RL 884 Waveguide: A Program for Equivalent Circuit Representation of Two-Dimensional Rectangular Waveguide Discontinutities Katherine J. Herrick The University of Michigan Radiation Laboratory Dept. of Electrical Engineering and Computer Science Ann Arbor, MI 48109 RL-884 = RL-884

Waveguide: A Program for Equivalent Circuit Representation of Two-Dimensional Rectangular Waveguide Discontinutities Katherine J. Herrick The University of Michigan Radiation Laboratory Dept. of Electrical Engineering and Computer Science Ann Arbor, MI 48109

Introduction Waveguide is a Fortran program which will characterize simple twodimensional rectangular waveguide discontinuities, using quasi-static equivalent circuit methods. Specifically, it calculates the normalized reactance for a change in width or the normalized susceptance for a change in height at a waveguide junction. The circuit models examined are found in Marcuvitz's Waveguide Handbook.1 The equations used to derive these models may be found in Appendix A. Description of Equivalent Circuits Waveguide is. largely based on equations for two-terminal structures (equivalent to one-port circuits) found in Marcuvitz's Waveguide Handbook. In general, a rectangular waveguide has a width, x, and a height, y, with propagation in the negative z direction (Fig 1 & 2). This program characterizes a waveguide discontinuity. The input section preceding the discontinuity is a uniform rectangular waveguide of width a and height b. Dominant mode propagation is assumed in this section. The output region is a section of rectangular waveguide of width a' and height b', whose dimensions are such that the operating frequency is above the cut-off frequency of the dominant mode. B/Yo, calculated for a change in height at the waveguide junction, represents the ratio between the susceptance and the characteristic admittance of the guide. Likewise, X/Zo, calculated for a change in width, represents the ratio between the reactance and the characteristic impedance of the guide. Four types of equivalent circuit models are examined. The first case is that of an axially symmetrical junction of two rectangular guides of unequal widths but equal heights (Fig. la). This case has dominant mode propagation in the large rectangular guide and no propagation in the small guide. By terminating the circuit with the inductive characteristic impedance of the smaller guide, an equivalent circuit is obtained (Fig. Ic). It is valid within the following parameters. First, the ratio of the width of the first waveguide to the wavelength must lie between 0.5 1 Marcuvitz, N., Waveguide Handbook. McGraw-Hill Book Company, Inc., New York, 1951. 1

I/ K 4s axI -5ymrH'u-tcai ~A-p a-,, y in uwidT 0 --- (rrpLLZe lo. oyY~oe4T- L ~+~p(-,ct,t"L- L&'C~ ~ ix c.- eq UJvaoiervL.0 I rCV -t-tW

I b~ i npvt4 inpPu~t >1 Cx -5 rncA-rTl CoLl b K(, ) I 2VCfQ C:> - I inp b ~2. ~tVO-fl.~c~ - 0 ---- - --- cwTGJN, In V,-C(E<Vt+ FuI}Ltr-c 2.

and 1.5 (0.5 < a/i <1.5). Second, the wavelength must be larger than twice the width of the second guide (X > 2a'), such that the second guide is at cut-off. The second case is that of an axially asymmetrical junction of two rectangular guides of unequal widths but equal heights (Fig. lb). This, again, has the H10-mode in the large rectangular guide and no propagation in the small guide. The equivalent circuit (Fig. Ic) is valid within the same parameters with the following exception. The ratio of the input width to the wavelength must lie between 0.5 and 1.0 (0.5 < a/X <1.0). The third case is that of an axially symmetrical junction of two rectangular guides of equal widths but unequal heights (Fig. 2a). This is equivalent to the FIlo-modes in both rectangular guides. The equivalent capacitive circuit (Fig. 2c) is valid while the ratio of the height of the first guide to the wavelength of the guide is less than 1 (b/Xg <1.). The equations used were obtained by the equivalent static method employing a static aperture field due to the incidence of the two lowest modes and is correct to within 1% in the range previously specified. The fourth case is that of an asymmetrical junction of two rectangular guides of equal widths but unequal heights (Fig. 2b). This is simply the asymmetric version of the third case and is also correct to within 1% in the range b/Xg <1. Its equivalent circuit representation is shown in Figure 2c. Review: Rectangular Waveguides Waveguide characterizes a hollow rectangular waveguide. The dominant mode is the TE10 mode, having the lowest cutoff frequency. Using x as the width and iy as the height of the guide, the general equation for cutoff frequency is: fc= l/(2i) f (ny)2(m/x)2 Since the cutoff frequency is a function of the modes and guide dimensions, the physical size of the waveguide will determine the propagation of the modes. Assuming x>y, we take m = 1 and n = 0, and fc = c/2x. 2

