Thomas Clupper - 10/15/07
Description of transmission line models and where they came from:
First, let’s start with the references (In chronological order):
1) “Formulas for the Skin Effect”, Wheeler, Proc. IRE, vol 30, Sept. 1942, pp. 412-424.
2) “Characteristic Impedance of the Shielded-Strip Transmission Line”, Cohn, IRE Trans. MTT, vol 2, July 1954, pp. 52-55.
3) “Problems in Strip Transmission Lines”, Cohn, IRE Trans MTT, vol 3, March 1955, pp. 119-126.
4) Stripline Circuit Design, Howe, 1974, Artech House, Chapter 1.
5) Transmission-Line properties of a Strip Line Between Parallel Planes”, Wheeler, IEEE Trans MTT, vol 26, no 11, Nov 1978.
6) “A Designers Guide to Stripline Circuits”, Bahl and Garg, Microwaves, 1978, pp. 90-96.
7) Computer-Aided Design of Microwave Circuits, Gupta, Garg, and Chadha, Artech House, 1981.
8) Transmission Line Design Handbook, Wadell, Artech House, 1991.
I noticed the stripline transmission line is not nearly as well researched and documented as the microstrip line. The above references pretty much cover the span of what’s written about the subject. In general, my approach to developing a model is to first calculate the characteristic impedance of the line (Zc) and the effective dielectric constant (εr eff). From there, I can calculate the velocity of propagation (Vp) and the distributed capacitance (C) and inductance (L). The next step will be to calculate the R and G terms from the attenuation constants. This is where the literature somewhat falls apart.
For impedance, I use the expression developed by Wheeler [5], which was also used in [7] and [8]. The εr eff = εr because stripline basically has TEM propagation, this also means that calculating the G term is straight forward (G = ω·C·tanδ). The R term (which is also used to calculate the internal inductance term (Li), was derived from the attenuation constant (αc) taken from Howe [4], which was taken from Cohn [3]. Other authors [6] and [7] refer to Cohn [3] for the treatment of conductor loss, although I could not follow their logic. However, there seems to be a problem with Cohn’s formulas for the narrow strip case (as best that I can tell). I could not get it to produce reasonable results, so I was forced to use the wide strip formulas for all cases.
There is a later reference (“An Exact TEM Calculation of Loss in a Stripline of Arbitrary Dimensions”, Rawal and Jackson, IEEE Trans MTT, vol 39, no 4, April 1991, pp. 694-699) that has better loss models, but since it is not closed form, I did not use them.
I ran a series of simulations comparing the results of TLineSim to Eagleware Genesys. Table 1 shows the impedance and insertion loss results for a 12 inch stripline. Note: the system impedance (Z0) was changed to 25 and 75 ohms for the line impedances of Zc=25 and 75 ohms respectively. This eliminated any ripple in the insertion loss curve. Also, note that the 75 ohm lines were out of the range of an accurate loss model (i.e. w/(h-t) < 0.35).
