Toroid turns ratios

I wanted to put up a T2FD (Tilted Terminated Folded Dipole) so I spent a few days reading web pages and PDFs first. I saw a lot of semi-technical speculation because the basic design has been around since 1949. But I wanted to model it first, so I looked in the sample antennas that come with MMANA-GAL and sure enough there was one in there that looked pretty good, and running through the modeling with the plots and calculations looked good too.

But it's a 600 ohm design with a 650 ohm terminating resistor and I want to feed it with RG6 which is 75 ohms. OT: I bought a 500 foot roll for $32 at Home Depot. So I needed to come up with an 8:1 balun design.

With baluns, at least the ones wound on toroids, the impedance ratio is the square of the turns ratio. For a 2:1 turns ratio you get 4:1 impedance, for 3:1 turns you get 9:1 impedance. So how can you get something like 8:1 but stick to whole turns? By using higher numbers of turns.

I wrote this little program (see the bottom of this page) with 2 loops from 2 to 50 which just calculates the turns ratio then squares it to get the impedance ratio. Sheer blind and dumb iteration. Then I sorted the output by impedance ratio by doing:

 sort -n -k 4 < turns.txt > turns2.txt
(Unix sort, not DOS/Windows; different syntax probably)

So how do you use this? Look down the right column until you find the impedance ratio you want, then look back at the first two (leftmost) columns to find the turns. There were a lot of duplicates but I left them in (reciprocals and multiple solutions). For an 8:1 impedance ratio I picked 17 turns on the primary, 6 turns on the secondary.

You should run whatever you find here through a calculator to check it before you use it, this just saves a lot of "hit or miss" speculation. I couldn't do I:J so I did I/J, some of it looks funny, but reciprocals are in here too. There are 2300+ lines here:

On impedance matching, I did find that the impedance can be considered to be the geometric mean of the DC resistance and the reactance at the lowest frequency you intend to use it for. You can look up a formula to figure out the inductance per turn on the core you plan to use, then calculate an inductive reactance from that. Try to figure out which side is more critical and calculate a number of turns for that, then use the impedance ratio to figure the number turns for the second winding. It seems to work.