Likewise, the cutoff wavelength, c =2/(n/)2 + (m/x)2 = 2x The wavelength of the guide is given by: kg = Xo/l-(fjf)2, ko = vpdf Theory: Waveguide Disconituities as Equivalent Circuits The waveguide structures dicusses in this report consist of two uniform sections of different dimensions separated by a discontinuity. Generally, the fields within a uniform section of waveguide may be described by the dominant mode. At a discontinuity, however, a complete description of the fields within that region requires an infinite number of nonpropagating modes in addition to the dominant propagating mode. These additional modes are evanescent and decay exponentially in both directions away from the discontinuity, resulting in a localized storage of reactive energy. Thus, the fields at the discontinuity may be approximated as lumped parameter circuit models. Getting Started with Waveguide To run Waveguide, follow these instructions: 1. Type the executable to initiate the program and a <CR>. On the apollos, using the Unix operating system, this would be: %' waveguide <CR> 2. Waveguide asks you to choose one of the four types of junctions. You may simply respond by typing the number of your choice (1,2,3, or 4) and <CR>. For example, if you were interested in a symmetrical junction, inductive equivalent circuit you would type 1 and a <CR>: %PLEASE SELECT WAVEGUIDE JUNCTION. 3

Two rectangular guides of: 1) Unequal widths but equal heights at an axially symmetrical junction (inductive equiv. ckt) 2) Unequal widths but equal heights at an axially asymmetrical junction (inductive equiv. ckt) 3) Equal widths but unequal heights at an axially symmetrical junction (capacitive equiv. ckt) 4) Equal widths but unequal heights at an axially asymmetrical junction (capacitive equiv. ckt) % 1<CR> 3. You will first be prompted for the dimensions of the first guide. Enter all guide dimensions in meters. If either of the inductive equivalent circuit cases is chosen (1 or 2), the restrictions on the dimensions of the second guide will be briefly stated and you must then enter an incremental change in the width of the second guide. For example, if you wanted to test a junction in which guide #1 was 19.55 cm X 15.3 cm with increments of.01: %Enter values for widthl and heightl(in meters): %.1955,.153 <CR> %Height2 will equal heightl. Width2 will range from 0.0 to widthl so 0.0 < alpha < 1.0. By what incremental change shall width2 increase? %.01 <CR> Alpha represents the ratio between width2 (a') and widthl (a). Alpha's limiting value is one to ensure the second waveguide's dimensions do not exceed the first. Conversely, if either of the capacitive equivalent circuit cases is chosen (3 or 4), the restrictions on the dimensions of the second guide will be briefly stated and you must enter an incremental change in height of the second guide. %oWidth2 will equal widthl. Height2 will range from 0.0 to heightl 4

so 0.0 < alpha < 1.0. By what incremental change shall height2 increase? %.01 <CR> In the capacitive case, alpha represents the ratio between height2 (b') and heightl (b). Again, alpha's limiting value is one to ensure the second waveguide's dimensions do not exceed the first. 4. Next, select frequency range (in GHz). For example, if your desired range was from.96 GHz to.97 GHz and you wanted to know values every.01 GHz: %Enter beginning freq.,ending freq., increments(GHz): %.96,.97,.01 5. For an inductive case, Waveguide treats input data in the following manner. At the initial frequency value, the ratio X/Zo is computed for each incremental change in width2. This incremental change is represented by alpha (a'/a), which ranges from 0 to 1. When alpha reaches its limit of 1, the frequency is incremented and the calculations are repeated. The program continues to loop until the limiting frequency is reached. 6. For a capacitive case, Waveguide treats input data in the following manner. At the initial frequency value, the ratio B/Yo is computed for each incremental change in height2. This incremental change is represented by alpha (b'/b), which ranges from 0 to 1. When alpha reaches its limit of 1, the frequency is incremented and the calculations are repeated. The program continues to loop until the limiting frequency is reached. 7. Results are stored in two files: info.dat and plot.dat. Info.dat contains general information such as the junction chosen, the guide dimensions, the frequency, alpha, and either X/Zo or B/Yo. Plot.dat is specifically for plotting. Column 1 contains values of alpha while 5

