Crystal Radio Capacitors and ESR

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In my ongoing effort to understand crystal radios and the theory that makes them work I have recently began looking into all things capacitance. The tuning capacitor is, next the coil itself, the principal component of the set. Since few builders fabricate their own tuning caps I suppose one takes them a bit for granted. I have been reading on the radio forums some threads concerning radio Q and the subject of the variable cap Q seems to be getting more attention. So, what is the deal?

I set about my study by gathering what I could find on the web concerning capacitor Q. The result was, not a whole lot, at least not for air-variable capacitors. The exception comes from Ben Tongue (not surprisingly) and I wish to extend special thanks to him for providing, as usual, essential measurements and background in his Article 24. The plots below show capacitor Q versus capacitance, and versus frequency for a small number of air-variable capacitors that I have in the box. Note that I made all my measurements with a calibrated inductor coil (236 uH at 1 Mhz) so the capacitance and frequency are related to each other by the resonance equation: f = 1 / 2pi sqrt(LC).

These are simple plots with log Q versus log capacitance in pF and versus log frequency in MHz. On each plot, a bit messy I admit, I show data for eight capacitors along with the Q versus f equation. You immediately see that for each capacitor the relation follows a power function and in all cases the function is good to R-squared at or above 99%. Additionally, I include the same data for two capacitors presented in Ben Tongue's excellent summary. It is also immediately obvious that not all caps are created equal!

I have named the caps in the following manner, the highest tested RF capacitance, materials of interest (A = aluminum plates, B = brass plates; C = ceramic insulators, P = phenolic insulators); finally something to do with the origin. Photos of the caps at the bottom of this page.

Identification of the caps as follows (from highest to lowest Q):

BT CapA		Ben Tongue's CapA is the best "Holy Grail" of the lot
383 AnC E	A Russian (I presume) eBay cap from Estonia
453 AnC Rus	A Russian cap frequently offered on eBay
512 BnC Am	A brass straignt-line frequency cap AMSOO
397 AnC E	Another Russian (I presume) eBay cap from Estonia, larger capacitance
356 BnC		Another Lovely straight-line frequency brass cap
354 InC M	Chinese caps (A and B) offered from Ming on eBay
404 AnC xss	A typical Crystal Set Society cap
479 SnM		Switched set of individual Silver Mica caps, not an air-variable

The following table summarizes the Q versus Capacitance (in pF) for each tested cap, arranged from highest to lowest quality.

Cap		Q vs C in pF		R2
BT A250		Q = 5683.3* C^0.1949	0.9340
390 AnC E	Q = 278.24* C^0.6372	0.7950
453 AnC Rus	Q = 45.665* C^0.8775	0.9968
512 BnC Am	Q = 47.095* C^0.8326	0.9928
397 AnC E	Q = 31.498* C^0.903	0.9867
356 BnC		Q = 9.6513* C^1.0455	0.9986
403 AnC EB	Q = 7.4831* C^1.0957	0.9949
337 AnC EB	Q = 6.6011* C^1.1043	0.9791
383 AnC EA	Q = 8.9089* C^1.0323	0.9921
354 InC MA	Q = 9.9566* C^0.9891	0.9859
365 InC MB	Q = 8.5243* C^1.0028	0.9953
404 AnP xss	Q = 10.364* C^0.9118	0.9940
Any discussion of component quality needs first to acknowledge that the fight to increase the component Q is in reality a fight to eliminate (not possible) or reduce as much as feasible the resistive losses associated with that component. It helps then to look not just at Q, but at how Q relates to the component series resistance Rs or ESR. There are two fundamental equations that describe this relation: 1) Q = cotan theta = 1/D. D is the dissipation factor and is an attribute of the dielectric material used in the construction of the capacitor. D = tan theta = ESR / Xc = 2pi f C * ESR. Solving for ESR we get: 2) ESR = D / 2pi f C. This equation tells us that capacitor losses expected due to resistance are dependant on both angular frequency and capacitance. It is important to remember that not all Q's are created equal. What we want is to keep the ESR to a minimum.