pri	sec	turns ratio	imp. ratio
2	49	0.040816	0.001666
2	48	0.041667	0.001736
2	47	0.042553	0.001811
2	46	0.043478	0.001890
2	45	0.044444	0.001975
2	44	0.045455	0.002066
2	43	0.046512	0.002163
2	42	0.047619	0.002268
2	41	0.048780	0.002380
2	40	0.050000	0.002500
2	39	0.051282	0.002630
2	38	0.052632	0.002770
2	37	0.054054	0.002922
2	36	0.055556	0.003086
2	35	0.057143	0.003265
2	34	0.058824	0.003460
2	33	0.060606	0.003673
3	49	0.061224	0.003748
2	32	0.062500	0.003906
3	48	0.062500	0.003906
3	47	0.063830	0.004074
2	31	0.064516	0.004162
3	46	0.065217	0.004253
2	30	0.066667	0.004444
3	45	0.066667	0.004444
3	44	0.068182	0.004649
2	29	0.068966	0.004756
3	43	0.069767	0.004867
2	28	0.071429	0.005102
3	42	0.071429	0.005102
3	41	0.073171	0.005354
2	27	0.074074	0.005487
3	40	0.075000	0.005625
2	26	0.076923	0.005917
3	39	0.076923	0.005917
3	38	0.078947	0.006233
2	25	0.080000	0.006400
3	37	0.081081	0.006574
4	49	0.081633	0.006664
2	24	0.083333	0.006944
3	36	0.083333	0.006944
4	48	0.083333	0.006944
4	47	0.085106	0.007243
3	35	0.085714	0.007347
2	23	0.086957	0.007561
4	46	0.086957	0.007561
3	34	0.088235	0.007785
4	45	0.088889	0.007901
2	22	0.090909	0.008264
3	33	0.090909	0.008264
4	44	0.090909	0.008264
4	43	0.093023	0.008653
3	32	0.093750	0.008789
2	21	0.095238	0.009070
4	42	0.095238	0.009070
3	31	0.096774	0.009365
4	41	0.097561	0.009518
2	20	0.100000	0.010000
3	30	0.100000	0.010000
4	40	0.100000	0.010000
5	49	0.102041	0.010412
4	39	0.102564	0.010519
3	29	0.103448	0.010702
5	48	0.104167	0.010851
2	19	0.105263	0.011080
4	38	0.105263	0.011080
5	47	0.106383	0.011317
3	28	0.107143	0.011480
4	37	0.108108	0.011687
5	46	0.108696	0.011815
2	18	0.111111	0.012346
3	27	0.111111	0.012346
4	36	0.111111	0.012346
5	45	0.111111	0.012346
5	44	0.113636	0.012913
4	35	0.114286	0.013061
3	26	0.115385	0.013314
5	43	0.116279	0.013521
2	17	0.117647	0.013841
4	34	0.117647	0.013841
5	42	0.119048	0.014172
3	25	0.120000	0.014400
4	33	0.121212	0.014692
5	41	0.121951	0.014872
6	49	0.122449	0.014994
2	16	0.125000	0.015625
3	24	0.125000	0.015625
4	32	0.125000	0.015625
5	40	0.125000	0.015625
6	48	0.125000	0.015625
6	47	0.127660	0.016297
5	39	0.128205	0.016437
4	31	0.129032	0.016649
3	23	0.130435	0.017013
6	46	0.130435	0.017013
5	38	0.131579	0.017313
2	15	0.133333	0.017778
4	30	0.133333	0.017778
6	45	0.133333	0.017778
5	37	0.135135	0.018262
3	22	0.136364	0.018595
6	44	0.136364	0.018595
4	29	0.137931	0.019025
5	36	0.138889	0.019290
6	43	0.139535	0.019470
2	14	0.142857	0.020408
3	21	0.142857	0.020408
4	28	0.142857	0.020408
5	35	0.142857	0.020408
6	42	0.142857	0.020408
7	49	0.142857	0.020408
7	48	0.145833	0.021267
6	41	0.146341	0.021416
5	34	0.147059	0.021626
4	27	0.148148	0.021948
7	47	0.148936	0.022182
3	20	0.150000	0.022500
6	40	0.150000	0.022500
5	33	0.151515	0.022957
7	46	0.152174	0.023157
2	13	0.153846	0.023669
4	26	0.153846	0.023669
6	39	0.153846	0.023669
7	45	0.155556	0.024198
5	32	0.156250	0.024414
3	19	0.157895	0.024931
6	38	0.157895	0.024931
7	44	0.159091	0.025310
4	25	0.160000	0.025600
5	31	0.161290	0.026015
6	37	0.162162	0.026297
7	43	0.162791	0.026501
8	49	0.163265	0.026656
2	12	0.166667	0.027778
3	18	0.166667	0.027778
4	24	0.166667	0.027778
5	30	0.166667	0.027778
6	36	0.166667	0.027778
7	42	0.166667	0.027778
8	48	0.166667	0.027778
8	47	0.170213	0.028972
7	41	0.170732	0.029149
6	35	0.171429	0.029388
5	29	0.172414	0.029727
4	23	0.173913	0.030246
8	46	0.173913	0.030246
7	40	0.175000	0.030625
3	17	0.176471	0.031142
6	34	0.176471	0.031142
8	45	0.177778	0.031605
5	28	0.178571	0.031888
7	39	0.179487	0.032216
2	11	0.181818	0.033058
4	22	0.181818	0.033058
6	33	0.181818	0.033058
8	44	0.181818	0.033058
9	49	0.183673	0.033736
7	38	0.184211	0.033934
5	27	0.185185	0.034294
8	43	0.186047	0.034613
3	16	0.187500	0.035156
6	32	0.187500	0.035156
9	48	0.187500	0.035156
7	37	0.189189	0.035793
4	21	0.190476	0.036281
8	42	0.190476	0.036281
9	47	0.191489	0.036668
5	26	0.192308	0.036982
6	31	0.193548	0.037461
7	36	0.194444	0.037809
8	41	0.195122	0.038073
9	46	0.195652	0.038280
2	10	0.200000	0.040000
3	15	0.200000	0.040000
4	20	0.200000	0.040000
5	25	0.200000	0.040000
6	30	0.200000	0.040000
7	35	0.200000	0.040000
8	40	0.200000	0.040000
9	45	0.200000	0.040000
10	49	0.204082	0.041649
9	44	0.204545	0.041839
8	39	0.205128	0.042078
7	34	0.205882	0.042388
6	29	0.206897	0.042806
10	48	0.208333	0.043403
5	24	0.208333	0.043403
9	43	0.209302	0.043807
4	19	0.210526	0.044321
8	38	0.210526	0.044321
7	33	0.212121	0.044995
10	47	0.212766	0.045269
3	14	0.214286	0.045918
6	28	0.214286	0.045918
9	42	0.214286	0.045918
8	37	0.216216	0.046749
10	46	0.217391	0.047259
5	23	0.217391	0.047259
7	32	0.218750	0.047852
9	41	0.219512	0.048186
10	45	0.222222	0.049383
2	9	0.222222	0.049383
4	18	0.222222	0.049383
6	27	0.222222	0.049383
8	36	0.222222	0.049383
11	49	0.224490	0.050396
9	40	0.225000	0.050625
7	31	0.225806	0.050989
10	44	0.227273	0.051653
5	22	0.227273	0.051653
8	35	0.228571	0.052245
11	48	0.229167	0.052517
3	13	0.230769	0.053254
6	26	0.230769	0.053254
9	39	0.230769	0.053254
10	43	0.232558	0.054083
7	30	0.233333	0.054444
11	47	0.234043	0.054776
4	17	0.235294	0.055363
8	34	0.235294	0.055363
9	38	0.236842	0.056094
10	42	0.238095	0.056689
5	21	0.238095	0.056689
11	46	0.239130	0.057183
6	25	0.240000	0.057600
7	29	0.241379	0.058264
8	33	0.242424	0.058770
9	37	0.243243	0.059167
10	41	0.243902	0.059488
11	45	0.244444	0.059753
12	49	0.244898	0.059975
10	40	0.250000	0.062500
11	44	0.250000	0.062500
12	48	0.250000	0.062500
2	8	0.250000	0.062500
3	12	0.250000	0.062500
4	16	0.250000	0.062500
5	20	0.250000	0.062500
6	24	0.250000	0.062500
7	28	0.250000	0.062500
8	32	0.250000	0.062500
9	36	0.250000	0.062500
12	47	0.255319	0.065188
11	43	0.255814	0.065441
10	39	0.256410	0.065746
9	35	0.257143	0.066122
8	31	0.258065	0.066597
7	27	0.259259	0.067215
12	46	0.260870	0.068053
6	23	0.260870	0.068053
11	42	0.261905	0.068594
10	38	0.263158	0.069252
5	19	0.263158	0.069252
9	34	0.264706	0.070069
13	49	0.265306	0.070387
12	45	0.266667	0.071111
4	15	0.266667	0.071111
8	30	0.266667	0.071111
11	41	0.268293	0.071981
7	26	0.269231	0.072485
10	37	0.270270	0.073046
13	48	0.270833	0.073351
12	44	0.272727	0.074380
3	11	0.272727	0.074380
6	22	0.272727	0.074380
9	33	0.272727	0.074380
11	40	0.275000	0.075625
8	29	0.275862	0.076100
13	47	0.276596	0.076505
10	36	0.277778	0.077160
5	18	0.277778	0.077160
12	43	0.279070	0.077880
7	25	0.280000	0.078400
9	32	0.281250	0.079102
11	39	0.282051	0.079553
13	46	0.282609	0.079868
10	35	0.285714	0.081633
12	42	0.285714	0.081633
14	49	0.285714	0.081633
2	7	0.285714	0.081633
4	14	0.285714	0.081633
6	21	0.285714	0.081633
8	28	0.285714	0.081633
13	45	0.288889	0.083457
11	38	0.289474	0.083795
9	31	0.290323	0.084287
14	48	0.291667	0.085069
7	24	0.291667	0.085069
12	41	0.292683	0.085663
10	34	0.294118	0.086505
5	17	0.294118	0.086505
13	44	0.295455	0.087293
8	27	0.296296	0.087791
11	37	0.297297	0.088386
14	47	0.297872	0.088728
12	40	0.300000	0.090000
3	10	0.300000	0.090000
6	20	0.300000	0.090000
9	30	0.300000	0.090000
13	43	0.302326	0.091401
10	33	0.303030	0.091827
14	46	0.304348	0.092628
7	23	0.304348	0.092628
11	36	0.305556	0.093364
15	49	0.306122	0.093711
12	39	0.307692	0.094675
4	13	0.307692	0.094675
8	26	0.307692	0.094675
