ZF 3HP transmission

History

The Ravigneaux planetary gearset types were first introduced in 1963 and produced through the mid-seventies and has been used in a wide range of cars. There are versions for longitudinal and transverse engines.

Planetary gearset concept

Planetary gearset concept: gear ratio quality^[a]

In-Depth Analysis[b] With Assessment And Torque Ratio[c] And Efficiency Calculation[d]		Planetary Gearset: Teeth[e]		Count	Nomi- nal[f] Effec- tive[g]	Cen- ter[h]
		Ravigneaux				Avg.[i]
Model Type	Version First Delivery	S1[j] R1[k]	S2[l] R2[m]	Brakes Clutches	Ratio Span	Gear Step[n]
Gear		R		1	2	3
Gear Ratio[b]		i R {\displaystyle {i_{R}}} [b]		i 1 {\displaystyle {i_{1}}} [b]	i 2 {\displaystyle {i_{2}}} [b]	i 3 {\displaystyle {i_{3}}} [b]
Step[n]		− i R i 1 {\displaystyle -{\frac {i_{R}}{i_{1}}}} [o]		i 1 i 1 {\displaystyle {\frac {i_{1}}{i_{1}}}}	i 1 i 2 {\displaystyle {\frac {i_{1}}{i_{2}}}} [p]	i 2 i 3 {\displaystyle {\frac {i_{2}}{i_{3}}}}
Δ Step[q][r]					i 1 i 2 : i 2 i 3 {\displaystyle {\tfrac {i_{1}}{i_{2}}}:{\tfrac {i_{2}}{i_{3}}}}
Shaft Speed		i 1 i R {\displaystyle {\frac {i_{1}}{i_{R}}}}		i 1 i 1 {\displaystyle {\frac {i_{1}}{i_{1}}}}	i 1 i 2 {\displaystyle {\frac {i_{1}}{i_{2}}}}	i 1 i 3 {\displaystyle {\frac {i_{1}}{i_{3}}}}
Δ Shaft Speed[s]		0 − i 1 i R {\displaystyle 0-{\tfrac {i_{1}}{i_{R}}}}		i 1 i 1 − 0 {\displaystyle {\tfrac {i_{1}}{i_{1}}}-0}	i 1 i 2 − i 1 i 1 {\displaystyle {\tfrac {i_{1}}{i_{2}}}-{\tfrac {i_{1}}{i_{1}}}}	i 1 i 3 − i 1 i 2 {\displaystyle {\tfrac {i_{1}}{i_{3}}}-{\tfrac {i_{1}}{i_{2}}}}
Torque Ratio[c]		μ R {\displaystyle \mu _{R}} [c]		μ 1 {\displaystyle \mu _{1}} [c]	μ 2 {\displaystyle \mu _{2}} [c]	μ 3 {\displaystyle \mu _{3}} [c]
Efficiency η n {\displaystyle \eta _{n}} [d]		μ R i R {\displaystyle {\frac {\mu _{R}}{i_{R}}}} [d]		μ 1 i 1 {\displaystyle {\frac {\mu _{1}}{i_{1}}}} [d]	μ 2 i 2 {\displaystyle {\frac {\mu _{2}}{i_{2}}}} [d]	μ 3 i 3 {\displaystyle {\frac {\mu _{3}}{i_{3}}}} [d]
3HP 12	160 N⋅m (118 lb⋅ft) Small engines 1963	25 32	32 64	3 2	2.5600 2.0000[g][o]	1.6000
						1.6000[n]
Gear		R		1	2	3
Gear Ratio[b]		−2.0000[o][g] − 2 1 {\displaystyle -{\tfrac {2}{1}}}		2.5600 64 25 {\displaystyle {\tfrac {64}{25}}}	1.5200[p] 38 25 {\displaystyle {\tfrac {38}{25}}}	1.0000 1 1 {\displaystyle {\tfrac {1}{1}}}
Step		0.7825[o]		1.0000	1.6842[p]	1.5200
Δ Step[q]					1.1080
Speed		-1.2800		1.0000	1.6842	2.5600
Δ Speed		1.2800		1.0000	0.6842	0.8758
Torque Ratio[c]		–1.9600 –1.9400		2.4836 2.4457	1.4928 1.4792	1.0000
Efficiency η n {\displaystyle \eta _{n}} [d]		0.9800 0.9700		0.9702 0.9553	0.9821 0.9731	1.0000
3HP 12	Large Engines 1963	28 32	32 64	3 2	2.2857 2.0000[g][o]	1.5119
						1.5119[n]
Gear		R		1	2	3
Gear Ratio[b]		−2.0000[o][g] − 2 1 {\displaystyle -{\tfrac {2}{1}}}		2.2857 16 7 {\displaystyle {\tfrac {16}{7}}}	1.4286 10 7 {\displaystyle {\tfrac {10}{7}}}	1.0000 1 1 {\displaystyle {\tfrac {1}{1}}}
Step		0.8750[o]		1.0000	1.6000	1.4286
Δ Step[q]					1.1280
Speed		-1.1429		1.0000	1.6000	2.2857
Δ Speed		1.1429		1.0000	0.6000	0.6842
Torque Ratio[c]		–1.9600 –1.9400		2.2175 2.1836	1.4038 1.3914	1.0000
