K6JCA: Antenna Auto-tuner Design, Part 5: Directional Coupler Design

This, my fifth post for my Automatic Antenna Tuner, deals with the design of the Directional Coupler. (Its construction and testing will follow in a later blog post. And the previous post in this series can be found here.)

I plan to use a "Tandem Match" style coupler (for background, see here) which uses two transformers -- one senses the voltage across the transmission-line, the other senses the current traveling in the transmission-line:

The two transformers will be wound on Ferrite cores (because I can use fewer turns for a high inductance, compared to Iron Powder cores).

Let's get started...

First -- the specifications I'd like this coupler to meet:

Frequency Range: 3.5 - 30 MHz (and possibly 54 MHz)
Forward Power In, Continuous operation (e.g. carrier) at a maximum SWR of 3:1: 200 watts. (To ensure that the cores do not overheat).
Forward Power In, Peak, at an Infinite SWR: 800 watts. (To ensure that the cores do not saturate, assuming 100% AM modulation at carrier power = 200 watts).
Directivity: 40 dB, min.

These specs might change as I design and test the coupler, but they're a good starting point.

Next, I would like to select core size and type (e.g. mix 43, mix 61, or...?).

But how do I select the right cores? Research was required. And here are the pertinent facts that I found:

With a normal sine-wave signal above 100 KHz, both Iron Powder and Ferrite type cores will first be affected by overheating caused by core losses, rather than saturation (Amidon).

Per Amidon, a core's operating temperature (ambient temperature plus core temperature rise) should be limited to 100°C for intermittent operation and 75°C for continuous operation (Amidon).

Core saturation is a secondary cause of coil failure. Saturation will decrease the permeability of the core causing it to have impaired performance or to become inoperative. The safe operating total flux density level for most Ferrite Materials is typically 2000 gauss and 5000 gauss for Powdered-Iron. (Amidon).

The extrapolated AC flux density limits (in the table below) can be used for BOTH Iron Powder and Ferrite type cores as a guideline to avoid excessive heating. These figures may vary slightly according to the type of the material being used. (Amidon)

(click on image to enlarge)

The permeability is at its maximum just below the Curie temperature. Above the Curie temperature, the core permeability sharply disappears, and the material is no longer magnetic while it is above the Curie Temperature. (Magnetics)

In all ferrites, saturation flux density (Bsat) declines as the temperature goes up. (Magnetics)

Per Fair-rite: "Strong magnetic fields or excessive mechanical stresses may result in irreversible changes in permeability or losses." (This warning is explicitly stated for materials 61 and 67).

And per VK2WB:

The permeability of all ferrite materials is frequency dependent (it decreases with frequency), so the manufacturer's quoted A_L values (determined at approximately 10 KHz), should not be use used for calculations involving RF frequencies. (VKHam)

Many ferrite cores can permanently change their permeability after being subjected to relatively high power (flux) levels. (VKHam)

Calculating Inductance and Flux Density:

This design uses two transformers. One senses current through the transmission line, the other sense voltage across the transmission line.

Because the Directional Coupler precedes the Tuning Network in the RF path from transmitter to antenna, I'm assuming it will see the following voltages and currents:

During normal operation, I'll specify that the SWR seen by the transmitter will be no worse than 3:1, thus the 3:1 SWR requirement for core heating at the average power of 200 watts being delivered to the load.

When the tuner first senses SWR (e.g. with the Tuning Network untuned), the SWR could be very large. But assume maximum power, when tuning, is low. And for saturation purposes assume Power is at its maximum (800 watts, assuming AM modulation with a carrier power of 200 watts) with an infinite SWR.

With these givens, I can calculate the maximum voltage and current that I would expect to see on the transmission line:

Vmax = (200 * 50)^(1/2) *(1 + (SWR-1)/(SWR+1)) = 150 Vrms
Vpk = ((800 * 50)^(1/2)) * 1.414*(1 + (Infinty-1)/(Infnity+1)) = 566 Volts
Imax = (200 / 50)^(1/2) *(1 + (1-SWR)/(1+SWR)) = 3.00 Arms

Recommended Maximum Flux Densities:

Here are recommended Maximum Flux Density Limits to ensure that cores do not overheat (these are the limits that the Mini Ring Core Calculator uses and which are based on (and interpolated from) Amidon's recommendations.)

And to verify that the cores do not saturate, I will assume that the Saturation Flux Density is 2000 Gauss for Ferrite material, per the Amidon note, above. Therefore, I would like the maximum Flux Density in this design, under peak conditions, to be well under 2000 Gauss.

