AD2CC: 500 Watt Antenna Tuner Part 4 - Capacitors and Inductors

Following K6JCAs design approach, I will be using a low pass L-C network topology. I will also be using relays to put capacitors in parallel to build the desired capacitor value and inductors in series for the same purpose. These are the necessary tasks:

Select the value of the capacitors and inductors in the design
Select capacitors that meet the design requirements
Design and test the inductors that meet the design requirements.

Capacitor and Inductor Values

By running the simulator, I got the maximum values of capacitors and inductors for each band to meet the design requirements. The values of importance to us are the values at the low end of the band (the higher value - and I work a lot of CW). Since we are not designing for the 160 meter band, we will skip those values and start with 2,728 pF as the maximum capacitance and 6,821 nH as the maximum inductance that needs to be synthesized.

Band	C (pF)	L (nH)
160m low	5,305	13,263
160m high	4,775	11,937
80m low	2,728	6,821
80m high	2,387	5,968
40m low	1,364	3,410
40m high	1,308	3,270
20m low	682	1,705
20m high	665	1,664
17m low	529	1,321
17m high	526	1,314
15m low	455	1,137
15m high	445	1,113
12m low	384	959
12m high	382	955
10m low	341	853
10m high	322	804
6m low	191	477
6m high	177	442

To build a "power of 2" set of values, we will divide the maximum value eight times and then sum the resulting 8 values to see how close we can get to the target value.

C (pf)	L (nH)
2,728	6,821

1,364	3,411
682	1,705
341	853
171	426
85	213
43	107
21	53
11	27
Sum	Sum
2,718	6,795

The sum of 6,795 is close enough to 6,821 and the sum of 2,718 close enough to 2,728 to be a good starting point. Since I am not going to design for the 160 meter band, I will drop the data for this band from here on.

Maximum Voltages and Currents

Let's look at the maximum voltages and currents (from the simulator) that the inductors and capacitors must handle:

	200 W	500 W	800 W
Vpk L	475	750	950
Vrms L	336	530	672
Irms L	6.2	9.9	12.5
Vpk C	442	697	883
Vrms C	313	493	624
Irms C	6.9	10.8	13.7

We know that the 800 volts inductor peak voltages occur on voice peaks, but when a relay is open, we have to select relays that don't break down at these levels. The same is true of relays in the capacitor legs of the circuit.

Capacitor Selection

For capacitor selection, I will follow the outline of this paper https://www.avx.com/docs/techinfo/RFMicrowaveThinFilm/energytf.pdf

For capacitors, we have a number of limitations. One is the maximum voltage which needs to be at least 1,000 volts. The other is maximum energy stored. We can calculate the maximum energy stored as (using the data sheet specified maximum DC voltage for the largest value capacitor):

\begin{equation}E=\frac {1}{2}CV ^{2}=\frac {1}{2}(3411 \times 10 ^{-12})(1000 ^{2})=1.7 \times 10 ^{-3}\end{equation}

The voltage across the capacitor is a modulated sinusoidal function. For simplicity, I will just look at the carrier. In each half of the cycle, it stores energy in the capacitor and then removes it. So, we can calculate the energy stored in the capacitor as:

\begin{equation}E=\int _{0} ^{\pi}v(t)i(t)dt\end{equation}

And we have:

\begin{equation}v(t)=V\sin(\omega t)\end{equation}

\begin{equation}i(t)=C \frac{dv}{dt}=\omega CV \cos(\omega t)\end{equation}

\begin{equation}E=\int _{0} ^{\pi}\omega CV ^{2}\sin(\omega t)\cos(\omega t)dt\end{equation}

With change of variables:

\begin{equation}z=\omega t\ \ \ dz=\omega dt\ \ \ dt=\frac {dz}{\omega}\end{equation}

\begin{equation}E=\frac {\omega CV ^{2} }{\omega} \int _{0} ^{\pi \omega} \sin(z)\cos(z)dz=CV ^{2}\left [ \frac {\sin ^{2}(z)}{2} \right ] _{0} ^{\pi \omega}=\frac {CV ^{2}\sin ^{2}(\pi \omega)}{2}\end{equation}

Maximum power stored will be with the value of the sin function set at its maximum of 1 or:

\begin{equation}E=\frac {1}{2}CV ^{2}\end{equation}

So, the energy criteria becomes the same as voltage criteria, at least in this case.

