bend in microstrip lines

W

W

L L

C

W

W

α

b

a

c

b

EQUIVALENT LENGTH DESIGN EQUATIONS FOR RIGHT-

ANGLED MICROSTRIP BENDS

H.J. Visser*

*Holst Centre – TNO

P.O. Box 8550

5605 KN Eindhoven, The Netherlands

E-mail: [email protected]

Keywords: Antenna feeds, discontinuities, microstrip,

propagation, reflection.

Abstract

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 the 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%) mitered right-angled

bends in microstrip on FR4 and FR4 -like substrates in the

frequency range 868MHz – 60GHz. The simulations revealed

that for reflection levels below -15dB, up to 10GHz mitering

is unnecessary. For reflection levels below -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 with 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 bends, having the reference

planes at 0.2λg distance from the bend, λg 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.

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 amplifier. Such feeding

networks usually employ several right-angled bends that

complicate the feeding network design. The complications

include reflections at the bends and assessment of the bends’

electrical lengths.

2 Reflection

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].

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

Authorized licensed use limited to: Eindhoven University of Technology. Downloaded on January 22, 2010 at 10:15 from IEEE Xplore. Restrictions apply.

W

W

l

l l2

l2

W

W

l

l l2

l2

W

leq

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 in-

house 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

restrict ourselves to the α=450, b=0.5√2W (or 50%) miter, see figure 1c. The α=450, 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 Substrate

thickness εr tanδ

868MHz

2.45GHz

10.02GHz

20.04GHz

60.1GHz

3.2mm

3.3mm

1.8mm

1.0mm

0.4mm

1.6mm

1.6mm

0.762mm

0.422mm

0.168mm

4.28

4.28

3.66

3.66

3.66

0

0

0

0

0

Table 1: Microstrip and substrate parameters used in full-

wave simulations.

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 (S11) 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 reflection levels up to -15dB

Figure 2: Dimensions of microstrip structure being simulated.

Frequency Length S11 unmitered S11 mitered

868MHz

2.45GHz

10.02GHz

60.1GHz

60mm

20mm

6mm

1mm

-39dB

-19dB

-15dB

-12dB

-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.

Figure 3: Dimensions of microstrip structure being simulated.


a

b

In figure 3, the dotted lines indicate the terminal planes of the

right-angled bend. The length l2 is the distance between the

terminal planes and reference planes

Wll −=2 . (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

Figure 4: Determination of equivalent length for unmitered

right-angled bend. a. Centreline approach. b. Modified

centreline approach.

Wlleq += 21 2 . (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

Wlleq 22

12 22 += . (3)

To compare the equivalent lengths of equations (2) and (3)

with the actual structure, the signal transfer (S21) 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 leq1

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

less than one degree when l2 is approaching zero, i.e. the

reference planes are brought down to the terminal planes of

the bend.

l S21 phase

unmitered bend

S21 phase

centreline

S21 phase modified

centreline

80mm

60mm

40mm

20mm

10mm

8mm

6mm

5mm

4mm

65.130

140.710

-143.690

-67.880

-30.070

-22.430

-15.120

-11.410

-7.880

63.210

139.230

-145.390

-70.010

-32.040

-23.940

-16.530

-12.870

-9.220

65.050

140.790

-143.680

-68.040

-30.310

-22.200

-14.800

-11.140

-7.540

Table 3: Calculated S21 phases for unmitered right-angled

microstrip bend and equivalent length microstrip

transmission lines at 868MHz.

l S21 phase

unmitered bend

S21 phase

centreline

S21 phase modified

centreline

30mm

20mm

10mm

9mm

8mm

7mm

6mm

5mm

4mm

53.550

166.200

-84.260

-73.940

-63.010

-52.130

-41.360

-30.360

-19.450

51.280

160.200

-90.840

-79.750

-68.920

-58.000

-47.080

-36.130

-25.470

56.430

165.500

-85.470

-74.480

-63.620

-52.780

-41.800

-30.850

-19.990



transmission lines at 2.45GHz.

