Wiring Ecobee thermostat to Vulkan gas furnace with no C terminal

You have two 24VAC transformers here, and that's where the fun starts

Your system is unusual in that it has two 24V control transformers within it -- one for the electric side that powers the relay and contactor, and a second for the wood side that powers the servo motor. Furthermore, the transformer for the servo motor is half the size of the transformer for the electric side of this furnace (20VA vs 40VA), limiting its ability to power a power-hungry WiFi thermostat as well. Given that the wood-side transformer has its C wire exposed at the control terminal block and the electric side transformer doesn't, that leaves you with two options:

1. Replace the 20VA 24VAC transformer with a 40VA version (40VA HVAC control transformers are utterly bog standard) and then tap C at terminal 6 on the furnace's control terminal block, using a 24VAC SPST relay to allow the thermostat to control the electric side.

2. Tap the electric-side C terminal, likely at terminal 7 on the relay, and then use a 24VAC SPST relay to allow the thermostat to control the wood side.

Something like this would suffice for this application as far as relays go (technically, it's not UL listed for standalone use, but for mounting into a furnace, that's not a big deal).

I have two of these thermostats installed. When you set a programed time for the thermostat, you can also select which sensor is active during that time. For instance, I have a sensor in the bedrooms and one in the main living area and then the thermostat itself. The thermostat is in an odd location and I don't like the temperature it ends up reading. So, for our "daytime" program, I have only the main living area sensor active. The system doesn't care about the other two temperatures and just turns on and off to make sure that one sensor is happy.

For the "sleep" part of the schedule, I have the two bedroom sensors active, so the temp of the main living area doesn't matter. I had a problem with one bedroom getting warmer than the other, but adjusting the dampers on the vents in the room corrected that so the rooms warm and cool at about the same rate.

There is also a setting to ensure the fan comes on for a minimum amount of time per hour. This can be set so the air circulates even if the active sensors are at the "correct" temperature.