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:

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).
- 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**. (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).

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

- 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.

- When the tuner first senses SWR (e.g. with the Tuning Network untuned), the SWR could be
*very*large. Assume Power is at its maximum (800 watts) when this occurs, for calculating a worst-case Flux Densities (to verify that the cores do not oversaturate).

- Note that Vmax = |Vf| + |Vr| and Imax = |If| + |Ir|. Therefore, at infinite SWR, because |Vf| = |Vr| and |If| = |Ir|, the maximum voltage and current will be 2*Vf and 2*If, where Vf and If are the voltage and current for a 1:1 SWR (i.e. into 50 ohms).

These power and SWR levels result in the following voltage and current requirements :

The "shaded" boxes contain the levels I will use. To summarize, they are:

- For core heating (at 200 watts): Vmax = 150 Vrms, Imax = 3.0 Arms
- For core saturation (at 800 watts): Vmax = 400 Vrms, Imax = 8.0 Arms

__Voltage-sense Transformer Requirements:__

The N-turn primary of the voltage-sense transformer connects across the transmission line, so I would like its impedance at the lowest frequency of 3.5 MHz to have an inductive reactance of at least 500 ohms (10 times greater than 50 ohms, the transmission line impedance -- note that this reactance, when placed in parallel across a 50 ohm load, would result in a Return Loss of 26 dB). So the inductance should be at least 23 uH.

As I just mentioned, I want to ensure that:

- The max RMS voltage across the voltage-sense transformer (at 200 watts, SWR = 3:1, tuner in bypass) doesn't overheat the transformer. (Flux Density to be below the recommended maximum).
- The max RMS voltage across the voltage-sense transformer (at 800 watts, SWR = Infinite, tuner in bypass) doesn't come close to saturating the transformer. (Flux Density to be below Saturation).

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

__Current-sense Transformer Requirements:__

I want to ensure that:

- The max RMS current through the current-sense transformer (at 200 watts, SWR = 3:1, tuner in bypass) doesn't overheat the transformer. (Flux Density to be below the recommended maximum).
- The max RMS current through the current-sense transformer (at 800 watts, SWR = infinite, tuner in bypass) doesn't come close to saturating the transformer. (Flux Density to be below Saturation).

(click on image to enlarge)

This 25 ohm resistor, connected to the transformer's N-turn secondary, "reflects" into the (1-turn) primary circuit, appearing as a 25 / N

^{2}resistor in parallel with the 1-turn primary of the transformer. So, the higher the turns ratio, the smaller this "reflected" resistor will be.

Here's the equivalent circuit (much simplified):

(click on image to enlarge)

L and Rloss are simply the inductance and loss (core) of the 1-turn primary. Note that if the 25 ohm termination resistor is

*not*placed across the secondary, there would be no "25 / N

^{2}" ohm reflected resistance across the primary. Without this shunt, all of the transmission line current will instead pass through the primary winding's Rloss and L -- and (from personal experience), if the current is high, the core can easily become too hot to touch.

But with the 25 ohm resistor attached, and assuming the "reflected" resistance is significantly smaller than the reactance of the 1-turn primary, almost all of the transmission-line current will shunt through the 25 / N

^{2}resistance and the core shouldn't overheat.

And thus, the voltage across the transformer's primary is essentially equal to Irms * 25 / N

^{2}. I'll use this voltage to calculate the current-sense transformer's Flux Density.

And for these calculations I will use Excel, rather than the "Mini Ring Core Calculator" program.

**IMPORTANT NOTE!!**

For

__Type-43 material__, the

**Mini Ring Core Calculator (V1.2)**uses

**values of A**

*smaller*_{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 FT114

**A**-61 core were lower by a factor of about 0.83.

**The Design Equations:**

**The Calculated Results:**Here's the spreadsheet data for 3.5 MHz:

(click on image to enlarge)

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

And here's the data at 30 MHz:

(click on image to enlarge)

Much better results at 30 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 * (Xl || Rf), where Rf is the secondary's load (Rtermination) reflected to the transformer's primary side).
- Please note that for "Peak Power" calculations, the SWR is assumed to be infinite.
- For "average power" operation, core heating is our 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.

Let's focus on 3.5 MHz, where more cores fail. Here's a reduced "summary" table comparing six of the most likely candidate cores. These results assume an "operational" SWR (max SWR of 3:1 during normal operation):

(click on image to enlarge)

If the maximum operational SWR is assumed to be 2:1 (rather than 3:1), then the results are:

(click on image to enlarge)

(

**A slight correction:**in the two tables above, the core listed as "26435002" should instead be "2643625002".)

**Conclusions:**Maximum Flux Density is well under 2000 in all cases.

If the Tuner is assumed to tune to at least a 2:1 SWR (a reasonable assumption), then five of the six cores in the last table above would be acceptable for the Voltage-sense transformer, using a winding ratio of 25:1 (i.e. Coupling Factor of -28 dB). Five cores decreases three cores if the tuner cannot tune to an SWR under 3:1.