l S21 phase

unmitered bend

S21 phase

centreline

S21 phase modified

centreline

14mm

12mm

10mm

8mm

7mm

6mm

5mm

4mm

3.5mm

3.3mm

3.1mm

3mm

2.9mm

2.7mm

2.5mm

2mm

-17.300

-92.200

-13.610

70.920

113.210

155.690

-162.150

-120.080

-99.550

-91.120

-82.730

-78.080

-74.530

-66.390

-57.810

-35.450

171.100

-105.100

-22.240

62.020

104.400

145.800

-173.200

-130.900

-108.230

-99.930

-91.610

-87.330

-83.250

-74.960

-66.660

-45.830

-178.100

-93.760

-11.480

73.220

115.700

156.600

-162.100

-119.500

-97.230

-88.920

-80.640

-76.420

-72.240

-63.920

-55.600

-34.570





a

b

l2

W/2

l S21 phase

unmitered bend

S21 phase

centreline

S21 phase modified

centreline

4mm

3mm

2.5mm

2mm

1.8mm

1.6mm

1.4mm

1.2mm

73.890

159.300

-158.420

-115.880

-99.150

-82.170

-64.880

-47.690

64.790

149.220

-168.660

-126.710

-109.650

-93.120

-76.270

-59.340

77.100

161.480

-156.460

-114.480

-97.640

-80.900

-63.920

-47.230




l S21 phase

unmitered bend

S21 phase

centreline

S21 phase modified

centreline

1.4mm

1.2mm

1mm

0.8mm

0.7mm

0.6mm

0.5mm

56.240

114.810

167.160

-140.940

-114.830

-88.480

-61.850

52.880

105.400

154.600

-155.500

-128.900

-102.500

-76.040

68.760

120.400

169.300

-139.700

-113.000

-86.700

-61.030




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 right-

angled bend tends to concentrate along the inner edge, see

figure 5.

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),

into a path following the inner edge more closely, equation

(3).

Figure 6: Current path approximations. a. Shortest path. b.

Shortest path and modiefied centreline path.

The shortest path length, lshort, follows from figure 6a

2

2

2

2l

Wlshort +

= . (4)

The equivalent length of the 50% mitered right-angled

microstrip bend, leqmit2, is now calculated as

shorteqeqmit lll 22 = , (5)

where leq2 is given by equation (3).

To compare this equivalent length with the actual mitered

right-angled bend, the signal transfer (S21) 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 leqmit1. The equivalent length leqmit1 is the equivalent

length based on the unmodified centreline approach, [7]

shorteqeqmit lll 11 = , (6)

where leq1 is given by 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, leqmit2, may generate results that

resemble the actual structure more closely than leqmit1. The

improvement, however, is not monotonously increasing with

decreasing length l as in the case of the unmitered bend.


l S21 phase

mitered bend

S21 phase

eqmit1

S21 phase

eqmit2

80mm

60mm

40mm

20mm

10mm

8mm

6mm

5mm

4mm

66.510

142.190

-142.490

-66.720

-28.910

-21.380

-13.980

-10.250

-6.530

65.600

139.750

142.680

-67.660

-30.690

-23.160

-16.970

-13.920

-8.320

66.450

142.560

-141.890

-66.820

-29.840

-22.290

-15.890

-12.960

-7.370

Table 8: Calculated S21 phases for mitered right-angled


transmission lines at 868MHz.

