50 Ω characteristic impedance system. The substrate permittivity is 2.5. The substrate height is 0.75mm. Solution: Start out by finding the coupling coefficient C of 10 dB gives c 10dB 1010 / 20 0.316 The even and odd mode characteristic impedances are 1 C 69.4 1 C Z 0e Z 0 Z 0O Z 0 1 C 36 1 C From figure 1, the dimension of microstrip coupled lines can be obtained S =0.12 h S 0.12h 0.09 W = 2.1 h W 2.1h 1.575 From figure 2, eff (e) 0.86 r 2.15 eff (o) 0.78 r 1.95 eff eff ( e ) eff ( o ) 2 1.431 o eff c 3 108 0.025mm f 12 10 9 0.025 0.01747 m 17.47 mm 1.431 l 4 4.37 mm 3.2 Broadband Coupled-line 10 dB directional coupler, 6 - 18 GHz, on a substrate with a 50 Ω characteristic impedance system. The substrate permittivity is 10 and the substrate height is 0.75 mm. Solution: Start by finding C1 C2 C3 since C = 0.316 10C1 C2 0 C2 2C1 0.316 It can be found C1 C3 0.0395 C2= 0.395 The even and odd mode characteristic impedances are Z 0e Z 0e 52 1 3 Z 0o Z 0o 48 1 3 Z 0e 76 2 Z 0o 32.9 2 From figure 1 S1 S 3 2.796 S1 = S3 = 2.097 h h W1 W3 2.8 W1 = 2.1 h h S2 0.07 h S2 = 0.0525 W2 2.1 W2 = 1.575 h For C1 and C3 =0.0395, to find coupling length eff (e) 0.87 r 2.175 eff ( o ) 0.82 r 2.05 eff ( e ) eff ( o ) eff 2 1.45 o c 3 108 0.025mm f 12 10 9 0 0.025 0.0172m 17.2mm 1.45 eff l 4 4.3mm For C2 = 0.395, to find coupling length eff ( e) 0.7 r 1.75 eff (o) 0.58 r 1.45 e f (fe ) e f (fo ) e f f 2 1.2 6 5 c 3 108 o 0.025mm f 12 10 9 o 0.025 0.01976m 19.76mm eff 1.265 l 4 4.94mm 3.3 Broadband Coupled-line 10 dB directional coupler, 6 - 18 GHz, on a substrate with a 50 Ω characteristic impedance system. The substrate permittivity is 10. The substrate height is 25 mil (0.635 mm). Solution: Start by finding C1 C2 C3 since C = 0.316 10C1 C2 0 C2 2C1 0.316 It can be found C1 C3 0.0395 C2= 0.395 The even and odd mode characteristic impedances are Z 0e Z 0e 52 1 3 Z 0o Z 0o 48 1 3 Z 0e 76 2 Z 0o 32.9 2 From figure 1 S1 S 3 2.11 S1 = S3 = 1.34 h h W1 h W3 0.99 h W1 = W3 = 0.63 S2 0.13 S2 = 0.083 h W2 0.75 W2 = 0.473 h For C1 and C3 = 0.0395, to find the coupling length eff (e) 0.7 r 7 eff ( o ) 0.61 r 6.1 eff ( e ) eff ( o ) eff 2 2.56 c 3 108 o 0.025mm f 12 10 9 0 0.025 0.00976m 9.76mm 2.56 eff l 4 2.44mm For C2 = 0.395, to find the coupling length eff ( e) 0.66 r 6.6 eff ( o ) 0.54 r 5.4 eff o eff ( e ) eff ( o ) 2 2.45 c 3 108 0.025m f 12 10 9 o 0.025m 10.2mm 2.45 eff l 4 2.55mm 4. Evaluation by simulations 4.1 Coupled-line 10 dB directional coupler, 11.5 -12.5 GHz a. Layout for single section directional coupler 2-D layout 3-D layout b. narrow band directional coupler performance Comments: the above graph of 10 dB directional coupler is based on minor tuning. Calculated value Actual value Discrepancy S 0.09 mm 0.09 mm No tuning W 1.575 mm 1.67 mm 0.095 mm L 4.37 mm 4.35 mm 0.02 mm 4.2 Broadband 10 dB directional coupler, 6- 18GHz, on a 2.5 permittivity and 0.75 mm height substrate a. Layout for 3-section directional couple 3-D layout 2-D layout b. Broadband directional coupler performance Comments: to obtain requested bandwidth, proper tuning is needed for this design. Calculated value Actual value Discrepancy S1, S3 2.097 mm 2.097 mm No tuning W1, W3 2.1 mm 1.46 mm 0.64 mm L1, L3 4.3 mm 1.29 mm 3.01 mm S2 0.0535 mm 0.0535 mm No tuning W2 1.575 mm 1.37 mm 0.25 mm L2 4.94 mm 4.94 mm No tuning It’s found that for the central section, S W &L is around theoretical value, however the length of each end section differs a lot from calculated value. The reason will be elaborated in the discussion part. 4.3 Broadband 10 dB directional coupler, 6- 18GHz, on a 10 permittivity and 0.635 mm height substrate a. Layout for 3-section directional couple 2-D layout 2-D layout b. Broadband directional coupler performance Comments: to obtain requested bandwidth, proper tuning is needed for this design. Calculated value Actual value Discrepancy S1, S3 1.34 mm 1.25 mm 0.09 mm W1, W3 0.63 mm 0.689 mm 0.059 mm L1, L3 2.44 mm 1.31 mm 1.13 mm S2 0.083 mm 0.082 mm 0.001 mm W2 0.473 mm 0.504 mm 0.031 mm L2 2.55 mm 2.87 mm 0.32 mm Same as previous design, L1, L3 differs greatly from calculated values. 