Estimates for a Lepelletier ZF 6HP26 :-
Drag in clutches, 3% losses
Gear inefficiencies 9% losses
Oil pump at 3000 RPM 12 bar, 5 litres per minute 5% losses
Total estimated losses 17%
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As I understand it, clutch losses seem to be about ½% per dragging clutch. Planetary gears seem to be about 96% efficiency (ie 4% losses) when not in direct drive, eg 92% losses when two planetary gearset are compounded. For the transmission above, the losses are so high due to the compounding effects in higher gears like 5th or 6th, compared to 3rd which is say 96% efficient (4% losses) but only used for half as long as 6th on average.
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I don't know if you count the dual clutch transmissions as automatics. Once something is drive-by-wire, the difference between manuals and automatics is eroded: (1) software prevents stupid gearchanges in manual modes, eg changing to first at Autobahn speeds, so real manual operation has been removed (2) paddle shifts or +/- controls can be added, so automatic operation has or can been extended.
These are more efficient than a manual if you count fuel economy or 0-60 times. This can be seen by comparing the figures for manual and dual clutch versions. But maybe its not fair to do that. Fuel economy is improved by better gear selection, and 0-60 times are improved by the lack of torque interruption.
You could estimate the inefficiency of automatics by looking at 0-60 times for manual and automatic version and by looking at top speeds.
When people talk about a car and horsepower, they are not talking about how much horsepower is used driving to the supermarket. They are talking about maximum horsepower.
Let's have a look at an example, albeit just a ZF CVT, not a real automatic.
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BMW Mini CVT.
PERFORMANCE, FUEL CONSUMPTION
AND CO2 EMISSIONS
MINI One: 1,598cc manual: 0-60: 10.6 seconds; top speed: 115mph; combined mpg 43.5; CO2 emissions 158g/km; Emissions Class EU4; Insurance Group 5
MINI One: 1,598cc CVT auto: 0-60: 12.4 seconds; top speed: 106mph; combined mpg 36.7; CO2 emissions 187g/km; Emissions Class EU4; Insurance Group 5
MINI One: 1,598cc chain cam petrol: 66kW (90bhp) at 5,500rpm/140Nm (103 lb ft) torque at 3,000rpm.
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106/115 = 0.92 = 92%
So the CVT's top speed is 8% less than than the manual's. The CVT presumably has an identical gear ratio available to the manual, so the lack of top speed is due to the CVT's inefficiency.
Top speed is governed by air resistance which is a square law, so an 8% drop in top speed means about a 16% drop in horsepower.
16% * 90bhp = 14.4.
So the CVT automatic mini loses about 14 horse power more than the loses on the manual, out of 90 BHP.
(The CVT 0-60 time is also about 15% slower than the manual despite the mini CVT always having the ideal ratio.)
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If you have a torque convertor on your automatic rather than a friction launch clutch, then you can add on losses for that too, if looking at fuel consumption type losses, but if looking at numbers to impress the public: horsepower, top speed, etc, the convertor would be locked up and you can forget about it.