EQUIVALENT LENGTH DESIGN EQUATIONS FOR RIGHTANGLED MICROSTRIP BENDS H.J. Visser* Holst Centre – TNO P.O. Box 8550 5605 KN Eindhoven, The Netherlands E-mail:
[email protected] *
Keywords: Antenna feeds, propagation, reflection.
discontinuities,
microstrip,
include reflections at the bends and assessment of the bends’ electrical lengths.
Abstract
2 Reflection
For printed antenna systems, microstrip feeding networks may become quite complex, including several right-angled bends. In designing feed networks we have to consider reflection levels at and electrical lengths of t he bends. Removing a part of the area of metallization in the bend’s corner can compensate for the excess capacitance and reduce the reflection level of the bend. Full wave simulations have been performed for unmitered and (50% ( 50%)) mitered right-angled bends in microstrip on FR4 and FR4 -like substrates in the frequency range 868MHz – 868MHz – 60GHz . The simulations revealed that for reflection levels below -15dB, -15dB, up to 10GHz mitering is unnecessary. For reflection levels below -20dB, -20dB, mitering must be applied for frequencies in excess of 2.5GHz . A slight modification of the centreline approach for unmitered bends leads to an equivalent electrical length for unmitered bends w ith an absolute accuracy of less than one degree for all frequencies and substrates, where the reference planes may be brought back all the way to the bend. Applying this modification to 50%-mitered 50%-mitered bends, having the reference planes at 0.2λ g distance from the bend, λ g being the wavelength in the substrate, leads to an absolute error in electrical length of less than two and a half degrees for all frequencies and substrates.
Due to charge accumulation at – particularly – the outer corner of a bend, an excess capacitance is created, while current interruptions give rise to excess inductances, [1]. Figure 1a shows a right-angled bend and the equivalent circuit is shown in figure 1b, [2]. W
W a
L
L C b
b W
α
1 Introduction Microstrip transmission lines are very popular due mainly to their ease of integration with common Printed Circuit Boards (PCBs). For wireless devices, all components as well as the antenna (e.g. PIFA or monopole) or part of the antenna (e.g. microstrip patch) can be mounted on top of the same PCB, the antenna being interconnected to the (active) elements by microstrip transmission lines. The antenna feeding network may become quite complex, e.g. for a dual polarised antenna or a balanced antenna that needs to be connected to an unbalanced unbalanced amplifier. Such feeding networks usually employ several right-angled bends that complicate the feeding network design. The complications
W c
Figure 1: Right-angled microstrip bend a. Unmitered bend, top view. b. Equivalent circuit c. Mitered bend, top view. Especially the excess capacitance may give rise to high reflection levels. Removing the area of metallization in the corner – a process known as chamfering or mitering – can compensate for the
excess capacitance. The equivalent circuit shown in figure 1b also holds for mitered bends as shown in figure 1c. Values for L and C in figure 1d are given in [2] for frequencies up to 14GHz . The design of a microstrip network may be accomplished by employing a full-wave analysis program, employing a microwave circuit simulator (that uses amongst others, equivalent circuits for microstrip bends) or employing inhouse written dedicated software for microstrip transmission lines and right-angled bends. However, full-wave and/or microwave circuit simulators are not at everyone’s disposal and the writing of dedicated software is a time-consuming process. Furthermore, accurate models for frequencies in excess of 14GHz are not readily available. Even if software is available, guidelines are still needed for a fast generation of initial designs. For an initial design it suffices to decide whether or not to apply a miter and to 0 restrict ourselves to the α =45 , b=0.5√ 2W (or 50%) miter, 0 see figure 1c. The α=45 , b=0.5√ 2W miter appears to be the optimum miter for wide lines and appears to improve the reflection over the unmitered case for all widths, heights, permittivities and frequencies considered, [3]. Reflection levels have been calculated employing a Method of Moments for unmitered and 50% mitered right-angled microstrip bends in 50Ω microstrip transmission lines. The microstrip parameters used in the simulations are shown in table 1. Frequency
Strip width
868MHz 2.45GHz 10.02GHz 20.04GHz 60.1GHz
3.2mm 3.3mm 1.8mm 1.0mm 0.4mm
Substrate thickness 1.6mm 1.6mm 0.762mm 0.422mm 0.168mm
ε r 4.28 4.28 3.66 3.66 3.66
tanδ 0 0 0 0 0
Table 1: Microstrip and substrate parameters used in fullwave simulations.