13	42	0.309524	0.095805
9	29	0.310345	0.096314
14	45	0.311111	0.096790
10	32	0.312500	0.097656
15	48	0.312500	0.097656
5	16	0.312500	0.097656
11	35	0.314286	0.098776
12	38	0.315789	0.099723
6	19	0.315789	0.099723
13	41	0.317073	0.100535
14	44	0.318182	0.101240
7	22	0.318182	0.101240
15	47	0.319149	0.101856
8	25	0.320000	0.102400
9	28	0.321429	0.103316
10	31	0.322581	0.104058
11	34	0.323529	0.104671
12	37	0.324324	0.105186
13	40	0.325000	0.105625
14	43	0.325581	0.106003
15	46	0.326087	0.106333
16	49	0.326531	0.106622
10	30	0.333333	0.111111
11	33	0.333333	0.111111
12	36	0.333333	0.111111
13	39	0.333333	0.111111
14	42	0.333333	0.111111
15	45	0.333333	0.111111
16	48	0.333333	0.111111
2	6	0.333333	0.111111
3	9	0.333333	0.111111
4	12	0.333333	0.111111
5	15	0.333333	0.111111
6	18	0.333333	0.111111
7	21	0.333333	0.111111
8	24	0.333333	0.111111
9	27	0.333333	0.111111
16	47	0.340426	0.115890
15	44	0.340909	0.116219
14	41	0.341463	0.116597
13	38	0.342105	0.117036
12	35	0.342857	0.117551
11	32	0.343750	0.118164
10	29	0.344828	0.118906
9	26	0.346154	0.119822
17	49	0.346939	0.120367
16	46	0.347826	0.120983
8	23	0.347826	0.120983
15	43	0.348837	0.121687
14	40	0.350000	0.122500
7	20	0.350000	0.122500
13	37	0.351351	0.123448
12	34	0.352941	0.124567
6	17	0.352941	0.124567
17	48	0.354167	0.125434
11	31	0.354839	0.125911
16	45	0.355556	0.126420
10	28	0.357143	0.127551
15	42	0.357143	0.127551
5	14	0.357143	0.127551
14	39	0.358974	0.128863
9	25	0.360000	0.129600
13	36	0.361111	0.130401
17	47	0.361702	0.130828
12	33	0.363636	0.132231
16	44	0.363636	0.132231
4	11	0.363636	0.132231
8	22	0.363636	0.132231
15	41	0.365854	0.133849
11	30	0.366667	0.134444
18	49	0.367347	0.134944
14	38	0.368421	0.135734
7	19	0.368421	0.135734
17	46	0.369565	0.136578
10	27	0.370370	0.137174
13	35	0.371429	0.137959
16	43	0.372093	0.138453
12	32	0.375000	0.140625
15	40	0.375000	0.140625
18	48	0.375000	0.140625
3	8	0.375000	0.140625
6	16	0.375000	0.140625
9	24	0.375000	0.140625
17	45	0.377778	0.142716
14	37	0.378378	0.143170
11	29	0.379310	0.143876
16	42	0.380952	0.145125
8	21	0.380952	0.145125
13	34	0.382353	0.146194
18	47	0.382979	0.146673
10	26	0.384615	0.147929
15	39	0.384615	0.147929
5	13	0.384615	0.147929
17	44	0.386364	0.149277
12	31	0.387097	0.149844
19	49	0.387755	0.150354
14	36	0.388889	0.151235
7	18	0.388889	0.151235
16	41	0.390244	0.152290
18	46	0.391304	0.153119
9	23	0.391304	0.153119
11	28	0.392857	0.154337
13	33	0.393939	0.155188
15	38	0.394737	0.155817
17	43	0.395349	0.156301
19	48	0.395833	0.156684
10	25	0.400000	0.160000
12	30	0.400000	0.160000
14	35	0.400000	0.160000
16	40	0.400000	0.160000
18	45	0.400000	0.160000
2	5	0.400000	0.160000
4	10	0.400000	0.160000
6	15	0.400000	0.160000
8	20	0.400000	0.160000
19	47	0.404255	0.163422
17	42	0.404762	0.163832
15	37	0.405405	0.164354
13	32	0.406250	0.165039
11	27	0.407407	0.165981
20	49	0.408163	0.166597
18	44	0.409091	0.167355
9	22	0.409091	0.167355
16	39	0.410256	0.168310
14	34	0.411765	0.169550
7	17	0.411765	0.169550
19	46	0.413043	0.170605
12	29	0.413793	0.171225
17	41	0.414634	0.171921
10	24	0.416667	0.173611
15	36	0.416667	0.173611
20	48	0.416667	0.173611
5	12	0.416667	0.173611
18	43	0.418605	0.175230
13	31	0.419355	0.175858
16	38	0.421053	0.177285
8	19	0.421053	0.177285
19	45	0.422222	0.178272
11	26	0.423077	0.178994
14	33	0.424242	0.179982
17	40	0.425000	0.180625
20	47	0.425532	0.181077
12	28	0.428571	0.183673
15	35	0.428571	0.183673
18	42	0.428571	0.183673
21	49	0.428571	0.183673
3	7	0.428571	0.183673
6	14	0.428571	0.183673
9	21	0.428571	0.183673
19	44	0.431818	0.186467
16	37	0.432432	0.186998
13	30	0.433333	0.187778
10	23	0.434783	0.189036
20	46	0.434783	0.189036
17	39	0.435897	0.190007
14	32	0.437500	0.191406
21	48	0.437500	0.191406
7	16	0.437500	0.191406
18	41	0.439024	0.192742
11	25	0.440000	0.193600
15	34	0.441176	0.194637
19	43	0.441860	0.195241
12	27	0.444444	0.197531
16	36	0.444444	0.197531
20	45	0.444444	0.197531
4	9	0.444444	0.197531
8	18	0.444444	0.197531
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19	22	0.863636	0.745868
38	44	0.863636	0.745868
32	37	0.864865	0.747991
13	15	0.866667	0.751111
26	30	0.866667	0.751111
39	45	0.866667	0.751111
33	38	0.868421	0.754155
20	23	0.869565	0.756144
40	46	0.869565	0.756144
27	31	0.870968	0.758585
34	39	0.871795	0.760026
41	47	0.872340	0.760978
14	16	0.875000	0.765625
21	24	0.875000	0.765625
28	32	0.875000	0.765625
35	40	0.875000	0.765625
42	48	0.875000	0.765625
7	8	0.875000	0.765625
43	49	0.877551	0.770096
36	41	0.878049	0.770970
29	33	0.878788	0.772268
22	25	0.880000	0.774400
37	42	0.880952	0.776077
15	17	0.882353	0.778547
30	34	0.882353	0.778547
38	43	0.883721	0.780963
23	26	0.884615	0.782544
31	35	0.885714	0.784490
39	44	0.886364	0.785640
16	18	0.888889	0.790123
24	27	0.888889	0.790123
32	36	0.888889	0.790123
40	45	0.888889	0.790123
8	9	0.888889	0.790123
41	46	0.891304	0.794423
33	37	0.891892	0.795471
25	28	0.892857	0.797194
42	47	0.893617	0.798551
17	19	0.894737	0.800554
34	38	0.894737	0.800554
43	48	0.895833	0.802517
26	29	0.896552	0.803805
35	39	0.897436	0.805391
44	49	0.897959	0.806331
18	20	0.900000	0.810000
27	30	0.900000	0.810000
36	40	0.900000	0.810000
9	10	0.900000	0.810000
37	41	0.902439	0.814396
28	31	0.903226	0.815817
19	21	0.904762	0.818594
38	42	0.904762	0.818594
29	32	0.906250	0.821289
39	43	0.906977	0.822607
10	11	0.909091	0.826446
20	22	0.909091	0.826446
30	33	0.909091	0.826446
40	44	0.909091	0.826446
41	45	0.911111	0.830123
31	34	0.911765	0.831315
21	23	0.913043	0.833648
42	46	0.913043	0.833648
32	35	0.914286	0.835918
43	47	0.914894	0.837030
11	12	0.916667	0.840278
22	24	0.916667	0.840278
33	36	0.916667	0.840278
44	48	0.916667	0.840278
45	49	0.918367	0.843399
34	37	0.918919	0.844412
23	25	0.920000	0.846400
35	38	0.921053	0.848338
12	13	0.923077	0.852071
24	26	0.923077	0.852071
36	39	0.923077	0.852071
37	40	0.925000	0.855625
25	27	0.925926	0.857339
38	41	0.926829	0.859012
13	14	0.928571	0.862245
26	28	0.928571	0.862245
39	42	0.928571	0.862245
40	43	0.930233	0.865333
27	29	0.931035	0.866825
41	44	0.931818	0.868285
14	15	0.933333	0.871111
28	30	0.933333	0.871111
42	45	0.933333	0.871111
43	46	0.934783	0.873819
29	31	0.935484	0.875130
44	47	0.936170	0.876415
15	16	0.937500	0.878906
30	32	0.937500	0.878906
45	48	0.937500	0.878906
46	49	0.938776	0.881299
31	33	0.939394	0.882461
16	17	0.941176	0.885813
32	34	0.941176	0.885813
33	35	0.942857	0.888980
17	18	0.944444	0.891975
34	36	0.944444	0.891975
35	37	0.945946	0.894814
18	19	0.947368	0.897507
36	38	0.947368	0.897507
37	39	0.948718	0.900066
19	20	0.950000	0.902500
38	40	0.950000	0.902500
39	41	0.951219	0.904819
20	21	0.952381	0.907029
40	42	0.952381	0.907029
41	43	0.953488	0.909140
21	22	0.954545	0.911157
42	44	0.954545	0.911157
43	45	0.955556	0.913086
22	23	0.956522	0.914934
44	46	0.956522	0.914934
45	47	0.957447	0.916704
23	24	0.958333	0.918403
46	48	0.958333	0.918403
47	49	0.959184	0.920033
24	25	0.960000	0.921600
25	26	0.961538	0.924556
26	27	0.962963	0.927298
27	28	0.964286	0.929847
28	29	0.965517	0.932224
29	30	0.966667	0.934444
30	31	0.967742	0.936524
31	32	0.968750	0.938477
32	33	0.969697	0.940312
33	34	0.970588	0.942042
34	35	0.971429	0.943673
35	36	0.972222	0.945216