l S21 phase

mitered bend

S21 phase

eqmit1

S21 phase

eqmit2

30mm

20mm

10mm

9mm

8mm

7mm

6mm

5mm

4mm

60.570

169.800

-81.660

-70.790

-59.740

-48.780

-37.950

-26.920

-16.060

60.030

169.000

-82.450

-71.680

-61.020

-50.460

-40.040

-30.350

-22.140

62.710

171.500

-79.960

-69.230

-58.600

-48.060

-37.700

-28.060

-19.560




l S21 phase

mitered bend

S21 phase

eqmit1

S21 phase

eqmit2

14mm

12mm

10mm

8mm

7mm

6mm

5mm

4mm

3.5mm

3.3mm

3.1mm

3mm

2.9mm

2.7mm

2.5mm

2mm

-171.400

-87.260

-5.430

79.280

121.270

163.420

-154.690

-112.380

-91.510

-83.080

-74.660

-70.320

-66.480

-57.960

-49.860

-26.420

-166.600

-83.210

-0.330

82.550

123.900

165.450

-153.560

-113.080

-102.640

-85.060

-77.420

-73.950

-70.190

-63.160

-56.220

-41.880

-160.930

-78.160

4.750

87.630

129.060

170.420

-150.140

-108.230

-88.220

-80.630

-72.890

-69.170

-65.200

-58.290

-51.430

-36.230




l S21 phase

mitered bend

S21 phase

eqmit1

S21 phase

eqmit2

4mm

3mm

2.5mm

2mm

1.8mm

1.6mm

1.4mm

1.2mm

83.340

168.050

-149.860

-107.010

-90.120

-72.990

-55.470

-37.930

85.100

168.550

-150.220

-109.270

-93.950

-78.340

-69.420

-51.160

90.820

174.080

-144.290

-104.010

-88.380

-73.290

-59.020

-44.730




l S21 phase

mitered bend

S21 phase

eqmit1

S21 phase

eqmit2

1.4mm

1.2mm

1mm

0.8mm

0.7mm

0.6mm

0.5mm

74.350

127.470

178.860

-129.240

-103.290

-76.740

-49.690

78.160

128.900

176.700

-132.800

-108.900

-86.780

-64.650

84.500

135.100

183.100

-126.200

-102.500

-78.650

-58.510




To derive practical guidelines for using the new equivalent

length approximation, leqmit2, 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, λg. This wavelength is given by

eff

gε

λλ 0= , (7)

where λ0 is the free space wavelength and εeff is the effective relative permittivity, that is given by, [8]

2

1

1012

1

2

1−

+

−+

+=

W

hrreff

εεε , (8)

where εr 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≈0.2λg, the absolute phase error is less than two and a half degrees for all frequencies and substrates

considered.


0

1

2

3

4

5

6

7

8

9

10

00.10.20.30.40.5

l/lambdag

delta (degrees)

868MHz del1

868MHz del2

Figure 7: Absolute phase differences at 868MHz.

0

1

2

3

4

5

6

7

8

9

10

00.10.20.30.40.5

l/lambdag

delta (degrees)

2.45GHz del1

2.45GHz del2

Figure 8: Absolute phase differences at 2.45GHz.

0

1

2

3

4

5

6

7

8

9

10

00.10.20.30.40.5

l/lambdag

delta (degrees)

10.02GHz del1

10.02GHz del2


0

1

2

3

4

5

6

7

8

9

10

00.10.20.30.40.5

l/lambdag

delta (degrees)

20.04GHz del1

20.04GHz del2


0

1

2

3

4

5

6

7

8

9

10

00.10.20.30.40.5

l/lambdag

delta (degrees)

60.1GHz del1

60.1GHz del2


4 Conclusions

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≈0.2λg with an absolute error of less than two and a half degrees.

References

[1] T.C. Edwards, “Foundations of Microstrip Circuit

Design”, John Wiley & Sons, Chichester UK, (1 981).

[2] M. Kirschning, R. H. Jansen and N. H. L. Koster,

“Measurement and Computer-Aided Modeling of

Microstrip Discontinuities by an Improved Resonator

Method”, IEEE MTT-S International Symposium Digest,

pp. 495-497, (1983).

[3] R. J. P. Douville and D. S. James, “Experimental Study

of Symmetric Microstrip Bends and Their

Compensation”, IEEE Trans. Microwave Theory and

Techniques, Vol. MTT-26, No. 3, pp. 175-182, (1978).

[4] T. S. Hong, W. E. McKinzie and N. G. Alexopoulos,

“Full-Wave Spectral-Domain Analysis of Compensation

of Microstrip Discontinuities Using Triangular

Subdomain Functions”, IEEE Trans. Microwave Theory

and Techniques, Vol. 40, No. 12, pp. 2137-2147, (1992).

[5] A. J. Rainal, “Reflections from Bends in Printed

Conductor”, IEEE Transactions on Components,

Hybrids, and Manufacturing Technology, Vol. 13, No.

2, pp. 407-413, (1990).

[6] J. X. Zheng and D. C. Chang, “Numerical Modeling of

Chamfered bends and Other Microstrip Junctions of

general Shape in MMIC”, IEEE MTT-S International

Symposium Digest, pp. 709-712, (1990).

[7] S. A. Bokhari, “Precision Delay Matching by Trace

length Control in Printed Circuit Boards”, Canadian

Conference on Electrical and Computer Engineering,

pp. 799-802, (2006).

[8] D.M. Pozar, “Microwave Engineering, 2nd edition”, John

Wiley & Sons, New York, (1979).


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bend in microstrip lines