l
W l2
l l2
W
Figure 2: Dimensions of microstrip structure being simulated. Frequency 868MHz 2.45GHz 10.02GHz 60.1GHz
Length 60mm 20mm 6mm 1mm
S 11 unmitered -39dB -19dB -15dB -12dB
S 11 mitered -39dB -29dB -33dB -25dB
Table 2: Reflection levels for unmitered and 50% mitered right-angled bends in microstrip. (VSWR ≤ 1.44) are considered to be acceptable, up to 10GHz mitering is not necessary. If reflection levels below -20dB (VSWR ≤ 1.22) are needed, mitering should be applied for frequencies in excess of 2.5GHz . These results agree with observations in [4,5]. For all right-angled bends simulated, for all frequencies, the radiation loss is negligible. These results agree with [3,4], [6]. Whether a miter is applied or not, a need exists for a simple procedure to assess the electrical length of a microstrip path including a right-angled bend. In the next section, equivalent microstrip transmission line lengths will be derived for unmitered and 50% mitered right-angled bends.
3 Equivalent length The electrical length of a right-angled microstrip bend may be determined by replacing the microstrip circuit containing the bend with a straight piece of microstrip transmission line having an equivalent length, see figure 3. l
The substrate parameters used are based on commercially available PCB material (FR4) and microwave laminate © (Rogers RO4003B ), wherein the loss tangent (tanδ ) has been set to zero. Thus, reflection level simulation results are not disturbed by loss effects. Reflections ( S 11) have been simulated for the configurations as shown in figure 2, for several values of microstrip transmission line length l in frequency bands around the central frequencies shown in table 1. The non-excited port has been match-terminated. The reflection levels for unmitered and 50% mitered bends are shown in table 2 for typical values of l at the central frequencies. Care has been taken to verify that these changes in reflection level are consistent with the observed changes over the frequency bands around these central frequencies. The simulations reveal that if reflecti on levels up to -15dB
W l2
l l2
W
leq
W
Figure 3: Dimensions of microstrip structure being simulated.
In figure 3, the dotted lines indicate the terminal planes of the right-angled bend. The length l 2 is the distance between the terminal planes and reference planes
l 2 = l − W .
(1)
For practical reasons – we want to design compact feeding networks – we want to have the reference planes as close as possible to the bend terminal planes. 3.1 Unmitered bend
In the centreline approach, as shown in figure 4a, the equivalent length is taken as the length of the centreline through the microstrip structure
less than one degree when l 2 is approaching zero, i.e. the reference planes are brought down to the terminal planes of the bend. l 80mm 60mm 40mm 20mm 10mm 8mm 6mm 5mm 4mm
S 21 phase unmitered bend 0 65.13 0 140.71 0 -143.69 0 -67.88 0 -30.07 0 -22.43 0 -15.12 0 -11.41 -7.880
S 21 phase centreline 0 63.21 0 139.23 0 -145.39 0 -70.01 0 -32.04 0 -23.94 0 -16.53 0 -12.87 -9.220
S 21 phase modified centreline 0 65.05 0 140.79 0 -143.68 0 -68.04 0 -30.31 0 -22.20 0 -14.80 0 -11.14 -7.540
Table 3: Calculated S 21 phases for unmitered right-angled microstrip bend and equivalent length microstrip transmission lines at 868MHz . l a
30mm 20mm 10mm 9mm 8mm 7mm 6mm 5mm 4mm
b
Figure 4: Determination of equivalent length for unmitered right-angled bend. a. Centreline approach. b. Modified centreline approach.
l eq1 = 2l 2 + W .
(2)
By slightly modifying the equivalent length as shown in figure 4b, we take the actual current flow better into account. The equivalent length is now given by
l eq 2 = 2l 2 +
1 2
2W .