36	37	0.972973	0.946676
37	38	0.973684	0.948061
38	39	0.974359	0.949375
39	40	0.975000	0.950625
40	41	0.975610	0.951814
41	42	0.976190	0.952948
42	43	0.976744	0.954029
43	44	0.977273	0.955062
44	45	0.977778	0.956049
45	46	0.978261	0.956994
46	47	0.978723	0.957900
47	48	0.979167	0.958767
48	49	0.979592	0.959600
10	10	1.000000	1.000000
11	11	1.000000	1.000000
12	12	1.000000	1.000000
13	13	1.000000	1.000000
14	14	1.000000	1.000000
15	15	1.000000	1.000000
16	16	1.000000	1.000000
17	17	1.000000	1.000000
18	18	1.000000	1.000000
19	19	1.000000	1.000000
2	2	1.000000	1.000000
20	20	1.000000	1.000000
21	21	1.000000	1.000000
22	22	1.000000	1.000000
23	23	1.000000	1.000000
24	24	1.000000	1.000000
25	25	1.000000	1.000000
26	26	1.000000	1.000000
27	27	1.000000	1.000000
28	28	1.000000	1.000000
29	29	1.000000	1.000000
3	3	1.000000	1.000000
30	30	1.000000	1.000000
31	31	1.000000	1.000000
32	32	1.000000	1.000000
33	33	1.000000	1.000000
34	34	1.000000	1.000000
35	35	1.000000	1.000000
36	36	1.000000	1.000000
37	37	1.000000	1.000000
38	38	1.000000	1.000000
39	39	1.000000	1.000000
4	4	1.000000	1.000000
40	40	1.000000	1.000000
41	41	1.000000	1.000000
42	42	1.000000	1.000000
43	43	1.000000	1.000000
44	44	1.000000	1.000000
45	45	1.000000	1.000000
46	46	1.000000	1.000000
47	47	1.000000	1.000000
48	48	1.000000	1.000000
49	49	1.000000	1.000000
5	5	1.000000	1.000000
6	6	1.000000	1.000000
7	7	1.000000	1.000000
8	8	1.000000	1.000000
9	9	1.000000	1.000000
49	48	1.020833	1.042101
48	47	1.021277	1.043006
47	46	1.021739	1.043951
46	45	1.022222	1.044938
45	44	1.022727	1.045971
44	43	1.023256	1.047053
43	42	1.023810	1.048186
42	41	1.024390	1.049375
41	40	1.025000	1.050625
40	39	1.025641	1.051939
39	38	1.026316	1.053324
38	37	1.027027	1.054785
37	36	1.027778	1.056327
36	35	1.028571	1.057959
35	34	1.029412	1.059689
34	33	1.030303	1.061524
33	32	1.031250	1.063477
32	31	1.032258	1.065557
31	30	1.033333	1.067778
30	29	1.034483	1.070155
29	28	1.035714	1.072704
28	27	1.037037	1.075446
27	26	1.038462	1.078402
26	25	1.040000	1.081600
25	24	1.041667	1.085069
49	47	1.042553	1.086917
24	23	1.043478	1.088847
48	46	1.043478	1.088847
47	45	1.044444	1.090864
23	22	1.045455	1.092975
46	44	1.045455	1.092975
45	43	1.046512	1.095187
22	21	1.047619	1.097506
44	42	1.047619	1.097506
43	41	1.048780	1.099941
21	20	1.050000	1.102500
42	40	1.050000	1.102500
41	39	1.051282	1.105194
20	19	1.052632	1.108033
40	38	1.052632	1.108033
39	37	1.054054	1.111030
19	18	1.055556	1.114197
38	36	1.055556	1.114197
37	35	1.057143	1.117551
18	17	1.058824	1.121107
36	34	1.058824	1.121107
35	33	1.060606	1.124885
17	16	1.062500	1.128906
34	32	1.062500	1.128906
33	31	1.064516	1.133195
49	46	1.065217	1.134688
16	15	1.066667	1.137778
32	30	1.066667	1.137778
48	45	1.066667	1.137778
47	44	1.068182	1.141012
31	29	1.068966	1.142687
46	43	1.069767	1.144402
15	14	1.071429	1.147959
30	28	1.071429	1.147959
45	42	1.071429	1.147959
44	41	1.073171	1.151695
29	27	1.074074	1.153635
43	40	1.075000	1.155625
14	13	1.076923	1.159763
28	26	1.076923	1.159763
42	39	1.076923	1.159763
41	38	1.078947	1.164127
27	25	1.080000	1.166400
40	37	1.081081	1.168736
13	12	1.083333	1.173611
26	24	1.083333	1.173611
39	36	1.083333	1.173611
38	35	1.085714	1.178776
25	23	1.086957	1.181474
37	34	1.088235	1.184256
49	45	1.088889	1.185679
12	11	1.090909	1.190083
24	22	1.090909	1.190083
36	33	1.090909	1.190083
48	44	1.090909	1.190083
47	43	1.093023	1.194700
35	32	1.093750	1.196289
23	21	1.095238	1.199546
46	42	1.095238	1.199546
34	31	1.096774	1.202914
45	41	1.097561	1.204640
11	10	1.100000	1.210000
22	20	1.100000	1.210000
33	30	1.100000	1.210000
44	40	1.100000	1.210000
43	39	1.102564	1.215648
32	29	1.103448	1.217598
21	19	1.105263	1.221607
42	38	1.105263	1.221607
31	28	1.107143	1.225765
41	37	1.108108	1.227904
10	9	1.111111	1.234568
20	18	1.111111	1.234568
30	27	1.111111	1.234568
40	36	1.111111	1.234568
49	44	1.113636	1.240186
39	35	1.114286	1.241633
29	26	1.115385	1.244083
48	43	1.116279	1.246079
19	17	1.117647	1.249135
38	34	1.117647	1.249135
47	42	1.119048	1.252268
28	25	1.120000	1.254400
37	33	1.121212	1.257117
46	41	1.121951	1.258775
18	16	1.125000	1.265625
27	24	1.125000	1.265625
36	32	1.125000	1.265625
45	40	1.125000	1.265625
9	8	1.125000	1.265625
44	39	1.128205	1.272847
35	31	1.129032	1.274714
26	23	1.130435	1.277883
43	38	1.131579	1.280471
17	15	1.133333	1.284444
34	30	1.133333	1.284444
42	37	1.135135	1.288532
25	22	1.136364	1.291322
33	29	1.137931	1.294887
41	36	1.138889	1.297068
49	43	1.139535	1.298540
16	14	1.142857	1.306122
24	21	1.142857	1.306122
32	28	1.142857	1.306122
40	35	1.142857	1.306122
48	42	1.142857	1.306122
8	7	1.142857	1.306122
47	41	1.146341	1.314099
39	34	1.147059	1.315744
31	27	1.148148	1.318244
23	20	1.150000	1.322500
46	40	1.150000	1.322500
38	33	1.151515	1.325987
15	13	1.153846	1.331361
30	26	1.153846	1.331361
45	39	1.153846	1.331361
37	32	1.156250	1.336914
22	19	1.157895	1.340720
44	38	1.157895	1.340720
29	25	1.160000	1.345600
36	31	1.161290	1.348595
43	37	1.162162	1.350621
14	12	1.166667	1.361111
21	18	1.166667	1.361111
28	24	1.166667	1.361111
35	30	1.166667	1.361111
42	36	1.166667	1.361111
49	42	1.166667	1.361111
7	6	1.166667	1.361111
48	41	1.170732	1.370613
41	35	1.171429	1.372245
34	29	1.172414	1.374554
27	23	1.173913	1.378072
47	40	1.175000	1.380625
20	17	1.176471	1.384083
40	34	1.176471	1.384083
33	28	1.178571	1.389031
46	39	1.179487	1.391190
13	11	1.181818	1.396694
26	22	1.181818	1.396694
39	33	1.181818	1.396694
45	38	1.184211	1.402355
32	27	1.185185	1.404664
19	16	1.187500	1.410156
38	32	1.187500	1.410156
44	37	1.189189	1.414171
25	21	1.190476	1.417234
31	26	1.192308	1.421598
37	31	1.193548	1.424558
43	36	1.194444	1.426697
49	41	1.195122	1.428316
12	10	1.200000	1.440000
18	15	1.200000	1.440000
24	20	1.200000	1.440000
30	25	1.200000	1.440000
36	30	1.200000	1.440000
42	35	1.200000	1.440000
48	40	1.200000	1.440000
6	5	1.200000	1.440000
47	39	1.205128	1.452334
41	34	1.205882	1.454152
35	29	1.206897	1.456599
29	24	1.208333	1.460069
23	19	1.210526	1.465374
46	38	1.210526	1.465374
40	33	1.212121	1.469238
17	14	1.214286	1.474490
34	28	1.214286	1.474490
45	37	1.216216	1.479182
28	23	1.217391	1.482042
39	32	1.218750	1.485352
11	9	1.222222	1.493827
22	18	1.222222	1.493827
33	27	1.222222	1.493827
44	36	1.222222	1.493827
49	40	1.225000	1.500625
38	31	1.225806	1.502602
27	22	1.227273	1.506198
43	35	1.228571	1.509388
16	13	1.230769	1.514793
32	26	1.230769	1.514793
48	39	1.230769	1.514793
37	30	1.233333	1.521111
21	17	1.235294	1.525952
42	34	1.235294	1.525952
47	38	1.236842	1.529778
26	21	1.238095	1.532880
31	25	1.240000	1.537600
36	29	1.241379	1.541023
41	33	1.242424	1.543618
46	37	1.243243	1.545654
10	8	1.250000	1.562500
15	12	1.250000	1.562500
20	16	1.250000	1.562500
25	20	1.250000	1.562500
30	24	1.250000	1.562500
35	28	1.250000	1.562500
40	32	1.250000	1.562500
45	36	1.250000	1.562500
5	4	1.250000	1.562500
49	39	1.256410	1.578567
44	35	1.257143	1.580408
39	31	1.258065	1.582726
34	27	1.259259	1.585734
29	23	1.260870	1.589792
24	19	1.263158	1.595568
48	38	1.263158	1.595568
43	34	1.264706	1.599481
19	15	1.266667	1.604445
38	30	1.266667	1.604445
33	26	1.269231	1.610947
47	37	1.270270	1.613587
14	11	1.272727	1.619835
28	22	1.272727	1.619835