The plot below summarizes the same data only this time plotting log Q versus log Rs (ESR). Here most of the data sets fall into a narrow zone running from High-Q / low ESR in the upper left to Low-Q / high ESR in the lower right. Ben Tongue's Holy Grail cap sits above the pack with an overall narrower range of ESR from max to min capacitance. No surprises, good caps are low-loss. I have also plotted lines of Q versus ESR at 1MHz for threee capacitance values, 10, 100, and 1000 pF. These lines indicate that Q is highest at high values of capacitance independant of individual capacitor loss. Note that all the caps measured straddle the 100 pF line. For typical MW coils with about 230 uH or so, 100 pF resonates around 1MHz.

From the datasets above it can be seen that high-Q caps are high-Q across the band, but Q varies and is highest at high capacitance, (lower frequency) and vice versa. All sets show a direct power-law relation between Q and f or ESR regardless of overall quality or type of capacitor. The data table from Ben Tongue (below) was found in his Article 24 and represent precision measuremnts by Bill Hebbert.

     pF	  	f	      Q
cap	tank	MHz	Cap A	Cap B
355	375	 520	19000	4750
170	190	 730	14000	3230
 94	114	 943	14400	1610
 81	101	1000	13148	1828	(my estimate)
 40	 60	1300	11600	 870
 15	 35	1710	 9800	 460

for I = 250 uH and 20pF stray in tank
Some notes on my results. It should be immediately clear that I have yet to snag an actual "holy grail" cap such as the one evaluated by Ben Tongue. Ben's good CapA has ceramic insulated stators and silver-plated brass plates with silver-plated wiper contacts. Essential features of a good cap. Inexpensive caps often have lossy phenolic insulators, something easily noticeable before you buy, look for this. I have noticed that on some caps the stator is held by ceramic "buttons" rather than a large ceramic plate and I have to wonder how effective this is. I would say that ceramic insulation is far more available that silver plating on the plates and wipers.

At the bottom end of the quality profile is the poor but readily available XSS variable cap (404 AnP). This has aluminum plates, phenolic insulation and a single aluminum wiper. An OK basic cap, but one easily beat if you are willing to make an effort. Next up in terms of quality was a big disappointment for me. These are two Chinese caps I recently purchased on eBay (354 and 365 InC). They are large and heavy units with big ceramic insulators and I figured this ought to be good. Both caps have decidedly lower Q that my vintage brass straight-line frequency caps. I do not think these will find their way into any radios here.

In the mid-Q range I find one brass cap (512 BnC) and two Russian caps (453 AnC, and 397 AnC). These all have ceramic insulation but no silver plating. The brass cap insulators are a black material that I call ceramic as it does not appear phenolic, but frankly I am unsure what it is. The cap is pretty good though. The two Russian caps are commonly available on eBay and are of two different styles. One somewhat large with thick "gun metal" type casing and the other a small delicate unit with a plastic dust-cover. Both are pretty good caps.

Finally and best cap that I have in my box is another Russian cap (383 AnC). This cap is clearly an unused NOS unit with clean shiny parts and button ceramic insulators. This is a slightly smaller version of one of the previous caps (397 AnC). The moderate Q version was also clean and NOS, so I do not expect all such caps to be automatically good Q. It is a pity as I have seen additional of these small caps on eBay, though clearly in used condition so I suspect they will have lower Q.

From all the above I would conclude that caps having ceramic insulation alone is not nirvana. Caps need also to be clean and ultimately need silver plating to truly aspire to "holy grail" status. A good reference worth looking at is a thread on The Radio Board: Improving 'Q' of capacitors.
The post shows how merely taking an older cap and cleaning it up (no small task) and replacing the insulators with HDPE sheet (high-density polyethylene) can take an original Q = 664 (1MHz) and raise it to Q = 12248 (1MHz). This value is on par with Ben Tongue's "holy grail" cap (Q = 13148 @ 1MHz) and either it was a dim fine cap to start with or something not fully kosher with the measurements. Still, the point is that improvements can be significant.


