I write this post as a new Thread, providing definitive answers for two questions which appear frequently - and have been answered incorrectly upon most occurrences. The way that it all works is complicated. Read carefully, and enjoy.
#1 "Will my Tow Vehicle (unmodified) re-charge the TM batteries while towing?" and
#2 "Will a 3-way Fridge, while running DC12V, really discharge the Trailer Batteries during Towing - even when the Tow Vehicle has tons of Alternator Capacity available to generate power?"
Short answers: NO, an unmodified Tow Vehicle will not significantly re-charge Trailer Batteries while towing; and YES, the Fridge will usually draw on the Trailer batteries "first", discharging them to around 50-60%. Details follow, and "solutions" are at the bottom.
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"Will my Tow Vehicle (unmodified) re-charge the Trailer batteries while towing, in less than 6-8 hours?"
The answer is NO, and upgrading to a slightly bigger "Trailer Battery Charge" wire will not help this by much. A massive wire upgrade is required.
In the Tow Vehicle, the actual voltage of "+12VDC" is determined by a regulator - typically integrated into the Alternator or an Engine Control Module ("ECM"). This sets the Output Voltage on the Alternator as a whole (it's variable). The regulator uses a "sense" wire connected into the under-hood fuse box bus. When voltage on the Sense wire is tested, the Alternator's regulating coils are fed more or less current to increase or decrease the output. (The regulating coils move magnets, creating more or less power within the Alternator as a whole.)
For a short period after running the Starter, the Alternator will run at very high levels of Voltage and Output current - creating a system Voltage well above 14 Volts, which will help the Tow Vehicle ("TV") battery recover quickly. But, after the TV Battery has recovered to a fairly high State-Of-Charge, the system Voltage will be allowed to drop - to a value which is typically less than 13.5 Volts.
Even with an somewhat upgraded "Trailer Battery Charge" wire, and with further upgrades to the Trailer Battery wiring connections, 13.5 Volts (or even less) will not provide significant charging power into the TM batteries. The main problem is Low Voltage right at the beginning, and avoiding some Voltage Drop along the wiring path doesn't provide much help with this problem.
If you use a really heavy cable, direct to the Trailer batteries form your TV battery, then you can charge them effectively - because the ECM "sees" the Trailer batteries as well as it sees the TV battery, and also because the TV battery is being discharged by through the "big" connection. The ECM will respond by asking for much higher power output and Voltage from the alternator, trying to bring State of Charge ("SOC") for the batteries back up to a well-charged state. But this is a solution for charging from the TV in camp - not driving down the road.
Without such a cable, or special equipment to create "Voltage Boost" from the TV to the Trailer - your batteries will (typically)only be charged at a rate of 4-8 Amps. That's typically divided among multiple batteries. It will take many hours to recharge "significantly discharged" batteries. And, if the Trailer contains a big electrical consumer, such as a Refrigerator running on 12VDC (and consuming A LOT more than 100 watts whenever the heater is "ON"), it will never happen at all.
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"Will the Fridge, while running DC12V, discharge the TM batteries during Towing"?
Generally YES, although this IS a function of Voltage Drop along the wires. Typical 3-way Refrigerators will draw about 12-16A from the "best" source of Voltage into the DC Power distribution "Load Center". (That's the box and circuit board with all the fuses.) Within the Trailer, Voltage can be provided from Trailer batteries at 12.8V down to about 12.4V (as the batteries discharge). At the TV fuse box, Voltage can be provided at about 13.5V. But the Trailer Batteries Batteries are the preferred source, unless wiring form the TV is substantially modified, or the 13.5 Source Voltage is increased.
Long calculations follow, you can skip over if you want:
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With large current, there is significant Voltage Drop between the Trailer batteries and the "12VDC Load Center within the converter unit. There is also Voltage drop from the Load Center to the Fridge, but this is the same in both configurations. (Power "from the TV" and power "from the Trailer batteries" uses the same pair of wires to reach the Fridge from the DC Load Center, unless you modify this).
For the "Trailer Battery" path, there is typically 7-12 feet of #10 between the battery and the DC Load Center (more with batteries on the tongue - less with batteries in the rear compartment, more close to the Fridge). We'll assume this path as a round trip of #10, even though the path from front-mounted batteries back to the DC Load Center "Grounding Bus" is actually made through the Frame). A 10ft round trip on #10 wire, carrying 12A, will loose about .25 Volts on this path. If the Trailer batteries fall from 12.8V to 12.4V during drastic discharge, they can maintain the Load Center at 12.55 -> 12.15 volts while the Fridge is running.