(3)
To compare the equivalent lengths of equations (2) and (3) with the actual structure, the signal transfer ( S 21) through the actual structure has been calculated by a Method of Moments for different values of l . The same has been done for straight pieces of microstrip trans mission line of physical lengths l eq1 and leq2 . The calculated phases are shown in tables 3 to 7 for the frequencies 868MHz , 2.45GHz , 10.02GHz , 20.04GHz and 60.1GHz , respectively. The tables clearly show the improvement of the modified centreline approach, equation (3), over the ‘traditional’ centreline approach, equation (2). The absolute phase error is
S 21 phase unmitered bend 0 53.55 0 166.20 0 -84.26 0 -73.94 0 -63.01 0 -52.13 -41.360 0 -30.36 0 -19.45
S 21 phase centreline 0 51.28 0 160.20 0 -90.84 0 -79.75 0 -68.92 0 -58.00 -47.080 0 -36.13 0 -25.47
S 21 phase modified centreline 0 56.43 0 165.50 0 -85.47 0 -74.48 0 -63.62 0 -52.78 -41.800 0 -30.85 0 -19.99
Table 4: Calculated S 21 phases for unmitered right-angled microstrip bend and equivalent length microstrip transmission lines at 2.45GHz . l 14mm 12mm 10mm 8mm 7mm 6mm 5mm 4mm 3.5mm 3.3mm 3.1mm 3mm 2.9mm 2.7mm 2.5mm 2mm
S 21 phase unmitered bend 0 -17.30 -92.200 0 -13.61 0 70.92 0 113.21 155.690 0 -162.15 0 -120.08 0 -99.55 0 -91.12 0 -82.73 0 -78.08 0 -74.53 -66.390 0 -57.81 0 -35.45
S 21 phase centreline 0 171.10 -105.100 0 -22.24 0 62.02 0 104.40 145.800 0 -173.20 0 -130.90 0 -108.23 0 -99.93 0 -91.61 0 -87.33 0 -83.25 -74.960 0 -66.66 0 -45.83
S 21 phase modified centreline 0 -178.10 0 -93.76 0 -11.48 0 73.22 0 115.70 156.600 0 -162.10 0 -119.50 0 -97.23 0 -88.92 0 -80.64 0 -76.42 0 -72.24 -63.920 0 -55.60 0 -34.57
Table 5: Calculated S 21 phases for unmitered right-angled microstrip bend and equivalent length microstrip transmission lines at 10.02GHz .
l 4mm 3mm 2.5mm 2mm 1.8mm 1.6mm 1.4mm 1.2mm
S 21 phase unmitered bend 73.890 0 159.30 0 -158.42 -115.880 0 -99.15 0 -82.17 0 -64.88 -47.690
S 21 phase centreline 64.790 0 149.22 0 -168.66 -126.710 0 -109.65 0 -93.12 0 -76.27 -59.340
S 21 phase modified centreline 77.100 0 161.48 0 -156.46 -114.480 0 -97.64 0 -80.90 0 -63.92 -47.230
into a path following the inner edge more closely, equation (3). W/2
l2
a
Table 6: Calculated S 21 phases for unmitered right-angled microstrip bend and equivalent length microstrip transmission lines at 20.04GHz . l 1.4mm 1.2mm 1mm 0.8mm 0.7mm 0.6mm 0.5mm
S 21 phase unmitered bend 0 56.24 0 114.81 0 167.16 -140.940 0 -114.83 0 -88.48 0 -61.85
S 21 phase centreline 0 52.88 0 105.40 0 154.60 -155.500 0 -128.90 0 -102.50 0 -76.04
S 21 phase modified centreline 0 68.76 0 120.40 0 169.30 -139.700 0 -113.00 0 -86.70 0 -61.03
Table 7: Calculated S 21 phases for unmitered right-angled microstrip bend and equivalent length microstrip transmission lines at 60.1GHz . 3.2 Mitered bend
For the mitered right-angled bend we follow the procedure as outlined in [7] for doubly chamfered bends. As in [7] we observe that also the current distribution through a rightangled bend tends to concentrate along the inner edge, see figure 5.
b
Figure 6: Current path approximations. a. Shortest path. b. Shortest path and modiefied centreline path. The shortest path length, l short , follows from figure 6a 2
W l short = + l 22 2
.