Images of the tested caps, same scale.

Note: A summary look at the cap Q taken (calculated) at 1 MHz

383 AnC E	5441 Q @ 1Mhz
512 BnC Am	2995
453 AnC Rus	2742
397 AnC E	2132
356 BnC		1280
383 AnC EA	1102
403 AnC EB	1245
337 AnC EB	1149
354 InC M	1015
479 SnM		1177
404 AnP xss	 733
498 AnC a	 622
498 AnC b	 487
A word of caution, that really great Estonia cap with the high Q seems kinda unique. I measured two additional similar caps (383 AnC EA, and 403/337 AnC EB (two gangs)) thinking I would get similar great results. What I got in fact were two mediocre results. If you plan to run out and buy these type of cap on ebay, donít say I didn't warn you!

kjs (updated 6/2016)


References:

AVX Basin Capacitor Formulas.
http://www.avx.com/docs/Catalogs/cbasic.pdf

Ben Tongue 1999, Sensitivity and selectivity issues in crystal radio sets including diode problems; measurements of the Q of variable and fixed capacitors, RF loss in slide switches and loss tangent of various dielectrics.
http://www.bentongue.com/xtalset/24Cmnts/24Cmnts.html

Wikipedia, Types of Capacitors, Ohmic Losses, Dissipation Factor and Q.
http://en.wikipedia.org/wiki/Types_of_capacitor#Ohmic_losses.2C_ESR.2C_dissipation_factor.2C_and_quality_factor

Dr. Gary L. Johnson, 2001. Lossy Capacitors
http://www.eece.ksu.edu/~gjohnson/tcchap3.pdf


A short note concerning my Q measurements as compared to those of Bill Hebbert and presented by Ben Tongue in his Article 24. By some chance, both Ben's work as well as my own include a commonly available 365 pF capacitor from the XSS and elsewhere. While his cap and my own are not the same unit, their properties ought to be similar enough. Therefore it is worth to take a look ar the two datasets together.

In the plot below of Q vs Capacitance I show Ben's three measurement sets as well as my own measuremnts on the same type cap. Ben's A250 (holy grail) is included just for show. His measurements on CapB were made with two separate inductors, one with 250 uH and one with 180 uH. My standard inductor has 225 uH. On the plot I use large markers for Ben's 250 uH (B250 - dark green) and my 225 uH (404AnP - red) data. It is clear that my Q measurements are systematically lower than his.

At this time I am uncertain as to whether my setup delivers lower Q measurements or if the partictular cap I measured was a bit more lossy than Ben's. This is a bit of ambiguity in my presentation that the reader ought to be aware of. Maybe my caps are better than presented? I do not know the answer but I include this plot just so you know.

A note concerning Loss Tangent

When looking into capacitor losses one often runs into the term "loss tangent". For many our eyes just glaze over and we read on understanding only that it has something interesting to do with theory. In the interest of knowing the code I give a brief explanation.

With a perfect lossless capacitor in AC operation (sine wave) current leads the voltage by 90 degrees. Where losses occur the current will lead by slightly less than this. If you imagine these losses as a resistance in series with a perfect capacitor, then this is called Effective Series Resistance, ESR and is valid in reference to a specific frequency: ESR = D / 2pi * f * C.

The lag angle between current and voltage is generally a fraction of a degree and is expressed as a positive value. The tangent of this lag angle is the "loss tangent" and is referred to more commonly as the Dissipation factor D. For example, polystyrene has a D of about 0.0001 for a lag angle of 0.006 degrees. Good dielectrics have very small loss tangents. Why do we express loss as D or loss tangent? Well, it related to the quality factor Q such that Q = 1/D.