column 2 contains values of X/Zo if a inductive case has been run, or B/Yo if an capacitive case has been run. Figures 6a., 6b., and 6c. are examples of info.dat, plot.dat, and an actual plot for case 1. Likewise, Figures 7, 8, and 9 represent files and plots for cases 2, 3, and 4 respectively. The plots presented in this report were generated using / /hybrid/ users/norman/plotting/ nplot. 6

Results for symmetric inductive case: Dimensions: f 0.9 600 0 0.96 00 0 0.96 00 0 0.96 00 0 0.96 00 0 0.9 6 000 0.96 00 0 0.96 00 0 0.96 00 0 0.96 00 0 0 ) 9 60 00 0. 9 600 0 0.1 96 000 0 ) 9 600 0 0. 9 6 0 00 w<OR== 2 *W2, 0.9 7 000 0.970 00 0 9 97 000 0. 97 0 00 0.97000 0.970C)0 0 Si 970 0 0 0 IQ 970 C) 0 0. 9 70 Cia 0 9 970 00 0. 97 0 00 0.97 00 0 0.9 700 0 0.97 00 0 0.97 00 0.1955m by W2 0.01 00 0 0.0 2 00 0 0.0 3 00 0 0.0 4 00 0 0.05 00 0 0.0 6 00 0 0.0 7 000 0.08 00 0 0.0 9 000 0.10 00 0 0. 1 100 0 0. 12 00 0 0.1 3 000 0. 14 00 0 0. 15 00 0,EQUIV. CK 0.01 00 0 0.0 2 00 0 0. 03 000 0. 04 000 0.05 00 0 0.0 600 0 0.0 700 0 0.08 00 0 0.09 00 0 0.10 00 0 0.I1100 0 0.12 00 0 0.13 00 0 0.14 00 0 0.15 00 0.1530m, W2 h alpha 0. 05115 0.10,230( 0.1 5 3 4 5 0.20 46 0 0.2 557 5 0.3 06-91 -0.35 8 06 0.40921 0.46036 0. 51 15 1 0.5 626 6 0.6 138 1 0.6 6 496 0.7 161 1 0.7 672 6 'T. NOT VALID 0.0 5 115 0.1 023 0 0.1 534 5 0.2 046 0 0.2 5 575 0 3 306 9 1 0.35 8 06 0.4 092 1 0.46 03 6 0.51 15 1 0.5 6 266 0.6138 1 0. 6 649 6 0.716111 0.7 672 6 byf.153Cm X/Zo 0 0 19 8 30 0 79 9 0.0 182 8. 0 33 19 3.0532 0 3.0 7 90 0. 1115 1 3.152 02 3.2 024 7. 2 65 83 3.34 7 09 3.45 54 8 3.61 07 2 3.8 670 0 1.4 8 512 0.0 02 03 0.0 0 822 0.01 88 1 0.0 3 416 0.0 547 9 0.0 8 141 0.1 149 8 0.1 56 89 0.2 09 19 0.27 5 09 0.36 00 5 0.47 42 9 0.64 03 7 0.92 36 8 1.69 74 7 w<OR= 2*W2,EQUIV. CKT NOT VALID Figure 6a. info.dat