Efficiency η n {\displaystyle \eta _{n}} [d]		0.9800 0.9700		0.9702 0.9553	0.9826 0.9740	1.0000
Actuated shift elements
Brake A[t]					❶
Brake B[u]					❶
Brake C[v]		❶		❶
Clutch D[w]				❶	❶	❶
Clutch E[x]		❶				❶
Geometric ratios: speed conversion
Gear Ratio[b] Ordinary[y] Elementary Noted[z]	i R = − R 2 S 2 {\displaystyle i_{R}=-{\frac {R_{2}}{S_{2}}}}	i 1 = R 2 S 1 {\displaystyle i_{1}={\frac {R_{2}}{S_{1}}}}		i 2 = R 2 ( S 1 + R 1 ) S 1 ( S 2 + R 2 ) {\displaystyle i_{2}={\frac {R_{2}(S_{1}+R_{1})}{S_{1}(S_{2}+R_{2})}}}		i 3 = 1 1 {\displaystyle i_{3}={\frac {1}{1}}}
	i R = − R 2 S 2 {\displaystyle i_{R}=-{\tfrac {R_{2}}{S_{2}}}}	i 1 = R 2 S 1 {\displaystyle i_{1}={\tfrac {R_{2}}{S_{1}}}}		i 2 = 1 + S 2 S 1 1 + S 2 R 2 {\displaystyle i_{2}={\tfrac {1+{\tfrac {S_{2}}{S_{1}}}}{1+{\tfrac {S_{2}}{R_{2}}}}}}
Kinetic ratios: torque conversion
Torque Ratio[c]	μ R = − R 2 S 2 η 0 {\displaystyle \mu _{R}=-{\tfrac {R_{2}}{S_{2}}}\eta _{0}}	μ 1 = R 2 S 1 η 0 3 2 {\displaystyle \mu _{1}={\tfrac {R_{2}}{S_{1}}}{\eta _{0}}^{\tfrac {3}{2}}}		μ 2 = 1 + S 2 S 1 η 0 1 + S 2 R 2 ⋅ 1 η 0 {\displaystyle \mu _{2}={\tfrac {1+{\tfrac {S_{2}}{S_{1}}}\eta _{0}}{1+{\tfrac {S_{2}}{R_{2}}}\cdot {\tfrac {1}{\eta _{0}}}}}}		μ 3 = 1 1 {\displaystyle \mu _{3}={\tfrac {1}{1}}}
Revised 14 January 2026 Nomenclature S n = {\displaystyle S_{n}=} sun gear: number of teeth R n = {\displaystyle R_{n}=} ring gear: number of teeth C n = {\displaystyle \color {gray}{C_{n}=}} carrier or planetary gear carrier (not needed) s n = {\displaystyle s_{n}=} sun gear: shaft speed r n = {\displaystyle r_{n}=} ring gear: shaft speed c n = {\displaystyle c_{n}=} carrier or planetary gear carrier: shaft speed With n = {\displaystyle n=} gear is i n = {\displaystyle i_{n}=} gear ratio or transmission ratio ω 1 ; n = ω t = {\displaystyle \omega _{1;n}=\omega _{t}=} shaft speed shaft 1: input (turbine) shaft ω 2 ; n = {\displaystyle \omega _{2;n}=} shaft speed shaft 2: output shaft T 1 ; n = T t = {\displaystyle T_{1;n}=T_{t}=} torque shaft 1: input (turbine) shaft T 2 ; n = {\displaystyle T_{2;n}=} torque shaft 2: output shaft μ n = {\displaystyle \mu _{n}=} torque ratio or torque conversion ratio η n = {\displaystyle \eta _{n}=} efficiency i 0 = {\displaystyle i_{0}=} stationary gear ratio η 0 = {\displaystyle \eta _{0}=} (assumed) stationary gear efficiency Gear ratio (transmission ratio) i n {\displaystyle i_{n}} — speed conversion — The gear ratio i n {\displaystyle i_{n}} is the ratio of input shaft speed ω 1 ; n {\displaystyle \omega _{1;n}} to output shaft speed ω 2 ; n {\displaystyle \omega _{2;n}} and therefore corresponds to the reciprocal of the shaft speeds i n = 1 ω 2 ; n ω 1 ; n = ω 1 ; n ω 2 ; n = ω t ω 2 ; n {\displaystyle i_{n}={\frac {1}{\frac {\omega _{2;n}}{\omega _{1;n}}}}={\frac {\omega _{1;n}}{\omega _{2;n}}}={\frac {\omega _{t}}{\omega _{2;n}}}} Torque ratio (torque conversion ratio) μ n {\displaystyle \mu _{n}} — Torque Conversion — The torque ratio μ n {\displaystyle \mu _{n}} is the ratio of output torque T 2 ; n {\displaystyle T_{2;n}} to input torque T 1 ; n {\displaystyle T_{1;n}} minus efficiency losses and therefore corresponds (apart from the efficiency losses) to