(4)
The equivalent length of the 50% mitered right-angled microstrip bend, l eqmit2, is now calculated as
l eqmit 2 = l eq 2 l short ,
(5)
where l eq2 is given by equation (3). To compare this equivalent length with the actual mitered right-angled bend, the signal transfer ( S 21 ) through the actual structure has been calculated by a Method of Moments for different values of l . The same has been done for straight pieces of microstrip transmission line of physical lengths leqmit2 and l eqmit1. The equivalent length l eqmit1 is the equivalent length based on the unmodified centreline approach, [7]
l eqmit 1 =
l eq1l short ,
(6)
where l eq1 is given by equation (2). Figure 5: Typical example of current distribution through a right-angled microstrip bend, dark: low current density, light: high current density. The current path therefore is not following the centreline, but deviates towards the shortest path, see figure 6a. For the unmitered right-angled bend we corrected for the actual current path by modifying the centreline path, equation (2),
The calculated phases are shown in tables 8 to 12 for the frequencies 868MHz , 2.45GHz , 10.02GHz , 20.04GHz and 60.1GHz , respectively. The tables show that the equivalent length based on the modified centreline approach, l eqmit2, may generate results that resemble the actual structure more closely than l eqmit1. The improvement, however, is not monotonously increasing with decreasing length l as in the case of the unmitered bend.
l 80mm 60mm 40mm 20mm 10mm 8mm 6mm 5mm 4mm
S 21 phase mitered bend 66.510 0 142.19 0 -142.49 -66.720 0 -28.91 0 -21.38 0 -13.98 -10.250 0 -6.53
S 21 phase eqmit1 65.600 0 139.75 0 142.68 -67.660 0 -30.69 0 -23.16 0 -16.97 -13.920 0 -8.32
S 21 phase eqmit2 66.450 0 142.56 0 -141.89 -66.820 0 -29.84 0 -22.29 0 -15.89 -12.960 0 -7.37
Table 8: Calculated S 21 phases for mitered right-angled microstrip bend and equivalent length microstrip transmission lines at 868MHz .
l 4mm 3mm 2.5mm 2mm 1.8mm 1.6mm 1.4mm 1.2mm
30mm 20mm 10mm 9mm 8mm 7mm 6mm 5mm 4mm
S 21 phase mitered bend 60.570 0 169.80 0 -81.66 0 -70.79 -59.740 0 -48.78 0 -37.95 0 -26.92 -16.060
S 21 phase eqmit1 60.030 0 169.00 0 -82.45 0 -71.68 -61.020 0 -50.46 0 -40.04 0 -30.35 -22.140
S 21 phase eqmit2 62.710 0 171.50 0 -79.96 0 -69.23 -58.600 0 -48.06 0 -37.70 0 -28.06 -19.560
Table 9: Calculated S 21 phases for mitered right-angled microstrip bend and equivalent length microstrip transmission lines at 2.45GHz .
l 14mm 12mm 10mm 8mm 7mm 6mm 5mm 4mm 3.5mm 3.3mm 3.1mm 3mm 2.9mm 2.7mm 2.5mm 2mm
S 21 phase mitered bend 0 -171.40 0 -87.26 0 -5.43 0 79.28 0 121.27 163.420 0 -154.69 0 -112.38 0 -91.51 0 -83.08 0 -74.66 0 -70.32 0 -66.48 -57.960 0 -49.86 0 -26.42
S 21 phase eqmit1 0 -166.60 0 -83.21 0 -0.33 0 82.55 0 123.90 165.450 0 -153.56 0 -113.08 0 -102.64 0 -85.06 0 -77.42 0 -73.95 0 -70.19 -63.160 0 -56.22 0 -41.88
S 21 phase eqmit2 0 -160.93 0 -78.16 0 4.75 0 87.63 0 129.06 170.420 0 -150.14 0 -108.23 0 -88.22 0 -80.63 0 -72.89 0 -69.17 0 -65.20 -58.290 0 -51.43 0 -36.23
Table 10: Calculated S 21 phases for mitered right-angled microstrip bend and equivalent length microstrip transmission lines at 10.02GHz .