Table 1: Stripline comparison between TLineSim and Eagleware Genesys
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Eagleware Geneys |
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TlineSim |
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w |
h |
t |
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Zc |
I Loss |
I Loss |
I Loss |
I Loss |
Zc |
I Loss |
I Loss |
I Loss |
I Loss |
|
mil |
mil |
mil |
dK |
Tand |
ohms |
300 MHz |
3 GHz |
10 GHz |
20 GHz |
ohms |
300 MHz |
3 GHz |
10 GHz |
20 GHz |
|
5.1 |
9.0 |
0.7 |
2.6 |
0.0040 |
50.01 |
0.766 |
2.790 |
5.904 |
9.398 |
49.98 |
0.748 |
2.770 |
5.885 |
9.379 |
|
10.1 |
16.0 |
0.7 |
2.6 |
0.0040 |
49.87 |
0.464 |
1.834 |
4.158 |
6.928 |
49.83 |
0.458 |
1.828 |
4.151 |
6.921 |
|
27.8 |
16.0 |
0.7 |
2.6 |
0.0040 |
24.99 |
0.390 |
1.600 |
3.730 |
6.324 |
24.97 |
0.386 |
1.596 |
3.726 |
6.319 |
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4.3 |
16.0 |
0.7 |
2.6 |
0.0040 |
74.82 |
0.569 |
2.166 |
4.764 |
7.786 |
74.77 |
0.532 |
2.069 |
4.5935 |
7.548 |
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4.7 |
12.0 |
0.7 |
4.0 |
0.0220 |
50.03 |
1.093 |
5.960 |
16.406 |
30.355 |
49.99 |
1.085 |
5.964 |
16.423 |
30.382 |
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10.1 |
23.0 |
0.7 |
4.0 |
0.0220 |
49.99 |
0.762 |
4.914 |
14.494 |
27.652 |
49.96 |
0.761 |
4.919 |
14.505 |
27.665 |
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30.9 |
23.0 |
0.7 |
4.0 |
0.0220 |
24.98 |
0.675 |
4.640 |
13.994 |
26.944 |
24.96 |
0.675 |
4.644 |
14.001 |
26.952 |
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3.6 |
23.0 |
0.7 |
4.0 |
0.0220 |
75.10 |
0.954 |
5.521 |
15.603 |
29.220 |
75.05 |
0.854 |
5.217 |
15.051 |
28.438 |
Here are the references that I used, and there are a lot more out there (In chronological order):
1) “Losses in Microstrip”, Pucel, Masse and Hartwig, IEEE Trans MTT, vol 16, no 6, June 1968, pp. 342-350 & 1064.
2) “Frequency Dependent Behavior of Microstrip”, Hartwig, Masse and Pucel, MTT Microwave Symposium Digest, vol 68, Issue 1, May 1968, pp. 110-116
3) Microwave Transmission-Line Impedance Data, Gunston, Van Nostrand Rainhold, 1972.
4) “Microstrip Dispersion”, Schneider, Proceedings of the IEEE, January 1972, pp. 144-146.
5) “Microstrip Dispersion Model”, Getsinger, IEEE Trans MTT, vol 21, no 1, Jan 1973, pp. 34-39.
6) “Transmission-Line Properties of a Strip on a Dielectric Sheet on a Plane”, Wheeler, IEEE Trans MTT, vol 25, no 8, Aug 1977, pp. 631-647.
7) “Accurate Models for Microstrip Computer-Aided Design”, Hammerstad and Jensen, MTT Microwave Symposium Digest, vol 80, Issue 1, May 1980, pp. 407-409.
8) “Losses of Microstrip Lines”, Denlinger, IEEE Trans MTT, vol 28, no 6, June 1980, pp. 513-522.
9) Computer-Aided Design, Gupta, Garg and Chadha, Artech House, 1981.
10) Foundations for Microstrip Design, Edwards, 1981
11) “Accurate Models for Effective Dielectric Constant of Microstrip with Validity up to Millimetre-Wave Frequencies”, Kirschning and Jansen, Electronic Letters, vol 18, no 6, March 1982, pp. 272-273.
12) “Measurement and Modeling of the Apparent Characteristic Impedance of Microstrip”, Getsinger, IEEE Trans MTT, vol 31, no 8, Aug 1983, pp. 624-632
13) “Design Rules for Microstrip Capacitance”, Bogatin, IEEE Trans on Components, Hybrids, and Manufacturing Technology, vol 11, no 3, Sept 1988, pp. 253-259.
14) “A Closed Form Analytical Model for the Electrical Properties of Microstrip Interconnects”, Bogatin, IEEE Trans on Components, Hybrids, and Manufacturing Technology, vol 13, no 2, June 1990, pp. 258-266.
15) Transmission Line Design Handbook, Wadell, Artech House, 1991.