Now compare with the use of the TV path: Within an unmodified TV, the "30A" fused "Trailer Battery Charge" wire is typically only 14 AWG, sometimes even 16AWG. (The grounding "return" wire is usually larger, but I'll ignore the difference, just as I ignored the difference for the "Battery Path" return to Battery "-" through the frame.) There's usually about 14 feet of this to reach the Bargeman Plug. At 12A, the Voltage drop in this segment is about 0.4V Volts.
The next segment is the Bargeman Cord inside the Trailer (reaching to the "Load Center" internally, and along the Trailer hitch A-frame outside). In a "typical configuration" this adds another 15 feet of #10.
With a starting Voltage of about 12.9V, the Voltage drop on this segment @ 12A current is almost another .4V, yielding 12.5V at the nut which terminates the Bargman "Trailer Battery Charge" wire. But from there, we have another 8 feet of #10 to actually reach the Load Center. The ending Voltage will be about 12.4VDC, if we lost nothing in all of the connectors and fuses. With some additional resistance occurring in various wire nuts, wiring harness pairs, and the Bargman connectors - I will further reduce this Voltage by an additional 0.1V (SWAG). Result = 12.3V.
The Voltage which can be supplied by TM Batteries (at the DC Load Center) falls from about 12.6V down to 12.2V while the Fridge is drawing DC from Trailer batteries exclusively. (In this scenario, the ending Voltage at the Batteries is about 12.4V, but Voltage drop occurs at high current from "Batteries" to "Load Center".)
The Voltage which can be supplied by the Bargeman path, if it was supporting the same load, is constant, but only about 12.3V. From a Battery Voltage (at the Batteries) of 12.8V, down to a Battery Voltage of about 12.5V the Trailer Batteries are the better source, and most of the power into the Fridge will be drawn the Trailer. vast majorpower will be drawn from them almost exclusively. The Bargeman-TV Path will begin to get used for a majority of the consumed power only after the Trailer Batteries have been "pulled down" to about 60% SOC.
Reducing Voltage drop under high loads does improve the "ending State of Charge" for running a Fridge, because the Fridge will begin to draw more power from the TV "sooner". Maybe a lot sooner, if your current TV wiring is exceptionally poor.
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The "fixes"
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The most effective fixes raise the TV Voltage in various ways, and run a higher-Voltage Bargeman (or Auxiliary) power wire into a Solar Controller, or into the Converter as 120VAC input:
#1: Buy a purpose-built Redarc or Ctek, which Boosts "Tow Vehicle Voltage" after it is received in the TM. They include the Charge Controller (to avoid over-charging your batteries). They're great, but they cost a lot. I owned a low-power Redarc in the past.
#2: Create a new "Charge from TV" method (using a 120VAC cable connnected to Converter AC input) fed by a 120VAC "Inverter" inside the TV. This is fairly cheap, but requires another cable between TV and TM at the hitch, and running all the way back to the Converter. (And it's a high-voltage cable... maybe not real safe in motion.)
#3: If you already have a Solar Controller, the you can do a similar "Voltage Bump" while still using the Bargman Cable "Trailer Battery Charge wire": Increase TV-supplied Voltage to 24V, and use a Relay in the Trailer to determine input Voltage and switch the TV "Trailer Battery Charge" wire to (a) it's original 12V connection; or (b) the Solar "Panel Input +" connection. This involves 3 Relays each in the TV and the Trailer, and should also involve a TV dashboard switch to turn it "On", because your TV needs to remain capable of towing other Trailers too (at "regular" 12V). Due to the high cost of installing a pretty good Solar Controller, it costs more than the 120VAC solution, and it requires Mods to the both the Trailer and TV electrical systems. But it avoids the second electrical cable between TV and Trailer, and doesn't need wiring upgrades - because most of the wires run at 2x Voltage, 1/2x current. (I have this, because it handles much more power than my previous Redarc cold handle, and it's more flexible.)
#4: Reduce Voltage drop by using a better "battery charge wire" within the TV, and (perhaps) going directly to the Trailer Batteries. (As discussed above for "ultra-high" charging capability; you can get some of the benefits by up-sizing smaller wires to #10 AWG.)
#5: Increase TV Voltage by modifying the Alternator Sense Wire. Simply insert a Diode into the wire. (A standard silicon diode, NOT a Shottky.) This will make the TV battery Voltage appear to be about .7 Volts lower than it really is - which will cause ECM to quickly raise operating Voltage to "charge the badly discharged battery".Very quick, very easy - but I do not recommend this, because it will overcharge both the Trailer and TV batteries if left in place for "too long". It also might be risky for some modern ECMs, and those computer modules can be very expensive.