S 21 phase eqmit1 85.100 0 168.55 0 -150.22 -109.270 0 -93.95 0 -78.34 0 -69.42 -51.160
S 21 phase eqmit2 90.820 0 174.08 0 -144.29 -104.010 0 -88.38 0 -73.29 0 -59.02 -44.730
Table 11: Calculated S 21 phases for mitered right-angled microstrip bend and equivalent length microstrip transmission lines at 20.04GHz .
l l
S 21 phase mitered bend 83.340 0 168.05 0 -149.86 -107.010 0 -90.12 0 -72.99 0 -55.47 -37.930
1.4mm 1.2mm 1mm 0.8mm 0.7mm 0.6mm 0.5mm
S 21 phase mitered bend 0 74.35 127.470 0 178.86 0 -129.24 0 -103.29 0 -76.74 0 -49.69
S 21 phase eqmit1 0 78.16 128.900 0 176.70 0 -132.80 0 -108.90 0 -86.78 0 -64.65
S 21 phase eqmit2 0 84.50 135.100 0 183.10 0 -126.20 0 -102.50 0 -78.65 0 -58.51
Table 12: Calculated S 21 phases for mitered right-angled microstrip bend and equivalent length microstrip transmission lines at 60.1GHz . To derive practical guidelines for using the new equivalent length approximation, l eqmit2, the absolute phase differences between the equivalent length transmission lines and the actual mitered bend structures have been evaluated as function of the length l , normalised to the wavelength in the substrate, λ g . This wavelength is given by
λ g =
λ 0 ε eff
,
(7)
where λ 0 is the free space wavelength and ε eff is the effective relative permittivity, that is given by, [8]
ε eff =
ε r + 1 2
+
ε r − 1 2
h
1 + 10 W
− 12
,
(8)
where ε r is the relative permittivity of the substrate, h is the thickness of the substrate and W is the microstrip width. The results are shown in the figures 7 to 11. The figures show that using the equivalent length based on the modified centreline approach leads to a better accuracy for a smaller separation of bend terminal planes and reference planes. For l ≈0 .2λ g , the absolute phase error is less than two and a half degrees for all frequencies and substrates considered.
10
868MHz del1 868MHz del2
10
9
60.1GHz del1
9
8
60.1GHz del2
8
7 6 5 4 3
) s e e r g e d (
7 6 5 4
a t l e d
3
2
2
1
1
0
0.5
0.4
0.3
0.2
0.1
) s e e r g e d ( a t l e d
0
0
0.5
0.4
l/lambdag
0.3
0.2
0.1
0
l/lambdag
Figure 7: Absolute phase differences at 868MHz .
Figure 11: Absolute phase differences at 60.1GHz . 10
2.45GHz del1
9
2.45GHz del2
8 7 6 5 4 3
4 Conclusions ) s e e r g e d ( a t l e d
2 1 0
0.5
0.4
0.3
0.2
0.1
0
Full wave simulations for microstrip bends on FR4 (like) substrates, reveal that mitering is necessary for frequencies in excess of (2.5GHz ) 10GHz to obtain reflection levels below (20dB) -15dB. Equation (3) calculates an equivalent length for an unmitered bend with an absolute error of less than one degree, having the reference planes at the bend terminal planes. For 50% mitered bends, equations (3), (4) and (5) calculate an equivalent length for l ≈0 .2λ g with an absolute error of less than two and a half degrees.
l/lambdag
References
Figure 8: Absolute phase differences at 2.45GHz . 10
10.02GHz del1
[1]
9
10.02GHz del2
8 7 6 5 4 3
) s e e r g e d ( a t l e d
[2]
[3]
2 1 0
0.5
0.4
0.3
0.2
0.1
0
[4]
l/lambdag
Figure 9: Absolute phase differences at 10.02GHz . 10
20.04GHz del1
9
20.04GHz del2
[5]
8 7 6 5 4 3
) s e e r g e d ( a t l e d
[6]
2 1 0
0.5
0.4
0.3
0.2
0.1
[7]
0
l/lambdag
Figure 10: Absolute phase differences at 20.04GHz . [8]
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