5. Discussion and recommendation 1. How the coupling coefficient is being affected by the width, spacing and length? Try to have an intuitive understanding while does not rely heavily on mathematics, and then verify it by software simulation. The 3-section coupler has been chosen instead of the single section model as the latter’s change is not as obvious as the former. The spacing will shift the whole curve up and down, or in other words, it change the value of C at center frequency, while it doesn’t change the shape of curve too much. Larger S means less coupling effect, or larger C. While in the other hand, if the spacing decreases, the coupling effect increases, or a smaller C. The width changes the curve a lot. when the width is increased, the flat broadband curve will concave a bit downward while if the width is decreased, the flat curve convex a bit upward around the center frequency. The length is a critical value related to frequency as it equals quarter wavelength. for smaller L, the curve will has a positive slope instead of being flat. While for a larger L, the slope is a little negative rather than being flat. As a result, large bandwidth could only be obtained once proper length has been chosen. 2. when designing the multisection coupler, it has been noted that for the left and right section, the length usually differs a lot from the theoretical value, try to explain this discrepancy and then come out with one possible solution. For an ideal multi-section directional coupler, the calculation is based on the assumption that the connections between each section do not have coupling effect, or at least be negligible. Original design with large bends Revised design with smaller bends However, in our design, the center spacing is less than one tenth of the two end spacing, which as a result, results in extended junction length between different sections. And in such a case, the coupling effect is no longer negligible. What’s more, as the spacing between two junctions are usually smaller, as shown from the graph above, the microstrip length is being decreased even further. One possible solution is to have smaller and thinner bends, as what’s shown in the graph. As all the bends provided in AWR software package are either bulky or EM based, so this solution is only theoretically true. However, as with smaller bends, this is no reason to doubt that the coupling effect due to the bends will be much smaller, which means the end section’s length is much more near what’s calculated. 3. When the directional coupler is being manufactured, some specifications for the coupler must be noted, such as the dimension of the coupler as it needs to be fit into a box and so on. Take the 3rd design as an example and then check whether your design can meet such specifications. If not, try to propose some possible solutions. The substrate chosen is 0.5 × 0.25 inch, as the coupler is around 0.5 inch long and 0.2 inch wide. It’s being noted that the distance between two adjacent ports is less than half a inch, which do not accord with the manufacturing requirement. However this problem can be overcome by adding more bends. Verified by simulation software, the coupling coefficient graph changes less than 3 %, which is, by and large, negligible. As a result, we conclude this design can meet the manufacturing specification and is a feasible design. 4. In the broadband directional coupler design, how can we achieve the required performance by implementing a Schiffman sawtooth pattern? A Schiffman sawtooth directional coupler uses multistage coupled lines, with a set of small zig-zag patterns joining together. Using this method, the mechanical length required for a given electrical length can be reduced. The Schiffman sawtooth pattern was designed using an empirical method. “It was not possible to analyze this structure using the available software tools”. [1] In microwave office software package, the zig-zag pattern isn’t provided as it’s too complicated, which was also verified by the microwave technician. So we didn’t include the realization of schiffman coupler in our design. However in the empirical design, the total length along the zig-zag path is made equal to the straight line length of this section when implementing an ordinary coupled line. This can reduce the total length by half. The spacing between straight lines is maintained between the interlocking teeth. 6. Reference 1.Liao, Samuel Y. “Microwave devices and circuits”, Prentice Hall, 1990 2.Trinogga, L. A. (Lothar Alfred), “Practical microstrip circuit design”, E. Horwood, 1991 3.Howe, Harlan H. “Stripline circuit design”, Artech House,1974 4.Pozar, David M. “Microwave engineering”, Wiley, 1998 5. Dana Brady, “The Design, Fabrication and Measurement of Microstrip Filter and Coupler Circuits”, CAP Wireless, Inc. 2002