Inductances for all six voltage-sense transformer candidates are well over 23 uH, so their reactance will be well above my minimum requirement of "10 times 50 ohms". But the real test will be after I build the coupler and can check its return-loss.

For the Voltage-sense transformer, the "Max Power at Infinite SWR" Flux Densities of all of the cores listed are well below saturation.

The Current-sense transformer Flux Densities are all well under the recommended limits. In fact, they are so low that I've only shown their values for worst-case operation. Note that they are low enough that an FT50-43 core (and perhaps a core even smaller) could be used.

__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**.

__25-Turn secondary:__Assume: 3 Arms primary current, 36" wire length, 26 AWG, 30 MHz. Secondary current is therefore 0.12A (i.e. 3A/25). 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:__

__25-Turn primary:__

- Average current is simply 150Vrms / XL (the reactance of the transformer primary). Let's take the FT87A-43 core as an example:
- At 3.5 MHz, Irms = 150 / (2 * pi * 3.5MHz * 605 uH) = 11 mA.
- Assume wire length = 36", wire gauge = 26 AWG: Total R at 3.5MHz = 0.38 ohms.
- Loss at 3.5 MHz: 0.011^2 * 0.38 =
**Negligible.** - Loss at 30 MHz: Irms = 1.3 mA. Total R, 26AWG @ 36" = 1 ohm. Loss =
**Negligible.**

__1-Turn Secondary:__

- Load across secondary is essentially 100 ohms (two 50 ohms in series).
- Vsecondary = 150 Vrms / N = 150 / 25 = 6 volts.
- Isecondary = 6 / 100 = 60 mA.
- Assume wire length = 2", 26 AWG.
- Loss at 3.5 MHz: Wire R = 0.02 ohms. Wire Loss = 0.06^2 * 0.02 =
**Negligible.** - Loss at 30 MHz: Wire R = 0.06 ohms. Wire Loss = 0.06^2 * 0.06 =
**Negligible.**

**26 AWG**.

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

For 25 turns, the Directional Coupler's "Coupling Factor" will be -28 dB.

Therefore, for continuous operation at a Power level of 200 watts (53 dBm), 25 dBm of power (0.32 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)

- Calculator, Inductor (including loss),
*Mini Ring Core Calculator*: http://www.electronicecircuits.com/electronic-software/mini-ring-core-calculator-program - Calculator, Ferrite Inductor, Excel,
*Ferrite Calcx*, VK2AB: http://www.vkham.com/resources/software/viewcategory/8-antennas-homebrew

**Links to my blog posts in this Auto-tuner series:**Part 1: Preliminary Specification

Part 2: Network Capacitor Selection

Part 3: Network Inductor Selection

Part 4: Relays and L-Network Schematic (Preliminary)

Part 5: Directional Coupler Design

Part 6: Notes on Match Detection

Part 7: The Build, Phase 1

Part 8: The Build, Phase 2 (Integration of Match Detection)

Part 9: The Build, Phase 3 (Incorporating a Microcontroller)

Part 10: The Final Schematics

**My Other Directional Coupler Posts:**Notes on HF Directional Couplers

Notes on the Bruene Coupler, Part 1

Notes on the Bruene Coupler, Part 2

Building an HF Directional Coupler

Notes on the Bird Wattmeter

Notes on the Monimatch

Notes on the Twin-lead "Twin-Lamp" SWR Indicator

**My Other Tuner posts:**A quick tutorial on Smith Chart basics: http://k6jca.blogspot.com/2015/03/a-brief-tutorial-on-smith-charts.html

The L-network: http://k6jca.blogspot.com/2015/03/notes-on-antenna-tuners-l-network-and.html

A correction to the usual L-network design constraints: http://k6jca.blogspot.com/2015/04/revisiting-l-network-equations-and.html

A look at highpass T-Networks: http://k6jca.blogspot.com/2015/04/notes-on-antenna-tuners-t-network-part-1.html

More on the W8ZR EZ-Tuner: http://k6jca.blogspot.com/2015/05/notes-on-antenna-tuners-more-on-w8zr-ez.html (Note that this tuner is also discussed in the highpass T-Network post).

The Elecraft KAT-500: http://k6jca.blogspot.com/2015/05/notes-on-antenna-tuners-elecraft-kat500.html

The Nye-Viking MB-V-A and the Rohde Coupler : http://k6jca.blogspot.com/2015/05/notes-on-antenna-tuners-nye-viking-mb-v.html

**Standard Caveat:**As always, I might have made a mistake in my equations, assumptions, drawings, or interpretations. If you see anything you believe to be in error or if anything is confusing, please feel free to contact me or comment below.

And so I should add -- this design and any associated information is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

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