the reciprocal of the shaft speeds too μ n = i n η n ; η 0 = ω 1 ; n η n ; η 0 ω 2 ; n = T 2 ; n η n ; η 0 T 1 ; n {\displaystyle \mu _{n}=i_{n}\eta _{n;\eta _{0}}={\frac {\omega _{1;n}\eta _{n;\eta _{0}}}{\omega _{2;n}}}={\frac {T_{2;n}\eta _{n;\eta _{0}}}{T_{1;n}}}} whereby η n ; η 0 {\displaystyle \eta _{n;\eta _{0}}} may vary from gear to gear according to the formulas listed in this table and 0 ≤ η n ; η 0 ≤ 1 {\displaystyle 0\leq \eta _{n;\eta _{0}}\leq 1} Efficiency The efficiency η n {\displaystyle \eta _{n}} is calculated from the torque ratio in relation to the gear ratio (transmission ratio) η n = μ n i n {\displaystyle \eta _{n}={\frac {\mu _{n}}{i_{n}}}} Power loss for single meshing gears is in the range of 1 % to 1.5 % helical gear pairs, which are used to reduce noise in passenger cars, are in the upper part of the loss range spur gear pairs, which are limited to commercial vehicles due to their poorer noise comfort, are in the lower part of the loss range Corridor for torque ratio and efficiency in planetary gearsets, the stationary gear ratio i 0 {\displaystyle i_{0}} is formed via the planetary gears and thus by two meshes for reasons of simplification, the efficiency for both meshes together is commonly specified there the efficiencies η 0 {\displaystyle \eta _{0}} specified here are based on assumed efficiencies for the stationary ratio i 0 {\displaystyle i_{0}} of η 0 = 0.9800 {\displaystyle \eta _{0}=0.9800} (upper value) and η 0 = 0.9700 {\displaystyle \eta _{0}=0.9700} (lower value) for both interventions together The corresponding efficiency for single-meshing gear pairs is η 0 1 2 {\displaystyle {\eta _{0}}^{\tfrac {1}{2}}} at 0.9800 1 2 = 0.98995 {\displaystyle 0.9800^{\tfrac {1}{2}}=0.98995} (upper value) and 0.9700 1 2 = 0.98489 {\displaystyle 0.9700^{\tfrac {1}{2}}=0.98489} (lower value) Layout Input and output are on opposite sides Planetary gearset 2 (the outer Ravigneaux gearset) is on the input (turbine) side Input (turbine) shafts is, if actuated S1 or S2 Output shaft is R2 (ring gear of the outer Ravigneaux gearset) Total ratio span (total gear ratio/total transmission ratio) nominal ω 2 ; n ω 2 ; 1 = ω 2 ; n ω 2 ; 1 ω 2 ; n ω 2 ; 1 ω 2 ; 1 ω 2 ; n = 1 ω 2 ; 1 1 ω 2 ; n = ω t ω 2 ; 1 ω t ω 2 ; n = i 1 i n {\displaystyle {\frac {\omega _{2;n}}{\omega _{2;1}}}={\frac {\frac {\omega _{2;n}}{\omega _{2;1}\omega _{2;n}}}{\frac {\omega _{2;1}}{\omega _{2;1}\omega _{2;n}}}}={\frac {\frac {1}{\omega _{2;1}}}{\frac {1}{\omega _{2;n}}}}={\frac {\frac {\omega _{t}}{\omega _{2;1}}}{\frac {\omega _{t}}{\omega _{2;n}}}}={\frac {i_{1}}{i_{n}}}} A wider span enables the downspeeding when driving outside the city limits increase the climbing ability when driving over mountain passes or off-road or when towing a trailer Total ratio span (total gear ratio/total transmission ratio) effective ω 2 ; n m a x ( ω 2 ; 1 ; | ω 2 ; R | ) = m i n ( i 1 ; | i R | ) i n {\displaystyle {\frac {\omega _{2;n}}{max(\omega _{2;1};|\omega _{2;R}|)}}={\frac {min(i_{1};|i_{R}|)}{i_{n}}}} The span is only effective to the extent that the reverse gear ratio matches that of 1st gear see also Standard