Finally, we need to concern ourselves with heating effects. The maximum rating of reasonably priced capacitors is 125 degrees C. So, I will design for 100 degrees C. Assuming that the internal temperature of the housing will be around 40 degrees C, the temperature rise that can be permitted is 60 degrees C. Many high quality surface mount RF type capacitors come in 1111 packages. The thermal resistance of this package is 67.7 degrees per watt (see the above mentioned paper) so the power dissipation is limited to 0.88 watts. Since the maximum RMS current through the capacitors is 6.8 amps, the maximum capacitor ESR has to be 19 milli Ohms or less.

Peak Capacitor Current (A)

	Cap Voltage (V)	3,000 pF	1,410 pF	682 pF	340 pF	173 pF	86 pF	43 pF	22 pF	12 pF
160m L	427	9.27	6.38	3.30	1.64	0.84	0.42	0.21	0.11	0.06
160m H	442	9.49	6.85	3.75	1.89	0.96	0.48	0.24	0.12	0.07
80m L	442	-	9.02	6.10	3.29	1.68	0.84	0.42	0.21	0.12
80m H	442	-	9.36	6.72	3.74	1.92	0.95	0.48	0.24	0.13
40m L	441	-	-	8.98	6.08	3.35	1.67	0.83	0.43	0.23
40m H	441	-	-	9.10	6.27	3.48	1.74	0.87	0.45	0.24
20m L	441	-	-	8.89	8.96	6.16	3.33	1.67	0.85	0.47
20m H	441	-	-	-	9.00	6.27	3.41	1.71	0.88	0.48
17m L	441	-	-	-	9.47	7.32	4.23	2.15	1.10	0.60
17m H	441	-	-	-	9.53	7.36	4.26	2.17	1.11	0.60
15m L	441	-	-	-	9.71	7.95	4.84	2.50	1.28	0.70
15m H	441	-	-	-	9.76	8.07	4.94	2.56	1.31	0.71
12m L	441	-	-	-	9.53	8.62	5.59	2.96	1.52	0.83
12m H	441	-	-	-	9.57	8.65	5.62	2.98	1.52	0.83
10m L	442	-	-	-	8.86	9.03	6.12	3.33	1.71	0.93
10m H	441	-	-	-	-	9.10	6.39	3.52	1.81	0.99
6m L	441	-	-	-	-	9.55	8.58	5.61	3.05	1.65
6m H	439	-	-	-	-	9.10	8.81	5.97	3.28	1.79
Max	442	-	9.36	9.1	9.76	9.1	6.39	3.52	1.81	0.99

Searching through the Mouser catalog for bargain prices, most standard line of capacitors did not meet the requirements. By accident, I stumbled across the Vishay HiFreq series of capacitors and after I more carefully checked, these were the same capacitors that Jeff, K6JCA had used. Price and availability were also reasonable. The only parts that I found on Mouser meeting 1,000 or 1,500 volts DC specification were values up to 160 pF (DigiKey had none). These capacitors all come in 1111 surface mount package. Below is the ESR data for this family and package. ESR decreases with capacitance and frequency for values starting at 10 pF. But it is not well specified for capacitance values larger than 47 pF and frequencies lower than 100 MHz. ESR for higher value capacitors at lower frequencies will be lower and for lower value capacitors, the current is much lower (above table).

The current rating curve is also helpful. Current rating goes down with frequency but it goes up with capacitance. Below 30 MHz, it will be a bit of guesswork.

This is what I found based on availability (first number from the table above, second number or numbers are from the list of available parts with some additional data). Per the above table for capacitor currents, as capacitance and frequency decrease, so does the current handling of the capacitor. Fortunately, the current demand on the capacitor also decreases with the same two factors.

11 pF: 12 pF (max current at 10 meters, 0.7 Arms, goes down to 85 mArms at 80 meter)
21 pF: 22 pF (max current at 10 meters, 1.3 Arms, goes down to 170 mArms at 80 meter)
43 pF: 43 pF (2.5 Arms at 10 meters, goes down to 340 mArms at 80 meters)
85 pF: 2 x 43 pF = 86 pF (2.3 Arms per capacitor at 10 meter, 340 mArms per capacitor at 80 meter)
171 pF: 180 pF (6.4 A rms)
341 pF: 180 pF + 150 pF = 330 pF (3.8 Arms to 180 pF & 3.2 Arms to 150 PF)
682 pF: 3 x 180 pF + 150 pF = 690 pF (safe with four capacitors sharing 6.5 Arms)
1,364 pF: 7 x 180 pF + 150 pF = 1,410 pF (safe with eight capacitors sharing 6.6 Arms)