42	33	1.272727	1.619835
37	29	1.275862	1.627824
23	18	1.277778	1.632716
46	36	1.277778	1.632716
32	25	1.280000	1.638400
41	32	1.281250	1.641602
18	14	1.285714	1.653061
27	21	1.285714	1.653061
36	28	1.285714	1.653061
45	35	1.285714	1.653061
9	7	1.285714	1.653061
49	38	1.289474	1.662742
40	31	1.290323	1.664932
31	24	1.291667	1.668403
22	17	1.294118	1.674740
44	34	1.294118	1.674740
35	27	1.296296	1.680384
48	37	1.297297	1.682980
13	10	1.300000	1.690000
26	20	1.300000	1.690000
39	30	1.300000	1.690000
43	33	1.303030	1.697888
30	23	1.304348	1.701323
47	36	1.305556	1.704475
17	13	1.307692	1.710059
34	26	1.307692	1.710059
38	29	1.310345	1.717004
21	16	1.312500	1.722656
42	32	1.312500	1.722656
46	35	1.314286	1.727347
25	19	1.315789	1.731302
29	22	1.318182	1.737603
33	25	1.320000	1.742400
37	28	1.321429	1.746174
41	31	1.322581	1.749220
45	34	1.323529	1.751730
49	37	1.324324	1.753835
12	9	1.333333	1.777778
16	12	1.333333	1.777778
20	15	1.333333	1.777778
24	18	1.333333	1.777778
28	21	1.333333	1.777778
32	24	1.333333	1.777778
36	27	1.333333	1.777778
4	3	1.333333	1.777778
40	30	1.333333	1.777778
44	33	1.333333	1.777778
48	36	1.333333	1.777778
8	6	1.333333	1.777778
47	35	1.342857	1.803265
43	32	1.343750	1.805664
39	29	1.344828	1.808561
35	26	1.346154	1.812130
31	23	1.347826	1.816635
27	20	1.350000	1.822500
23	17	1.352941	1.830450
46	34	1.352941	1.830450
42	31	1.354839	1.835588
19	14	1.357143	1.841837
38	28	1.357143	1.841837
34	25	1.360000	1.849600
49	36	1.361111	1.852623
15	11	1.363636	1.859504
30	22	1.363636	1.859504
45	33	1.363636	1.859504
41	30	1.366667	1.867778
26	19	1.368421	1.872576
37	27	1.370370	1.877915
48	35	1.371429	1.880816
11	8	1.375000	1.890625
22	16	1.375000	1.890625
33	24	1.375000	1.890625
44	32	1.375000	1.890625
40	29	1.379310	1.902497
29	21	1.380952	1.907030
47	34	1.382353	1.910900
18	13	1.384615	1.917160
36	26	1.384615	1.917160
43	31	1.387097	1.924037
25	18	1.388889	1.929012
32	23	1.391304	1.935728
39	28	1.392857	1.940051
46	33	1.393939	1.943067
14	10	1.400000	1.960000
21	15	1.400000	1.960000
28	20	1.400000	1.960000
35	25	1.400000	1.960000
42	30	1.400000	1.960000
49	35	1.400000	1.960000
7	5	1.400000	1.960000
45	32	1.406250	1.977539
38	27	1.407407	1.980796
31	22	1.409091	1.985537
24	17	1.411765	1.993080
48	34	1.411765	1.993080
41	29	1.413793	1.998811
17	12	1.416667	2.006944
34	24	1.416667	2.006944
44	31	1.419355	2.014568
27	19	1.421053	2.019391
37	26	1.423077	2.025148
47	33	1.424242	2.028466
10	7	1.428571	2.040816
20	14	1.428571	2.040816
30	21	1.428571	2.040816
40	28	1.428571	2.040816
43	30	1.433333	2.054445
33	23	1.434783	2.058601
23	16	1.437500	2.066406
46	32	1.437500	2.066406
36	25	1.440000	2.073600
49	34	1.441176	2.076990
13	9	1.444444	2.086420
26	18	1.444444	2.086420
39	27	1.444444	2.086420
42	29	1.448276	2.097503
29	20	1.450000	2.102500
45	31	1.451613	2.107180
16	11	1.454545	2.115702
32	22	1.454545	2.115702
48	33	1.454545	2.115702
35	24	1.458333	2.126736
19	13	1.461538	2.136095
38	26	1.461538	2.136095
41	28	1.464286	2.144133
22	15	1.466667	2.151111
44	30	1.466667	2.151111
47	32	1.468750	2.157227
25	17	1.470588	2.162630
28	19	1.473684	2.171745
31	21	1.476190	2.179138
34	23	1.478261	2.185255
37	25	1.480000	2.190400
40	27	1.481481	2.194787
43	29	1.482759	2.198573
46	31	1.483871	2.201873
49	33	1.484848	2.204775
12	8	1.500000	2.250000
15	10	1.500000	2.250000
18	12	1.500000	2.250000
21	14	1.500000	2.250000
24	16	1.500000	2.250000
27	18	1.500000	2.250000
3	2	1.500000	2.250000
30	20	1.500000	2.250000
33	22	1.500000	2.250000
36	24	1.500000	2.250000
39	26	1.500000	2.250000
42	28	1.500000	2.250000
45	30	1.500000	2.250000
48	32	1.500000	2.250000
6	4	1.500000	2.250000
9	6	1.500000	2.250000
47	31	1.516129	2.298647
44	29	1.517241	2.302022
41	27	1.518519	2.305898
38	25	1.520000	2.310400
35	23	1.521739	2.315690
32	21	1.523810	2.321995
29	19	1.526316	2.329640
26	17	1.529412	2.339100
49	32	1.531250	2.344727
23	15	1.533333	2.351111
46	30	1.533333	2.351111
43	28	1.535714	2.358418
20	13	1.538462	2.366864
40	26	1.538462	2.366864
37	24	1.541667	2.376736
17	11	1.545455	2.388430
34	22	1.545455	2.388430
48	31	1.548387	2.397503
31	20	1.550000	2.402500
45	29	1.551724	2.407848
14	9	1.555556	2.419753
28	18	1.555556	2.419753
42	27	1.555556	2.419753
39	25	1.560000	2.433600
25	16	1.562500	2.441406
36	23	1.565217	2.449905
47	30	1.566667	2.454444
11	7	1.571429	2.469388
22	14	1.571429	2.469388
33	21	1.571429	2.469388
44	28	1.571429	2.469388
41	26	1.576923	2.486686
30	19	1.578947	2.493075
49	31	1.580645	2.498439
19	12	1.583333	2.506944
38	24	1.583333	2.506944
46	29	1.586207	2.516052
27	17	1.588235	2.522491
35	22	1.590909	2.530992
43	27	1.592593	2.536351
16	10	1.600000	2.560000
24	15	1.600000	2.560000
32	20	1.600000	2.560000
40	25	1.600000	2.560000
48	30	1.600000	2.560000
8	5	1.600000	2.560000
45	28	1.607143	2.582908
37	23	1.608696	2.587902
29	18	1.611111	2.595679
21	13	1.615385	2.609468
42	26	1.615385	2.609468
34	21	1.619048	2.621315
47	29	1.620690	2.626635
13	8	1.625000	2.640625
26	16	1.625000	2.640625
39	24	1.625000	2.640625
44	27	1.629630	2.655693
31	19	1.631579	2.662050
49	30	1.633333	2.667778
18	11	1.636364	2.677686
36	22	1.636364	2.677686
41	25	1.640000	2.689600
23	14	1.642857	2.698980
46	28	1.642857	2.698980
28	17	1.647059	2.712803
33	20	1.650000	2.722500
38	23	1.652174	2.729679
43	26	1.653846	2.735207
48	29	1.655172	2.739596
10	6	1.666667	2.777778
15	9	1.666667	2.777778
20	12	1.666667	2.777778
25	15	1.666667	2.777778
30	18	1.666667	2.777778
35	21	1.666667	2.777778
40	24	1.666667	2.777778
45	27	1.666667	2.777778
5	3	1.666667	2.777778
47	28	1.678571	2.817602
42	25	1.680000	2.822400
37	22	1.681818	2.828512
32	19	1.684211	2.836565
27	16	1.687500	2.847656
49	29	1.689655	2.854935
22	13	1.692308	2.863905
44	26	1.692308	2.863905
39	23	1.695652	2.875236
17	10	1.700000	2.890000
34	20	1.700000	2.890000
46	27	1.703704	2.902606
29	17	1.705882	2.910035
41	24	1.708333	2.918403
12	7	1.714286	2.938776
24	14	1.714286	2.938776
36	21	1.714286	2.938776
48	28	1.714286	2.938776
43	25	1.720000	2.958400
31	18	1.722222	2.966049
19	11	1.727273	2.983471
38	22	1.727273	2.983471
45	26	1.730769	2.995562
26	15	1.733333	3.004444
33	19	1.736842	3.016620
40	23	1.739130	3.024575
47	27	1.740741	3.030178
14	8	1.750000	3.062500
21	12	1.750000	3.062500
28	16	1.750000	3.062500
35	20	1.750000	3.062500
42	24	1.750000	3.062500
49	28	1.750000	3.062500
7	4	1.750000	3.062500
44	25	1.760000	3.097600
37	21	1.761905	3.104308
30	17	1.764706	3.114187
23	13	1.769231	3.130177
46	26	1.769231	3.130177
39	22	1.772727	3.142562
16	9	1.777778	3.160494
32	18	1.777778	3.160494
48	27	1.777778	3.160494
41	23	1.782609	3.177694
25	14	1.785714	3.188776
34	19	1.789474	3.202216
43	24	1.791667	3.210069
18	10	1.800000	3.240000
27	15	1.800000	3.240000
36	20	1.800000	3.240000
45	25	1.800000	3.240000
9	5	1.800000	3.240000
47	26	1.807692	3.267751
38	21	1.809524	3.274376
29	16	1.812500	3.285156
49	27	1.814815	3.293553
20	11	1.818182	3.305785
40	22	1.818182	3.305785
31	17	1.823529	3.325259
42	23	1.826087	3.334594
11	6	1.833333	3.361111
22	12	1.833333	3.361111
33	18	1.833333	3.361111
44	24	1.833333	3.361111
46	25	1.840000	3.385600
35	19	1.842105	3.393352
24	13	1.846154	3.408284
48	26	1.846154	3.408284