5. 1150892E-02 1. 9765685E-03 0.1023018 7.9938713E-03 0.1534527 1.8279992E-02 0.2046036 3.3186160E-02 0.2557545 5.3203840E-02 0.3069053 7.9003461E-02 0.3580562 0.1115088 0.4092071 0.1520242 0.4603580 0.2024691 0.5115089 0.2658284 0.5626597 0.3470l 16 0.6138107 0.4554784 0.6649615 0.6107232 0.7161124 0.8670046 0.7672634 1.485118 5.1150892E-02 2.0332176E-03 0.1023018 8.2244696E-03 0.1534527 1.8812237E-02 0.2046036 3.4163967E-02 0.2557545 5.4794662E-02 0.3069053 8.1408598E-02 0.3580562 0.1149799 0.4092071 0.1568911 0.4603580 0.2091908 0.5115089 0.2750949 0.5626597 0.3600459 0.6138107 0.4742852 0.6649615 0.6403701 0.7161124 0.9236836 0.7672634 1.697470 2.00, (,,,,, r r -.-rr-.Fx --- |..| 1.80 1.60_ =.96 GHz 1.40 -- =.97GHz/ 1.20 - 1.00 - 0.800 - 0.600 0.400 0.200 0.000... I I I -I-I I... I. 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 alpha (a'/a) Figure 6b. plot.dat Figure 6c. X/Zo vs alpha

Results for Dimensions: f 0.96 00 0 0. 96 00 0 0.96 00 0 0.96 00 0 0.96 00 0 0.96 00 0 0.'960 00 0.9 60 00 0.9 60 00 0. 9 6000 0. 9 6 00 0 0.1 96 000 0 ) 9 600 0 0. 96 00 0 0.9 6 00 0 0 9 97 000 0. 97 00 0 0.97 00 0 0. 97 00 0 0 Si 97 000 0 Si 970 C) 0 0. 97000 0. 97 00 0 0. 97 0C00 0.97 0 00 0.97 00 0 0.9 7 000 0.97 00 0 0. 97 00 0 0.97 00 0 asymmetric inductive case: W2 0.0 0 000 0.0 000 0 0.00 00 0 0.00 00 0 0.00 00 0 0.00 00 0 0.00 00 0 0.00 00 0 0.00 00 0 0.00 00 0 0.00 00 0 0.00 00 0 0.00 00 0 0.00 00 0 0.00 00 0 0.00 00 0 0.00 00 0 0.00 00 0 0.00 00 0 0.00 00 0 0.00 0 00 0.00 00 0 0.00 00 0 0.0 000 0 0.00 00 0 0.00 0 00 0.00 00 0 0.00 00 0 0.00 00 0 0.00 00 0 a lphia 0.051-15 0.102-30C 0.153-45 0.204 60 0.25575, 0.30 691 0.35606 0. 409121 0. 46 C3 6 0.5 115 1 0.5 626 6 0. 6 138 1 0.66 49 6 0.7 161 1 0.76 72 6 0.05 11 5 0.1 023 0 0. 1534 5 0.2 046 0 0.255 7 5 0.3 069 1 0.3580C)6 0 4 4092 1 0.46 03 6 0.511 -1 1.5 0.6138 1 0 6 664 19 6 0.716 11 0 7 7672 6 X/Zo 0.0 000 1 0.0 082 0 0.0 167 4 0.12 39 5 0.3 862 1 0.58 12 3 0. 00 00 1 0.0 0 112 0.01'72-5 0.0 842 4 0.1 2 817 0.1 894 5 1.1 193 2 Figure 7a. if~a info.dat

5. 1150892E-02 1.3668611E-05 0.1023018 2.1785172E-04 0.1534527 1.0921899E-03 0.2046036 3.4085822E-03 0.2557545 8.2008895E-03 0.3069053 1.6742714E-02 0.3580562 3.0548261E-02 0.4092071 5.1418442E-02 0.4603580 8.1572764E-02 0.5115089 0.1239472 0.5626597 0.1828516 0. 6138107 0.2655055 0.6649615 0.3862067 0.7161124 0.5812325 0.7672634 1.004403 5.1150892E-02 1.4059674E-05 0.1023018 2.2411690E-04 0.1534527 1.1238480E-03 0.2046036 3.5084779E-03 0.2557545 8.4446706E-03 0.3069053 1.7249178E-02 0.3580562 3.1491984E-02 0.4092071 5.3047806E-02 0.4603580 8.4240593E-02 n.5115089 0.1281 711 0.5626597 0.1894479 0.6138107 0.2759297 0.6649615 0.4036019 0.7161124 0.6150534 0.7672634 1.119317 2.00 1.80 1.60 1.40 1.20 1.00 0.800 0.600 0.400 0.200 0.00 0 0.20 0.30 0.4 0 0.50 0.60 0.70 0.80 0.90 1.00 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 alpha (a'/a) Figure 7b. plot.dat Figure 7c. X/Zo vs alpha