Modeled Data from plots shown above:

		
f	L	C	Qtank	Qcoil	Qcap	Rp	ESR	D		Qvsf			R≤
MHz	uH	pF				Mohm	ohm				
Brass n Ceramic 						356 BnC		Q = 1280.1 * f^-2.271	0.9991
1.878	267	27	142	258	315	0.99	10.03	0.003179			
1.461	252	47	174	256	541	1.25	4.27	0.001849			
1.183	241	75	190	245	835	1.50	2.14	0.001197			
1.013	234	105	198	235	1240	1.85	1.20	0.000806			
0.875	229	144	199	225	1735	2.19	0.726	0.000576			
0.769	225	190	198	216	2358	2.56	0.461	0.000424			
0.655	221	268	192	205	3238	2.94	0.280	0.000309			
0.572	217	356	188	196	4713	3.69	0.166	0.000212			
												
Aluminum n Ceramic Estonia A 					383 AnC E	Q = 5441 * f^-1.401	0.9901
2.587	295	13	196	223	1609	7.70	2.98	0.000622			
2.116	277	20	218	251	1671	6.15	2.20	0.000598			
1.460	251	47	237	256	3197	7.38	0.72	0.000313			
1.066	236	94	227	238	4676	7.40	0.34	0.000214			
0.827	227	163	215	221	7701	9.09	0.153	0.000130			
0.761	225	195	209	215	8135	8.74	0.132	0.000123			
0.636	220	285	199	203	10574	9.29	0.083	0.000095			
0.552	217	383	190	194	11912	8.96	0.063	0.000084			
												
Aluminum n Phenolic XSS						404 AnP xss	Q = 732.75 * f^-1.98	0.9922
2.019	273	23	101	255	169	0.58	20.52	0.005929			
1.176	241	76	169	245	547	0.97	3.25	0.001827			
0.957	232	119	181	231	845	1.18	1.65	0.001184			
0.783	226	183	186	217	1286	1.43	0.86	0.000778			
0.703	222	230	185	209	1604	1.58	0.613	0.000624			
0.632	220	288	182	202	1790	1.56	0.488	0.000559			
0.573	218	354	180	196	2149	1.68	0.365	0.000465			
0.539	216	404	176	192	2185	1.60	0.335	0.000458			
												
Aluminum n Ceramic Russia 3-gang 				453 AnC Rus	Q = 2742 * f^-1.911	0.9966
2.248	282	18	173	245	588	2.34	6.76	0.001699			
1.661	259	35	206	259	1007	2.72	2.69	0.000993			
1.191	241	74	218	246	1933	3.49	0.93	0.000517			
0.899	230	136	212	227	3329	4.33	0.39	0.000300			
0.748	224	202	206	214	5435	5.73	0.194	0.000184			
0.642	220	279	197	203	6451	5.73	0.138	0.000155			
0.574	218	354	191	196	7358	5.77	0.107	0.000136			
0.510	215	453	185	189	9783	6.74	0.070	0.000102			
												
Brass n Ceramic Amsoo						512 BnC Am	Q = 2294.7 * f^-1.806	0.9915
2.057	274	22	179	253	607	2.15	5.84	0.001647			
1.763	263	31	195	259	787	2.29	3.70	0.001270			
1.318	246	59	212	251	1356	2.76	1.50	0.000737			
1.021	235	104	214	236	2332	3.51	0.65	0.000429			
0.776	225	187	206	216	4371	4.80	0.251	0.000229			
0.669	221	255	198	206	4866	4.53	0.191	0.000206			
0.564	217	367	189	195	6005	4.62	0.128	0.000167			
0.481	214	512	181	185	7834	5.07	0.083	0.000128			
												
