I've noticed that on our VW TDI (ALH) with an automatic transmission when coasting down a hill the amount of fuel injected drops to zero. That's cool for fuel economy, but very puzzling to me. I'm wondering how the engine stays running? I'm under the impression that you can't push start an automatic, I guess I thought this meant that the wheels can't drive the engine through the transmission. If that is the case, why doesn't the engine stall when its fuel supply is cut off during coasting?
A classic automatic car transmission uses a hydraulic torque converter, which is the pendant of the clutch of a manual car.
Imagine this as two axial turbine wheels (like PC fans), one driven by the motor, the other connected to the gear.
When the motor runs, its turbine wheel pushes transmission liquid into the other wheel, generating torque there.
At idle, the motor-side turbine wheel generates only a very low pressure in the fluid, and the torque transferred to the gear-side turbine wheel is low to almost zero. That's exactly what you want when you're waiting in front of a traffic light.
When it's green and you press the gas pedal, motor RPM increases and the turbine produces higher pressure, which will now generate a considerable torque at the second turbine wheel. And the larger the difference in RPM of both wheels, the higher the transferred torque.
Now, there's always a certain difference in RPM of both wheels, which also depends on the torque demand. That's bad because power is RPM times torque, i.e. the motor generates more power than the gear receives. And since power is fuel consumption, this is not very efficient. The solution in modern cars is to lock up motor and gear mechanically at some point via a lock up clutch, making both wheels running at the same RPM.
Now, when motor and gear is locked up mechanically, fuel injection can be stopped, and the motor is cranked through the transmission.
And you can't push-start an automatic car since that low RPM -> no torque transfer applies to the reverse way, too, plus the converter is designed to transfer power from the motor to the gear, not the other way around. It would be possible to push-start an automatic if the lock-up clutch could be engaged with the motor not running / the car standing.
By the way, recent development is to extend the start-stop-system which stops the motor when the car stands to the times when the car cruises without pressing the gas pedal. It's assumed the driver just wants to maintain speed at that moment, while the cranked motor brakes it. Decoupling and stopping the motor can save fuel. (Don't do this manually - you will loose brake vacuum and steering servo.)
Oh, and of course all this is not restricted to a diesel engine, it's even not restricted to hydraulic torque converters.
For example, the clutch of an automatic scooter works like a drum brake, with the drum connected to the rear wheel, and the shoes on a rotating plate connected to the motor. The shoes are kept back by springs, but higher RPM pushes them outwards, against the drum. It's impossible to push-start the motor, since this only moves the drum, not the shoes. On the other side, once the clutch is engaged at some higher speed, the motor can be switched off, but will continue to crank.