37	20	1.850000	3.422500
13	7	1.857143	3.448980
26	14	1.857143	3.448980
39	21	1.857143	3.448980
41	22	1.863636	3.473140
28	15	1.866667	3.484444
43	23	1.869565	3.495274
15	8	1.875000	3.515625
30	16	1.875000	3.515625
45	24	1.875000	3.515625
47	25	1.880000	3.534400
32	17	1.882353	3.543253
49	26	1.884615	3.551775
17	9	1.888889	3.567901
34	18	1.888889	3.567901
36	19	1.894737	3.590028
19	10	1.900000	3.610000
38	20	1.900000	3.610000
40	21	1.904762	3.628118
21	11	1.909091	3.644628
42	22	1.909091	3.644628
44	23	1.913043	3.659735
23	12	1.916667	3.673611
46	24	1.916667	3.673611
48	25	1.920000	3.686400
25	13	1.923077	3.698225
27	14	1.928571	3.719388
29	15	1.933333	3.737778
31	16	1.937500	3.753906
33	17	1.941176	3.768166
35	18	1.944444	3.780864
37	19	1.947368	3.792244
39	20	1.950000	3.802500
41	21	1.952381	3.811791
43	22	1.954545	3.820248
45	23	1.956522	3.827977
47	24	1.958333	3.835069
49	25	1.960000	3.841600
10	5	2.000000	4.000000
12	6	2.000000	4.000000
14	7	2.000000	4.000000
16	8	2.000000	4.000000
18	9	2.000000	4.000000
20	10	2.000000	4.000000
22	11	2.000000	4.000000
24	12	2.000000	4.000000
26	13	2.000000	4.000000
28	14	2.000000	4.000000
30	15	2.000000	4.000000
32	16	2.000000	4.000000
34	17	2.000000	4.000000
36	18	2.000000	4.000000
38	19	2.000000	4.000000
4	2	2.000000	4.000000
40	20	2.000000	4.000000
42	21	2.000000	4.000000
44	22	2.000000	4.000000
46	23	2.000000	4.000000
48	24	2.000000	4.000000
6	3	2.000000	4.000000
8	4	2.000000	4.000000
49	24	2.041667	4.168403
47	23	2.043478	4.175803
45	22	2.045455	4.183884
43	21	2.047619	4.192744
41	20	2.050000	4.202500
39	19	2.052632	4.213296
37	18	2.055556	4.225308
35	17	2.058824	4.238754
33	16	2.062500	4.253906
31	15	2.066667	4.271111
29	14	2.071429	4.290816
27	13	2.076923	4.313610
25	12	2.083333	4.340278
48	23	2.086957	4.355388
23	11	2.090909	4.371901
46	22	2.090909	4.371901
44	21	2.095238	4.390023
21	10	2.100000	4.410000
42	20	2.100000	4.410000
40	19	2.105263	4.432133
19	9	2.111111	4.456790
38	18	2.111111	4.456790
36	17	2.117647	4.484429
17	8	2.125000	4.515625
34	16	2.125000	4.515625
49	23	2.130435	4.538753
32	15	2.133333	4.551111
47	22	2.136364	4.564050
15	7	2.142857	4.591837
30	14	2.142857	4.591837
45	21	2.142857	4.591837
43	20	2.150000	4.622500
28	13	2.153846	4.639053
41	19	2.157895	4.656510
13	6	2.166667	4.694445
26	12	2.166667	4.694445
39	18	2.166667	4.694445
37	17	2.176471	4.737024
24	11	2.181818	4.760331
48	22	2.181818	4.760331
35	16	2.187500	4.785156
46	21	2.190476	4.798186
11	5	2.200000	4.840000
22	10	2.200000	4.840000
33	15	2.200000	4.840000
44	20	2.200000	4.840000
42	19	2.210526	4.886426
31	14	2.214286	4.903061
20	9	2.222222	4.938272
40	18	2.222222	4.938272
49	22	2.227273	4.960744
29	13	2.230769	4.976331
38	17	2.235294	4.996540
47	21	2.238095	5.009070
18	8	2.250000	5.062500
27	12	2.250000	5.062500
36	16	2.250000	5.062500
45	20	2.250000	5.062500
9	4	2.250000	5.062500
43	19	2.263158	5.121884
34	15	2.266667	5.137778
25	11	2.272727	5.165289
41	18	2.277778	5.188272
16	7	2.285714	5.224490
32	14	2.285714	5.224490
48	21	2.285714	5.224490
39	17	2.294118	5.262976
23	10	2.300000	5.290000
46	20	2.300000	5.290000
30	13	2.307692	5.325444
37	16	2.312500	5.347656
44	19	2.315789	5.362881
14	6	2.333333	5.444445
21	9	2.333333	5.444445
28	12	2.333333	5.444445
35	15	2.333333	5.444445
42	18	2.333333	5.444445
49	21	2.333333	5.444445
7	3	2.333333	5.444445
47	20	2.350000	5.522500
40	17	2.352941	5.536332
33	14	2.357143	5.556122
26	11	2.363636	5.586777
45	19	2.368421	5.609418
19	8	2.375000	5.640625
38	16	2.375000	5.640625
31	13	2.384615	5.686390
43	18	2.388889	5.706790
12	5	2.400000	5.760000
24	10	2.400000	5.760000
36	15	2.400000	5.760000
48	20	2.400000	5.760000
41	17	2.411765	5.816609
29	12	2.416667	5.840278
46	19	2.421053	5.861496
17	7	2.428571	5.897959
34	14	2.428571	5.897959
39	16	2.437500	5.941406
22	9	2.444444	5.975308
44	18	2.444444	5.975308
49	20	2.450000	6.002500
27	11	2.454545	6.024794
32	13	2.461539	6.059172
37	15	2.466667	6.084445
42	17	2.470588	6.103806
47	19	2.473684	6.119113
10	4	2.500000	6.250000
15	6	2.500000	6.250000
20	8	2.500000	6.250000
25	10	2.500000	6.250000
30	12	2.500000	6.250000
35	14	2.500000	6.250000
40	16	2.500000	6.250000
45	18	2.500000	6.250000
5	2	2.500000	6.250000
48	19	2.526316	6.382271
43	17	2.529412	6.397924
38	15	2.533333	6.417778
33	13	2.538461	6.443787
28	11	2.545455	6.479339
23	9	2.555556	6.530864
46	18	2.555556	6.530864
41	16	2.562500	6.566406
18	7	2.571429	6.612245
36	14	2.571429	6.612245
49	19	2.578947	6.650970
31	12	2.583333	6.673611
44	17	2.588235	6.698962
13	5	2.600000	6.760000
26	10	2.600000	6.760000
39	15	2.600000	6.760000
47	18	2.611111	6.817901
34	13	2.615385	6.840237
21	8	2.625000	6.890625
42	16	2.625000	6.890625
29	11	2.636364	6.950413
37	14	2.642857	6.984694
45	17	2.647059	7.006920
16	6	2.666667	7.111111
24	9	2.666667	7.111111
32	12	2.666667	7.111111
40	15	2.666667	7.111111
48	18	2.666667	7.111111
8	3	2.666667	7.111111
43	16	2.687500	7.222656
35	13	2.692308	7.248521
27	10	2.700000	7.290000
46	17	2.705882	7.321799
19	7	2.714286	7.367347
38	14	2.714286	7.367347
49	18	2.722222	7.410494
30	11	2.727273	7.438016
41	15	2.733333	7.471111
11	4	2.750000	7.562500
22	8	2.750000	7.562500
33	12	2.750000	7.562500
44	16	2.750000	7.562500
47	17	2.764706	7.643599
36	13	2.769231	7.668639
25	9	2.777778	7.716049
39	14	2.785714	7.760204
14	5	2.800000	7.840000
28	10	2.800000	7.840000
42	15	2.800000	7.840000
45	16	2.812500	7.910156
31	11	2.818182	7.942149
48	17	2.823529	7.972318
17	6	2.833333	8.027778
34	12	2.833333	8.027778
37	13	2.846154	8.100592
20	7	2.857143	8.163265
40	14	2.857143	8.163265
43	15	2.866667	8.217778
23	8	2.875000	8.265625
46	16	2.875000	8.265625
49	17	2.882353	8.307959
26	9	2.888889	8.345679
29	10	2.900000	8.410000
32	11	2.909091	8.462810
35	12	2.916667	8.506945
38	13	2.923077	8.544378
41	14	2.928571	8.576530
44	15	2.933333	8.604445
47	16	2.937500	8.628906
12	4	3.000000	9.000000
15	5	3.000000	9.000000
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21	7	3.000000	9.000000
24	8	3.000000	9.000000
27	9	3.000000	9.000000
30	10	3.000000	9.000000
33	11	3.000000	9.000000
36	12	3.000000	9.000000
39	13	3.000000	9.000000
42	14	3.000000	9.000000
45	15	3.000000	9.000000
48	16	3.000000	9.000000
6	2	3.000000	9.000000
9	3	3.000000	9.000000
49	16	3.062500	9.378906
46	15	3.066667	9.404445
43	14	3.071429	9.433674
40	13	3.076923	9.467456
37	12	3.083333	9.506945
34	11	3.090909	9.553719
31	10	3.100000	9.610000
28	9	3.111111	9.679012
25	8	3.125000	9.765625
47	15	3.133333	9.817778
22	7	3.142857	9.877551
44	14	3.142857	9.877551
41	13	3.153846	9.946746
19	6	3.166667	10.027778
38	12	3.166667	10.027778
35	11	3.181818	10.123967
16	5	3.200000	10.240000
32	10	3.200000	10.240000
48	15	3.200000	10.240000
45	14	3.214286	10.331633
29	9	3.222222	10.382716
42	13	3.230769	10.437870
13	4	3.250000	10.562500
26	8	3.250000	10.562500
39	12	3.250000	10.562500
49	15	3.266667	10.671111
36	11	3.272727	10.710744
23	7	3.285714	10.795918
46	14	3.285714	10.795918
33	10	3.300000	10.890000
43	13	3.307692	10.940828
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20	6	3.333333	11.111111
30	9	3.333333	11.111111
40	12	3.333333	11.111111
47	14	3.357143	11.270409
37	11	3.363636	11.314050
27	8	3.375000	11.390625