Resullrs for symmetric capacitive case: )imensions:.1955m by.1530m,.1955m by H2 f H2 alpha B/Yo 0.96000 0.01000 0.06536 1.40867 0.96000 0.02000 0.13072 1.00134 0.96000 0.03000 0.19608 0.76406 0.96000 0.04000 0.26144 0.59722 0.96000 0.05000 0.32680 0.46992 0.96000 0.06000 0.39216 0.36855 0.96000 0.07000 0.45752 0.28590 0.96000 0.08000 0.52288 0.21771 0.96000 0.09000 0.58824 0.16125 0.96000 0.10000 0.65359 0.11467 0.96000 0.11000 0.71895 0.07677 0.96000 0.12000 0.78431 0.04672 0.96000 0.13000 0.84967 0.02403 0.96000 0.14000 0.91503 0.00852 0.96000 0.15000 0.98039 0.00060 alpha (H2/H1) has reached limiting value of 1 0.97000 0.01000 0.06536 1.45082 0.97000 0.02000 0.13072 1.03174 0.97000 0.03000 0.19608 0.78749 0.97000 0.04000 0.26144 0.61567 0.97000 0.05000 0.32680 0.48450 0.97000 0.06000 0.39216 0.38000 0.97000 0.07000 0.45752 0.29478 0.97000 0.08000 0.52288 0.22446 0.97000 0.09000 0.58824 0.16622 0.97000 0.10000 0.65359 0.11819 0.97000 0100 011000 0.71895 0.07911 0.97000 0.12000 0.78431 0.04813 0.97000 0.13000 0.84967 0.02475 0.97000 0.14000 0.915C3 0.00877 0.97000 0.15000 0.98039 0.00061 tlpha (H2/H1) has reached limiting value of 1 Figure 8a. info.dat

1 cr 0.06536 1.40867.U 0.13072 1.00134 0.19608 0.76406 1.35 0.26144 0.59722 0.32680 0.46992 1.20 \ =.96 GHz 0.39216 0.36855 -- =.97GHz 0.45752 0.28590 1.05 0.52288 0.21771 0.58824 0.16125 0900 0.65359 0.11467 0.71895 0.07677 0 750 0.78431 0.04672 0.84967 0.024030 0.91503 0.00852 0.600 0.98039 0.00060 0.06536 1.45082 0.450 0.13072 1.03174 0.19608 0.78749 0.300 0.26144 0.61567 0.32680 0.48450 0.150 0.39216 0.38000 0.45752 0.29478 0.000 1. 1 0.52288 0.22446 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0.58824 0.16622 0.65359 0.11819 0.71895 0.07911 alpha(b'/b) 0.78431 0.04813 0.84967 0.02475 0.91503 0.00877 0.98039 0.00061 Figure 8b. plot.dat Figure 8c. B/Yo vs alpha

Results for asymmetric capacitive case: Dimensions:.1955m by.1530m,.1955m by H2 f H2 alpha B/Yo 0.96000 0.01000 0.06536 3.05568 0.96000 0.02000 0.13072 2.22648 0.96000 0.03000 0.19608 1.72974 0.96000 0.04000 0.261]44 1.36893 0.96000 0.05000 0.32680 1.08505 0.96000 0.06000 0.39216 0.85339 0.96000 0.07000 0.45752 0.66133 0.96000 0.08000 0.52288 0.50135 0.96000 0.09000 0.58824 0.36847 0.96000 0.10000 0.65359 0.25922 0.96000 0.11000 0.71895 0.17116 0.96000 0.12000 0.78431 0.10250 0.96000 0.13000 0.84967 0.05181 0.96000 0.14000 0.91503 0.01805 0.96000 0.15000 0.98039 0.00124 alpha (H2/H1) has reached limiting value of 1 0.97000 0.01000 0.06536 3.16757 0.97000 0.02000 0.130'72 2.31323 0.97000 0.03000 0.19608 1.80007 0.97000 0.04000 0.26144 1.42618 0.97000 0.05000 0.32680 1.13116 0.97000 0.06000 0.39216 0.88985 0.97000 0.07000 0.45752 0.68948 0.97000 0.08000 0.52288 0.52245 0.97000 0.09000 0.58824 0.38367 0.97000 0.10000 0.65359 0.26963 0.97000 0.11000 0.71895 0.17781 0.97000 0.12000 0.78431 0.10633 0.97000 0.13000 0.84967 0.05366 0.97000 0.14000 0.91503 0.01866 0.97000 0.15000 0.98039 0.00128 alpha (H2/H1) has reached limiting value of 1 Figure 9a. info.dat