Inductor Design and Core Selection

Now on to inductors. I do my work on Macs and Mini Ring calculator only runs on Windows, so I had to figure something else out. Chapter One of the RF Circuit Design book by Chris Bowick came to the rescue (this book is available online as a pdf download). For air core solenoid inductors (to minimize losses), the formula is:

\begin{equation}L= \frac {0.394 r ^{2} N ^{2}}{9r+10l}\end{equation}

Where:

r = the coil radius in cm
l = the coil length in cm
L= the inductance in micro Henries

Subject to the condition that:

\begin{equation}l > 0.67r\end{equation}

Solving the above equation for N we have:

\begin{equation}N=\frac {1}{r}\sqrt {\frac {L(9r+10l)}{0.394}}\end{equation}

Here is the procedure that I used to arrive at the solution using a spreadsheet:

Set l = r and calculate turns from the above formula
Independently calculate coil length that might be different than l (I used a packing factor of 1.2 to allow for a bit of loose winding).
Use goal seek to set the number of turns using K6JCA's design as a rough objective
Replace the value of l with the computed value (rounded) of the coil length
Redo goal seek to recompute r for the same number of turns.
All the time, have a check cell testing for meeting the length - radius relationship stated above

The solution for the lower value inductors was straight forward. It took some experimentation with different number of turns to arrive at what I think is a reasonable radius, length and number of turns.

Wire diameter has a large impact on inductor geometry and it is driven by skin effect. This is how I approached it.

Also from the RF Circuit Design book, I have a table of magnet wire diameters coated and bare and resistance per thousand ft. With a little bit of arithmetic, I calculated the resistivity in Ohms-Meter.

The formula for skin depth in a copper conductor is:

\begin{equation}\delta = \frac {0.066}{\sqrt{f}}\end{equation}

And the resistance of the wire at frequency is:

\begin{equation}R=\frac {l}{\sigma (2\pi a \delta + \pi \delta ^{2})}=\frac {lr}{2\pi a \delta + \pi \delta ^{2}}\end{equation}

where:

l = length of wire in meters

r = resistivity (1/sigma) in Ohms - meter

sigma = conductivity in Siemens/meter

a = radius of the wire in meters

delta = skin depth in meters

Jeff, K6JCA has already done the work and decided that inductors up to 800 nH can be practically wound with air core. So, I will follow his approach.

Table below is the other half of the simulator output for the inductor voltages. There is an interplay between voltage and frequency as they impact current and skin effect for calculating power. So, I ended up building a spreadsheet table for inductors up to 800 nH to find the combinations that yield maximum power dissipation. It is after the inductor voltage table.

For wire gauge, I ran a number of different scenarios through the spreadsheet and eventually decided on 12 AWG magnet wire. In some cases, 14 AWG dissipated over 2 watts and I have no analytical way of knowing if that is a reasonable power level. Another reason is that K6JCA had used 12 AWG and that made it more of a safe decision.

Peak Inductor Voltage (V)

	Ind Current	6,400 nH	3,200 nH	1,600 nH	800 nH	400 nH	200 nH	100 nH	50 nH	25 nH
160m L	8.8	430	291	159	80	40	20	10	5	2
160m H	8.8	447	314	176	89	44	22	11	6	3
80m L	8.8	467	426	285	155	78	39	19	10	5
80m H	8.8	0	447	314	176	89	44	22	11	6
40m L	8.8	0	467	426	285	155	78	39	19	10
40m H	8.8	0	448	432	294	161	81	41	20	10
20m L	8.8	0	0	467	426	285	155	78	39	19
20m H	8.8	0	0	460	428	290	158	80	40	20
17m L	8.8	0	0	0	457	341	196	100	50	25
17m H	8.8	0	0	0	460	341	198	101	50	25
15m L	8.8	0	0	0	473	373	225	117	58	29
15m H	8.8	0	0	0	470	376	229	119	59	30
12m L	8.8	0	0	0	473	404	259	138	69	35
12m H	8.8	0	0	0	475	405	260	138	69	35
10m L	8.8	0	0	0	467	425	285	155	78	39
10m H	8.8	0	0	0	445	435	297	164	82	41
6m L	8.8	0	0	0	0	475	405	260	138	69
6m H	8.8	0	0	0	0	468	420	277	149	75
Max	8.8	467	467	467	475	435	297	164	82	41

And as it can be seen in the table below, the maximum power happens at different bands for each inductor.