High-stability Invar n Ceramic Ming A				354 InC MA	Q = 1014.5 * f^-2.145	0.9871
21.769	263	31	142	259	315	0.92	9.29	0.003173
1.475	252	46	164	256	455	1.06	5.14	0.002199
1.252	244	66	178	248	631	1.21	3.04	0.001585
0.980	233	113	184	233	880	1.26	1.63	0.001136
0.845	228	156	192	222	1389	1.68	0.871	0.000720
0.724	223	216	192	211	2050	2.08	0.496	0.000488
0.637	220	284	190	203	3161	2.78	0.278	0.000316
0.562	217	369	184	195	3351	2.57	0.229	0.000298
			
Aluminum n Ceramic Estonia B					397 AnC E	Q = 2132 * f^-1.976	0.9875
2.313	284	17	152	242	411	1.70	10.06	0.002436
1.940	270	25	177	257	573	1.88	5.74	0.001745
1.239	243	68	213	248	1499	2.84	1.26	0.000667
0.978	233	114	210	233	2168	3.10	0.66	0.000461
0.847	228	155	208	222	3125	3.79	0.388	0.000320
0.733	224	211	199	212	3105	3.20	0.332	0.000322
0.598	218	324	192	199	5687	4.67	0.144	0.000176
0.543	216	397	188	193	8446	6.24	0.087	0.000118
										
High-stability Invar n Ceramic Ming B				365 InC MB	Q = 927.01 * f^-2.170	0.9960
1.697	261	34	140	259	306	0.85	9.08	0.003269
1.434	250	49	162	255	441	0.99	5.12	0.002270
1.231	243	69	169	247	535	1.01	3.51	0.001868
1.045	236	98	184	237	824	1.28	1.88	0.001213
0.887	230	140	191	226	1244	1.59	1.029	0.000804
0.762	225	194	191	215	1685	1.81	0.639	0.000594
0.636	220	284	186	203	2338	2.06	0.376	0.000428
0.565	217	365	184	195	3386	2.61	0.228	0.000295

Aluminum n Ceramic Estonia					383 AnC EA	Q = 1102.1 * f^-2.266	0.9938
0.558	217	375	186	194	4652	3.54	0.16	0.000215			
0.608	219	313	189	200	3382	2.83	0.25	0.000296			
0.714	223	223	194	210	2472	2.47	0.40	0.000404			
0.879	229	143	190	225	1240	1.57	1.02	0.000807			
1.032	235	101	191	236	988	1.51	1.543	0.001012			
1.256	244	66	181	249	666	1.28	2.887	0.001502			
1.976	271	24	118	256	220	0.74	15.338	0.004554			
2.469	290	14	95	232	160	0.72	28.201	0.006268			

Aluminum n Ceramic Estonia					403 AnC EB	Q = 1245 * f ^-2.405	0.9962
2.490	291	14	85	230	135	0.61	33.69	0.007404			
2.136	277	20	118	251	223	0.83	16.65	0.004474			
1.734	262	32	148	259	346	0.99	8.24	0.002887			
1.314	246	60	177	251	605	1.23	3.35	0.001652			
1.012	234	106	193	235	1078	1.61	1.383	0.000928			
0.848	228	154	197	222	1716	2.08	0.708	0.000583			
0.624	219	297	192	201	3979	3.42	0.216	0.000251			
0.549	217	388	187	193	5828	4.36	0.128	0.000172			

Aluminum n Ceramic Estonia					337 AnC EB	Q = 1149 * f ^-2.429	0.9810
2.395	287	15	94	237	156	0.68	27.62	0.006390			
2.039	274	22	111	254	197	0.69	17.83	0.005084			
1.525	254	43	155	257	390	0.95	6.24	0.002565			
1.200	242	73	184	246	730	1.33	2.49	0.001369			
1.013	234	105	190	235	1005	1.50	1.484	0.000995			
0.865	229	148	198	224	1742	2.16	0.714	0.000574			
0.703	222	231	190	209	2052	2.02	0.479	0.000487			
0.598	218	325	191	199	5355	4.39	0.153	0.000187			



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Sunday, 09-Dec-2018 13:45:19 CST

Comments and/or Correspondance to: kevin smith


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