0.06536 0.13072 0.19608 0.26144 0.32680 0.39216 0.45752 0.52288 0.58824 0.65359 0.71895 0.78431 0.84967 0.91503 0.98039 0.06536 0.13072 0.19608 0.26144 0.32680 0.39216 0.45752 0.52288 0.58824 0.65359 0.71895 0.78431 0.84967 0.91503 0.98039 3.05568 2.22648 1.72974 1.36893 1.08505 0.85339 0.66133 0.50135 0.36847 0.25922 0.17116 0.10250 0.05181 0.01805 0.00124 3.16757 2.31323 1.80007 1.42618 1.13116 0.88985 0.68948 0.52245 0.38367 0.26963 0.17781 0.10633 0.05366 0.01866 0.00128 3.50 r 3.15 2.80 - 2.45 - 2.10 1.75 1.40 1.05 0.700 0.350 0.000 L 0,00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 alpha (b'/b) Figure 9b. plot.dat Figure 9c. B/Yo vs alpha

Appendix A: Equations as Presented by Marcuvitz'

. I>, I 11Q) H Q H I P I Q, 11 7 I I w I tll -ciz, >1 i: 00 0 p I 11 J — 'I-, I I WI tll. p >,I Q I +, tl_, I - i z, L, i t 1) I I -. - 1 k, -, - t) i i i 1 I I I +1 ' ~ K &i; ii (0 -II-4 QI- r< QI' Q1 I i -, Q 1-1 Q_ I.-I p I —, 5 I 0 -pto I:rI J t-j + - -HI + II (0 +1 + - (- _ H + + 'U1 "-, — I IK7< -K — 'C I (0 LQK- (0 & I0 K I I - i 1; I I I~-ft'-K I -l I — I - - I Iz - -, -.t Z, 1- 1 I 1I -I-I i i I — l % I I I, -t( I'D I I I -D i I -4 -1 1 " I.;I - i CD t o C/ 4 0-0 r" 5 n r7l n C4 ril i - I - 11 I I - -,, i i; - — I I 7 -l< ,-, t I, - Q i — T- I I - p -I - I II m 1 1-9 t"D I WI tl I

(1 a = -- a at (J \2a The functions E((x), E(oc'), F(cx), and F(oc') represent the complete Legendre elliptic integrals of the first, F, and second, E, kind. CASE 2) ASYMMETRICAL INDUCTIVE CASE x 7( * -- I - IO -- -,.- A) I - I --- A( ) A, < ii - - K 2; N22 \i X WHERE TH-I FOLILOWING IEQUATIONS ARE THE SAMNE AS IN CASE 1: A'l I -- 1. ' 1 -1 A~ 4- 2<1 - )CA"I' ( V. -- L I 4-12N22 -I 2(A' +I- )C'N 12 +~(2( A 22 " C2 (o - f,'!, - x '. Ai '1' (712 (1 -/- (1 - R2) -- 72' a' = / 1 - a2. Q -1 - I () a af (X a -; a XQ /I x Kx\2

AND R 1, R2, T 1, N1I1I, N22, N 12, AND Q' DIFFER: [2( 2( + 1 2+ I1 a2L 3 (~~a2) +- 1 -V 2 7 "<) <~a>2K + 232a ( ) 1 6a2 7Q I(+f 'W/' + a 2 i 2LcxL - I (I 12 ((Q 22\x,/ / 1

CASE 3) SYMMETRICAL CAPACITIVE CASE [13 2) [ C ( 1- I +a)( + -4)+2' + 2C )7, = ) ( A' - C-, - I/ (4 ( I - y) (+ ' - 1 3 A'( WHERE: i X 1 -- (-) I CASE 4) ASYIMMETRICAL CAPACITIVE CASE Same equatilon for B/Y) as for CASE 3. Only difference:,r, = g/2.