Air Core Inductor Wire Loss (W)

Band	800 nH	400 nH	200 nH	100 nH	50 nH	25 nH
80m L	1.65	1.08	0.66	0.40	0.29	0.18
80m H	1.74	1.15	0.68	0.44	0.29	0.21
40m L	1.97	1.51	0.93	0.60	0.37	0.25
40m H	1.96	1.53	0.94	0.62	0.39	0.24
20m L	1.55	1.81	1.30	0.84	0.56	0.32
20m H	1.51	1.80	1.30	0.86	0.56	0.35
17m L	1.22	1.76	1.41	0.95	0.62	0.38
17m H	1.23	1.75	1.43	0.96	0.62	0.38
15m L	1.04	1.68	1.49	1.03	0.67	0.41
15m H	1.00	1.66	1.49	1.04	0.67	0.43
12m L	0.81	1.53	1.53	1.12	0.73	0.46
12m H	0.81	1.53	1.53	1.11	0.73	0.46
10m L	0.66	1.42	1.55	1.18	0.79	0.48
10m H	0.55	1.36	1.54	1.21	0.79	0.49
6m L	-	0.74	1.31	1.39	1.03	0.63
6m H	-	0.64	1.26	1.41	1.07	0.67
Max power	1.97	1.81	1.55	1.41	1.07	0.67

This is the final air core inductor design with 12 AWG wire and winding factor of 1.2

	25 nH	50 nH	100 nH	200 nH	400 nH	800 nH
Coil Diameter (mm)	7.3	8.1	9.5	12.4	12.8	13.2
Coil Length (mm)	5.1	7.6	10.2	12.7	20.3	25.4
Turns	2	3	4	5	8	12

...and here they are.

As you can see, they are a bit different than the above table. During testing, I had to play with them to get them closer to the desired values. I will discuss the test data in a separate page.

I also designed the 1,600 nH and 3,200 inductors as air core solenoids. They ended up being quite large. The first one came out as 3/4" x 1 5/8" and the second one as 7/8" x 2". So, like K6JCA, I will try toroids. As a side note, the reason I have the air core coil dimensions in millimeters is because it is much easier to use calipers using 0.1 mm increments, but to get a sense of how big something is, inches seem to be more useful (at least for an American).

So, I go back to the RF Circuit Design book and also use the Micrometals online inductor designer and analysis tools. Powdered iron is much more suitable for high power RF applications (see page 13 of the book). Figure 1-26 shows what material is suitable for what frequency ranges. These two inductors come into play in the 80 to 20 meter bands (see the inductor voltage table above). That is in the 3.5 MHz to 14.35 MHz. From figure 1-26 in the RF Circuit Design book we see that materials 3, 15, and 1 don't cover high enough frequencies and materials 10, 12, and 0 don't cover low enough frequencies. That leaves 2 and 6 material. From the graphs in the book, the recommended frequency range for the 6 material is in the 10 - 50 MHz range. The recommended frequency range for 2 material is 2-30 MHz. So, 2 material is the material of choice.

The formula for the number of turns from the RF design book is:

\begin{equation}N = 100 \times \sqrt {\frac {L}{A _{L}}}\end{equation}

Where N is the number of turns and L is inductance in micro Henries.

Important to note that Amidon publishes its Al values in micro Henries per 100 turns squared while Micrometals publishes them in nano Henries per turns squared. So, while using the Micrometals data, drop the 100 in the formula and use nano Henries.

The other limiting factor is the maximum temperature rise. Mini Ring calculator has published its formulas online and they are the same as the formulas on the Micrometals iron powder core data sheets. Micrometals includes the parameters for these formulas on the data sheet, but I did not find what numbers Mini Ring calculator inputs into their formulas. I ran a set of numbers through the Micrometals online inductor analyzer and also with formulas on a spreadsheet and compared them to the Mini Ring calculator data that Jeff, K6JCA published in his blog. They are vary different. I will have to find why at some point, but for now, I will use the Micrometals numbers since they have an Amazon online store and the parts are easy to buy.

I worked my way through various sizes of the 2 material. T157-2 has total loss of over 7 watts and core temperature rise of 55 degrees. In a 40 degree environment, that is a bit too much. So, on to the next bigger and available size. T200-2 is also available on Amazon at a reasonable price and with acceptable performance. That is the solution.

L (uH)	3.16	1.63
f (MHz)	3.5	3.5
Irms (A)	4.28	5.73
N (turns)	16	11.5
Delta T (deg C)	40	41
Core Loss (W)	5.6	5.1
Wire loss (W)	1.6	2.1
Total loss (W)	7.2	7.2
Wire length (cm)	55.8	19.1

Next step is to build and evaluate the inductors and here they are. Test data including derived equivalent shunt capacitance in a separate page (this page is already too long).