Calculating Flux Densities

Flux-density calculations will use the following equation:

The Calculated Results:

Please refer to this blog post for additional details in calculating the maximum flux densities of the current-sense and voltage-sense transformers:

Calculating Tandem-Match Flux Densities

Here's the spreadsheet data for 3.5 MHz:

Values in RED signify a failure to meet my criteria (e.g. Flux Density is above 80 Gauss on 3.5 MHz).

You can see that the FT50 Mix 43 core is satisfactory for the current-sense transformer, and the 2643625002 core, at 74 gauss, is satisfactory for the voltage-sense transformer.

And here's the data at 28 MHz:

Much better results at 28 MHz, even though the Flux Density Limit has dropped from 80 to 30 Gauss.

Some additional notes on the spreadsheets:

The numbers in parenthesis, such as "L(2)" or "B(1)", simply refer to the "number" of the equation used (e.g. equation L2 or equation B1 -- see the equations, above).
The "Vrms" value for the Current-Sense transformer is the voltage drop across this transformer's single-turn primary (i.e. Vrms = Irms * (jXl || Rf), where Rf is the secondary's load (Rtermination) reflected to the transformer's primary side). In the tables, above, Rf dominates.
Please note that for transformer "Peak Power" calculations, the SWR is assumed to be infinite.
For "average power" operation, core heating is my concern. Thus, for Pavg, Flux Density, B(1), is compared against "Bmax" (e.g. 80 gauss at 3.5 MHz). If the calculated Flux Density exceeds Bmax, the result appears RED.
For "peak power" calculations (i.e. worst-case) heating isn't really the concern (because this condition should be a short-lived event). Instead, I check that the core does not saturate. Therefore, for Ppk, Flux Density is compared against Bsat (e.g. 2000 gauss). If the calculated Flux Density exceeds Bsat, the result appears RED.

Conclusions:

Maximum Flux Density (for saturation determination) is under 2000 gauss in all cases.

At a 3:1 SWR, then four of the Mix 43 cores in the Mix 43 table above would be acceptable for the Voltage-sense transformer, using a winding ratio of 24:1 (i.e. Coupling Factor of -27.6 dB).

The Current-sense transformer Flux Densities are all well under the recommended limits. Note that they are low enough that an FT50-43 core could be used.

IMPORTANT NOTE!!

If using the Mini Ring Core Calculator

For Type-43 material, the Mini Ring Core Calculator (V1.2) uses smaller values of A_L than those published by Fair-rite or Amidon.

Therefore, the Calculator's Inductance and Flux Density results will be lower (by a factor of about 0.85, per my experience) from those calculated using the manufacturer's A_L.

Also -- although the Calculator's results for mix-61 material generally track my results obtained via published equations, I did notice that the Calculator's Flux results specifically for the FT114A-61 core were lower by a factor of about 0.83.

Wire Gauge:

Next, I'd like to look at wire gauge for the transformer windings.

Current-sense Transformer:

1-Turn primary: 14, 12 or 10 AWG for its primary. Note: for a 3 Amp RMS current, the power loss on, say, 2 inches of 14AWG wire at 30 MHz (worst loss due to skin effect) would be 1/8 watt.

24-Turn secondary: Assume: 3 Arms primary current, 36" wire length, 26 AWG, 30 MHz. Secondary current is therefore 0.12A (i.e. 3A/24). Using an on-line AC Resistance Calculator with wire length of 914.4 mm and diameter to 0.4049 mm (using this AWG to mm Converter), the secondary's resistance would be 1.044 ohms. Power loss is (I^2)*R and therefore the Secondary's Loss = 0.015 watt.

So the Current-sense primary winding would be 14 AWG (and probably larger).
And the secondary winding would be 26 AWG.

Voltage-sense Transformer:

24-Turn primary:

The current flowing in the voltage-sense transformer's 24-turn primary is:
i = Vload * | (Rload - R) / (R*Rload*(2*(n^2) + 1) | (see: Calculating Tandem-Match Flux Densities)
Assuming 200 watts delivered to the load resistance of 16.67 ohms (i.e. 3:1 SWR), the Average current in the 24-turn winding of the voltage-sense transformer is simply 173 Vrms / |50*150*(2*(24^2)+1)/(150-50)| . This value equals 47 mA.
Assume wire length = 36", wire gauge = 26 AWG: Total R at 3.5MHz = 0.38 ohms.
Loss at 3.5 MHz: 0.047^2 * 0.38 = Negligible.
Loss at 30 MHz: Total R, 26AWG @ 36" = 1 ohm. Loss = 2 mW = Negligible.