16) Microstrip Circuits, Gardoil, John Wiley and Sons, 1994.
17) Microstrip Lines and Slotlines, Gupta, Garg, Bahl and Bhartia, Artech House 1996.
First, I have to say that rummaging through all of this literature to get simple expressions for a microstrip model was no easy task. There are a lot of errors and omissions that make the problem difficult. With that said, I chose the best formulas that would accurately model a microstrip line. My initial models used expressions from [13] and [14] for conductor loss, but I found them inaccurate and ended up going back to the literature to replace them. The procedure for calculating the R,L,G and C goes something like this:
1) Calculate the effective width of the strip (Weff) due to non-zero thickness. This was taken from [16] who got it from [3]. (I may try a different one in the future).
2) Calculate the effective dielectric constant (εr eff) due to non-TEM propagation. This was taken from [16] , who got it from [7].
3) Calculate the frequency dispersion corrected εr eff (εr eff-f) from [16], who got it from [11].
4) Calculate the characteristic impedance (Zc) from [16], who got it from [7].
5) Next calculate the Velocity of propagation (Vp) and Electrical length (ELen) from εr eff-f and length of segment.
6) From Zc and Vp, I can now calculate C and L.
7) Calculate G from αd taken from [16].
8) Calculate R and Li from αc taken from [9].
I ran a series of simulations comparing the results of TLineSim to Eagleware Genesys. Table 2 shows the impedance and insertion loss results for a 12 inch microstrip. Note: the system impedance (Z0) was changed to 25 and 75 ohms for the line impedances of Zc=25 and 75 ohms respectively. This eliminated any ripple in the insertion loss curve.
Table 2: Microstrip line comparison between TLineSim and Eagleware Genesys
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Eagleware |
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TLineSim |
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w |
h |
t |
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Zc |
I Loss |
I Loss |
I Loss |
Zc |
I Loss |
I Loss |
I Loss |
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mils |
mils |
mils |
dK |
Tand |
ohms |
300 MHz |
3 GHz |
20 GHz |
ohms |
300 MHz |
3 GHz |
20 GHz |
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21.3 |
8.0 |
0.7 |
2.60 |
0.0040 |
50.02 |
0.283 |
1.180 |
4.797 |
49.89 |
0.286 |
1.202 |
4.856 |
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56.2 |
8.0 |
0.7 |
2.60 |
0.0040 |
24.99 |
0.277 |
1.191 |
5.017 |
24.97 |
0.278 |
1.203 |
5.020 |
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10.4 |
8.0 |
0.7 |
2.60 |
0.0040 |
74.91 |
0.303 |
1.225 |
4.780 |
74.57 |
0.305 |
1.246 |
4.866 |
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15.7 |
8.0 |
0.7 |
4.00 |
0.0220 |
50.00 |
0.577 |
3.757 |
21.540 |
49.67 |
0.595 |
3.868 |
21.828 |
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43.6 |
8.0 |
0.7 |
4.00 |
0.0220 |
25.00 |
0.594 |
4.010 |
23.506 |
24.94 |
0.601 |
4.056 |
23.346 |
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7.1 |
8.0 |
0.7 |
4.00 |
0.0220 |
74.86 |
0.615 |
3.741 |
20.607 |
73.99 |
0.628 |
3.867 |
21.161 |
Coaxial transmission lines are probably the easiest to model. Many field theory books cover the subject. One good one is Fields and Waves in Communication Electronics, by Ramo, Whinnery and Van Duzer, John Wiley and Sons. In the 2nd edition, on page 252, is a Table (Table 5.11b) that lists the R, L, G and C for a coax line. These formulas were used in TLineSim.
Rectangular waveguide is also covered in most field theory books. The model used in TLineSim comes from Fields and Waves in Communication Electronics, by Ramo, Whinnery and Van Duzer, John Wiley and Sons. There are also some other references that where used in the waveguide model that I will publish later.