44	13	3.384615	11.455622
17	5	3.400000	11.560000
34	10	3.400000	11.560000
41	12	3.416667	11.673611
24	7	3.428571	11.755102
48	14	3.428571	11.755102
31	9	3.444444	11.864198
38	11	3.454545	11.933885
45	13	3.461539	11.982248
14	4	3.500000	12.250000
21	6	3.500000	12.250000
28	8	3.500000	12.250000
35	10	3.500000	12.250000
42	12	3.500000	12.250000
49	14	3.500000	12.250000
7	2	3.500000	12.250000
46	13	3.538461	12.520710
39	11	3.545455	12.570248
32	9	3.555556	12.641975
25	7	3.571429	12.755102
43	12	3.583333	12.840278
18	5	3.600000	12.960000
36	10	3.600000	12.960000
47	13	3.615385	13.071006
29	8	3.625000	13.140625
40	11	3.636364	13.223141
11	3	3.666667	13.444445
22	6	3.666667	13.444445
33	9	3.666667	13.444445
44	12	3.666667	13.444445
48	13	3.692308	13.633136
37	10	3.700000	13.690000
26	7	3.714286	13.795918
41	11	3.727273	13.892562
15	4	3.750000	14.062500
30	8	3.750000	14.062500
45	12	3.750000	14.062500
49	13	3.769231	14.207101
34	9	3.777778	14.271605
19	5	3.800000	14.440000
38	10	3.800000	14.440000
42	11	3.818182	14.578512
23	6	3.833333	14.694445
46	12	3.833333	14.694445
27	7	3.857143	14.877551
31	8	3.875000	15.015625
35	9	3.888889	15.123457
39	10	3.900000	15.210000
43	11	3.909091	15.280992
47	12	3.916667	15.340278
12	3	4.000000	16.000000
16	4	4.000000	16.000000
20	5	4.000000	16.000000
24	6	4.000000	16.000000
28	7	4.000000	16.000000
32	8	4.000000	16.000000
36	9	4.000000	16.000000
40	10	4.000000	16.000000
44	11	4.000000	16.000000
48	12	4.000000	16.000000
8	2	4.000000	16.000000
49	12	4.083333	16.673611
45	11	4.090909	16.735537
41	10	4.100000	16.809999
37	9	4.111111	16.901234
33	8	4.125000	17.015625
29	7	4.142857	17.163265
25	6	4.166667	17.361111
46	11	4.181818	17.487604
21	5	4.200000	17.639999
42	10	4.200000	17.639999
38	9	4.222222	17.827160
17	4	4.250000	18.062500
34	8	4.250000	18.062500
47	11	4.272727	18.256199
30	7	4.285714	18.367348
43	10	4.300000	18.490000
13	3	4.333333	18.777779
26	6	4.333333	18.777779
39	9	4.333333	18.777779
48	11	4.363636	19.041323
35	8	4.375000	19.140625
22	5	4.400000	19.360001
44	10	4.400000	19.360001
31	7	4.428571	19.612246
40	9	4.444445	19.753086
49	11	4.454545	19.842976
18	4	4.500000	20.250000
27	6	4.500000	20.250000
36	8	4.500000	20.250000
45	10	4.500000	20.250000
9	2	4.500000	20.250000
41	9	4.555555	20.753086
32	7	4.571429	20.897959
23	5	4.600000	21.160000
46	10	4.600000	21.160000
37	8	4.625000	21.390625
14	3	4.666667	21.777779
28	6	4.666667	21.777779
42	9	4.666667	21.777779
47	10	4.700000	22.090000
33	7	4.714286	22.224489
19	4	4.750000	22.562500
38	8	4.750000	22.562500
43	9	4.777778	22.827160
24	5	4.800000	23.040001
48	10	4.800000	23.040001
29	6	4.833333	23.361111
34	7	4.857143	23.591837
39	8	4.875000	23.765625
44	9	4.888889	23.901234
49	10	4.900000	24.010000
10	2	5.000000	25.000000
15	3	5.000000	25.000000
20	4	5.000000	25.000000
25	5	5.000000	25.000000
30	6	5.000000	25.000000
35	7	5.000000	25.000000
40	8	5.000000	25.000000
45	9	5.000000	25.000000
46	9	5.111111	26.123457
41	8	5.125000	26.265625
36	7	5.142857	26.448980
31	6	5.166667	26.694445
26	5	5.200000	27.040001
47	9	5.222222	27.271605
21	4	5.250000	27.562500
42	8	5.250000	27.562500
37	7	5.285714	27.938776
16	3	5.333333	28.444445
32	6	5.333333	28.444445
48	9	5.333333	28.444445
43	8	5.375000	28.890625
27	5	5.400000	29.160000
38	7	5.428571	29.469387
49	9	5.444445	29.641975
11	2	5.500000	30.250000
22	4	5.500000	30.250000
33	6	5.500000	30.250000
44	8	5.500000	30.250000
39	7	5.571429	31.040817
28	5	5.600000	31.360001
45	8	5.625000	31.640625
17	3	5.666667	32.111111
34	6	5.666667	32.111111
40	7	5.714286	32.653061
23	4	5.750000	33.062500
46	8	5.750000	33.062500
29	5	5.800000	33.639999
35	6	5.833333	34.027779
41	7	5.857143	34.306122
47	8	5.875000	34.515625
12	2	6.000000	36.000000
18	3	6.000000	36.000000
24	4	6.000000	36.000000
30	5	6.000000	36.000000
36	6	6.000000	36.000000
42	7	6.000000	36.000000
48	8	6.000000	36.000000
49	8	6.125000	37.515625
43	7	6.142857	37.734695
37	6	6.166667	38.027779
31	5	6.200000	38.439999
25	4	6.250000	39.062500
44	7	6.285714	39.510204
19	3	6.333333	40.111111
38	6	6.333333	40.111111
32	5	6.400000	40.959999
45	7	6.428571	41.326530
13	2	6.500000	42.250000
26	4	6.500000	42.250000
39	6	6.500000	42.250000
46	7	6.571429	43.183674
33	5	6.600000	43.560001
20	3	6.666667	44.444443
40	6	6.666667	44.444443
47	7	6.714286	45.081635
27	4	6.750000	45.562500
34	5	6.800000	46.240002
41	6	6.833333	46.694443
48	7	6.857143	47.020409
14	2	7.000000	49.000000
21	3	7.000000	49.000000
28	4	7.000000	49.000000
35	5	7.000000	49.000000
42	6	7.000000	49.000000
49	7	7.000000	49.000000
43	6	7.166667	51.361111
36	5	7.200000	51.840000
29	4	7.250000	52.562500
22	3	7.333333	53.777779
44	6	7.333333	53.777779
37	5	7.400000	54.759998
15	2	7.500000	56.250000
30	4	7.500000	56.250000
45	6	7.500000	56.250000
38	5	7.600000	57.759998
23	3	7.666667	58.777779
46	6	7.666667	58.777779
31	4	7.750000	60.062500
39	5	7.800000	60.840000
47	6	7.833333	61.361111
16	2	8.000000	64.000000
24	3	8.000000	64.000000
32	4	8.000000	64.000000
40	5	8.000000	64.000000
48	6	8.000000	64.000000
49	6	8.166667	66.694443
41	5	8.200000	67.239998
33	4	8.250000	68.062500
25	3	8.333333	69.444443
42	5	8.400000	70.559998
17	2	8.500000	72.250000
34	4	8.500000	72.250000
43	5	8.600000	73.959999
26	3	8.666667	75.111115
35	4	8.750000	76.562500
44	5	8.800000	77.440002
18	2	9.000000	81.000000
27	3	9.000000	81.000000
36	4	9.000000	81.000000
45	5	9.000000	81.000000
46	5	9.200000	84.639999
37	4	9.250000	85.562500
28	3	9.333333	87.111115
47	5	9.400000	88.360001
19	2	9.500000	90.250000
38	4	9.500000	90.250000
48	5	9.600000	92.160004
29	3	9.666667	93.444443
39	4	9.750000	95.062500
49	5	9.800000	96.040001
20	2	10.000000	100.000000
30	3	10.000000	100.000000
40	4	10.000000	100.000000
41	4	10.250000	105.062500
31	3	10.333333	106.777779
21	2	10.500000	110.250000
42	4	10.500000	110.250000
32	3	10.666667	113.777779
43	4	10.750000	115.562500
22	2	11.000000	121.000000
33	3	11.000000	121.000000
44	4	11.000000	121.000000
45	4	11.250000	126.562500
34	3	11.333333	128.444443
23	2	11.500000	132.250000
46	4	11.500000	132.250000
35	3	11.666667	136.111115
47	4	11.750000	138.062500
24	2	12.000000	144.000000
36	3	12.000000	144.000000
48	4	12.000000	144.000000
49	4	12.250000	150.062500
37	3	12.333333	152.111115
25	2	12.500000	156.250000
38	3	12.666667	160.444443
26	2	13.000000	169.000000
39	3	13.000000	169.000000
40	3	13.333333	177.777771
27	2	13.500000	182.250000
41	3	13.666667	186.777771
28	2	14.000000	196.000000
42	3	14.000000	196.000000
43	3	14.333333	205.444443
29	2	14.500000	210.250000
44	3	14.666667	215.111115
30	2	15.000000	225.000000
45	3	15.000000	225.000000
46	3	15.333333	235.111115
31	2	15.500000	240.250000
47	3	15.666667	245.444443
32	2	16.000000	256.000000
48	3	16.000000	256.000000
49	3	16.333334	266.777771
33	2	16.500000	272.250000
34	2	17.000000	289.000000
35	2	17.500000	306.250000
36	2	18.000000	324.000000
37	2	18.500000	342.250000
38	2	19.000000	361.000000
39	2	19.500000	380.250000
40	2	20.000000	400.000000
41	2	20.500000	420.250000
42	2	21.000000	441.000000
43	2	21.500000	462.250000
44	2	22.000000	484.000000
45	2	22.500000	506.250000
46	2	23.000000	529.000000
47	2	23.500000	552.250000
48	2	24.000000	576.000000
49	2	24.500000	600.250000