1-Turn Secondary:

Assume current through secondary is current through primary times the turns ratio.
Iprimary = 47 mA
Isecondary = 47 mA * 24 = 1.1 A.
Assume wire length = 2", 26 AWG.
Loss at 3.5 MHz: Wire R = 0.02 ohms. Wire Loss = 1.1^2 * 0.02 = 24 mW = Negligible.
Loss at 30 MHz: Wire R = 0.06 ohms. Wire Loss = 1.1^2 * 0.06 = 72 mW = Negligible.

So the windings of the Voltage-sense transformer can both be 26 AWG.

Terminating Resistors:

Finally, a quick calculation to determine the power rating of the two 50 ohm terminating resistors.

For 24 turns, the Directional Coupler's "Coupling Factor" will be -27.6 dB.

Therefore, for continuous operation at a Power level of 200 watts (53 dBm), 25.4 dBm of power (about 0.35 watts) will need to be dissipated by the 50 ohm resistor on the "Vforward" side.

I'll spec the two terminating resistors to be identical, with this value: 50 ohms, 0.5 watts minimum, each.

Next Steps:

The next steps will include experimenting with cores and winding ratios, to see if I can get acceptable performance (e.g. I'd like Directivity to be at least 40 dB from 3.5 - 30 MHz). But this will have to wait until I get purchase some cores!

And an important further note regarding implementation:

Directional Couplers using the Tandem-Match topology (two toroidal transformers) typically use identical large cores for their two transformers. Per my spreadsheet, it is the voltage-sense transformer that dictates this large core size. The current-sense transformer's core can be much smaller. Going to a smaller core would save both money and circuit space.

In fact, if I instead use a capacitive voltage-divider (i.e. Bruene coupler) rather than the voltage-sense transformer to sense transmission line voltage, I believe even more space and money could be saved.

Hmmm...food for thought!

Other:

The previous post in this series can be found here.

And the next post in this series, part 6 (Notes on Match Detection), can be found here.

Resources:

Book, Ferromagnetic-Core Design and Application Handbook, 2nd Edition, M.F. Demaw (Available from Amidon)

Website, Manufacturer, Amidon: http://www.amidoncorp.com/.
Website, Manufacturer, Fair-rite: http://www.fair-rite.com/cgibin/catalog.pgm#select:freq1
Website, Ferrite-Core Design Information, VKHAM (VK2WB): http://www.vkham.com/resources/reference-info/ferromagnetic-information

Databook, Soft Ferrites and Accessories, 2013, Ferroxcube: http://www.ferroxcube.com/FerroxcubeCorporateReception/datasheet/FXC_HB2013.pdf (Good reference for design equations).
Data, Ferrite Materials, Fair-rite: http://www.fair-rite.com/newfair/materials.htm
Data, Ferrite Magnetic Properties, Fair-rite: http://www.fair-rite.com/newfair/materials_mag_prop.htm

App Note, Design, "Power Considerations (Iron Powder and Ferrite)," Amidon: http://www.amidoncorp.com/product_images/specifications/1-35.pdf
App Note, "Power Considerations (cont')", Amidon: http://www.amidoncorp.com/product_images/specifications/1-36.pdf
App Note, "Designing with Magnetic Cores at High Temperatures", Magnetics: http://www.mag-inc.com/design/design-guides/designing-with-magnetic-cores-at-high-temperatures
App Note, "How to Choose Ferrite Components for EMI Suppression," Fair-rite: http://www.fair-rite.com/newfair/pdf/CUP%20Paper.pdf (Good discussion on core-loss model based on permeability).
User's Guide, Soft Ferrites, Magnetic Materials Producers Association, 1998: http://www.magneticsgroup.com/pdf/mmpa%20SFG-98.pdf (Another source of good information and equations (e.g. Ae, le, and Ve)).
Article, In Compliance Magazine, Using Ferrites to Suppress EMI: http://incompliancemag.com/article/using-ferrites-to-suppress-emi/ (good equation derivation, but note an error in the permeability magnitude equation. 'j' should not be appear under the square-root sign in equation 4.)
Magnetics, Colorado Power Electronics Center, University of Colorado: http://ecee.colorado.edu/copec/book/slides/Ch12slide.pdf (Basic theory, Inductor design, Transformer design)