And the little C program I wrote to figure that out. There's also a tarball (782 bytes)

/*
    calculating turns ratios for toroid baluns
    
  I need to match a 600 ohm antenna (t2fd) to 75 ohm line (rg6), so I need 8:1
  (different t2fds have different impedances)
  
  impedance ratio is the square of turns ratio
  
  preferably sticking to integer numbers of turns
  
  closest are:
  48      17      2.823529        7.972318
  17      6       2.833333        8.027778  <-- looks best
  34      12      2.833333        8.027778 
  
*/

#include <stdio.h>
#include <stdlib.h>
#include <math.h>

int t1min = 2;   // winding 1 min turns
int t1max = 50;  // winding 1 max turns
int t2min = 2;   // winding 2 min turns
int t2max = 50;  // winding 2 max turns
FILE *opf;

void calc(void) {
  int i,j;
  float tr, ir;
  for (i=t1min; i<t1max; i++) {
    for (j=t2min; j<t2max; j++) {
      tr = (1.0 * i)/(1.0 * j);
      ir = ((1.0 * i)/(1.0 * j)) * ((1.0 * i)/(1.0 * j));
      fprintf(opf,"%i\t%i\t%f\t%f\n",i,j,tr,ir);
    }
  }  // for i
}

int main(void) {
  opf = fopen("turns.txt","w");
  if (opf != NULL) {
    calc();
    fclose(opf);
    printf("done\n");
  } else {
    printf("file create error\n");
  }
  return 0;
}

This is why I'm bothering, this plot below. This goes from 1.8 to 56 MHz and the SWR stays below 3.2. The antenna is 46 feet long and 18 inches wide. These are a popular all-purpose antenna, especially with the military. For transmitting use some percentage of power goes into the terminating resistor.

And if you scale it up by a factor of 2 so it might stand a chance of working at lower frequencies it looks like this:

Again slightly off topic but I've had fairly good luck winding baluns on toroids that came out of old switching power supplies. Specifically computer power supplies, and some of them are like an inch or more across so there's room to put some wire in there. Chokes for antenna feedlines too, although last time I used RG-59 in one there was only room for 3 turns. Your mileage may vary of course. If you can think of a good way to check it at the frequency you'll use it at by all means do it, but how to check it will depend on the equipment you've got, and there's more than one way.

I wrote and asked Joel Hallas, W1ZR ("The Doctor is In" at QST) about used toroids. My thinking was that testing them with an antenna analyzer as though they were an extension of the feedline should work. I didn't ask permission to put these responses on the web, but here they are anyway.

My original question:
I've read somewhere that junk box toroids are of little use because the characteristics can vary so much.  Does having an antenna/SWR analyzer change this?  Shouldn't it be possible to wind up a simple 1:1 unun and use the coax loss function to see if the core works at the intended frequency?
I've got an MFJ-269 analyzer and I've been wondering this.  I took a toroid out of an old AT computer power supply, unwound it, then put back 2 of the original windings at 9 turns each.  I checked one winding with the other open and got an indicated loss of 0.4 db at 3.5 mhz and 0.9 db at 59 mhz.  I measured the inductance at 3.89 mhz as 9.901 uh.  I'm not sure how much of this I can believe.  I was wondering if I could wind RG-58 into it to make a choke balun but after checking this way it looks like I could make a regular balun out of it.  I'm only running 100 watts.  Seems odd there's no mention of this in MFJ's manual.

Toroids aren't very expensive, but with Amidon having a $15 minimum order and $6.50 minimum shipping, one random 75 cent core for a project can cost a lot.  I've got a dozen or so old computer power supplies from dumpster-diving.

Joel Hallas' response:
Interesting question. You can use your antenna analyzer to get a good idea of the common mode impedance of turns on a toroid by using an antenna analyzer that can measure the actual complex impedance at a frequency.  Just hook the leads to the wire, or more usually the coax shield, at each end of the wrap.  If you are in the inductive range, the reactance should increase with increasing frequency.

Another measurement you can make is to determine the relative permittivity of the material. make a coil of some number of turns about the same size as would be on the core. Measure the reactance at a frequency and calculate the inductance. Put the same number of turns on the core and repeat the measurement and calculation. The ratio of the inductances is the relative permittivity of the core and you can use that to determine the inductance of some other number of turns for future ref.

Since the reactance of an inductor is directly proportional to the inductance -- the relative permittivity is also just the ratio of the impedances.

I'm copying Sr ARRL Lab Engineer Zack Lau in case he has any other thoughts. GL & 73, Joel
Joel R. Hallas, W1ZR
Technical Editor, QST

Zack Lau's response:
No, junk box toroids are perfectly serviceable for most hams.  But, if you write stuff up for publication, it is generally a good idea to stick to readily available parts, as some readers don't like authors who use "unobtainium".   The major issue is that there is no standard color code.

An important issue with baluns is the loss of the material. This can reduce the maximum impedance to perhaps 1kohm. Good enough for well matched 50 ohm antennas, but not center fed dipoles used on all bands.  Also,  ferrite is a poor conductor of heat, so high power ratings typically seen are for best conditions, not worst conditions.  All the kW ferrite baluns I've seen will burn up if connected to a 100 watt rig and a good transmatch (but no antenna). No way can they get rid of 90+ watts of heat.  Fortunately, the SWR drift as they heat up is a noticeable warning to most hams.

I'm not sure I phrased my question right. Not quite the answer I was hoping for, and it didn't make it into QST.

AB1JX / calcs