R:1 Digression Reverse gear is usually longer than 1st gear the effective span is therefore of central importance for describing the suitability of a transmission because in these cases, the nominal spread conveys a misleading picture which is only unproblematic for vehicles with high specific power Market participants Manufacturers naturally have no interest in specifying the effective span Users have not yet formulated the practical benefits that the effective span has for them The effective span has not yet played a role in research and teaching Contrary to its significance the effective span has therefore not yet been able to establish itself either in theory or in practice. End of digression Ratio span's center ( i 1 i n ) 1 2 {\displaystyle (i_{1}i_{n})^{\frac {1}{2}}} The center indicates the speed level of the transmission Together with the final drive ratio it gives the shaft speed level of the vehicle Average gear step ( ω 2 ; n ω 2 ; 1 ) 1 n − 1 = ( i 1 i n ) 1 n − 1 {\displaystyle \left({\frac {\omega _{2;n}}{\omega _{2;1}}}\right)^{\frac {1}{n-1}}=\left({\frac {i_{1}}{i_{n}}}\right)^{\frac {1}{n-1}}} There are n − 1 {\displaystyle n-1} gear steps between n {\displaystyle n} gears with decreasing step width the gears connect better to each other shifting comfort increases Sun 1: sun gear of gearset 1: inner Ravigneaux gearset Ring 1: ring gear of gearset 1: inner Ravigneaux gearset Sun 2: sun gear of gearset 2: outer Ravigneaux gearset Ring 2: ring gear of gearset 2: outer Ravigneaux gearset Standard 50:50 — 50 % is above and 50 % is below the average gear step — With steadily decreasing gear steps (yellow highlighted line Step) and a particularly large step from 1st to 2nd gear the lower half of the gear steps (between the small gears; rounded down, here the first 1) is always larger and the upper half of the gear steps (between the large gears; rounded up, here the last 1) is always smaller than the average gear step (cell highlighted yellow two rows above on the far right) lower half: smaller gear steps are a waste of possible ratios (red bold) upper half: larger gear steps are unsatisfactory (red bold) Standard R:1 — reverse and 1st gear have the same ratio — The ideal reverse gear has the same transmission ratio as 1st gear no impairment when maneuvering especially when towing a trailer a torque converter can only partially compensate for this deficiency Plus 11.11 % minus 10 % compared to 1st gear is good Plus 25 % minus 20 % is acceptable (red) Above this is unsatisfactory (bold) see also Total ratio span (total gear ratio/total transmission ratio) effective Standard 1:2 — gear step 1st to 2nd gear as small as possible — With continuously decreasing gear steps (yellow marked line Step) the largest gear step is the one from 1st to 2nd gear, which for a good speed connection and a smooth gear shift must be as small as possible A gear ratio of up to 1.6667 : 1 (5 : 3) is good Up to 1.7500 : 1 (7 : 4) is acceptable (red) Above is unsatisfactory (bold) From large to small gears (from right to left) Standard STEP — from large to small gears: steady and progressive increase in gear steps — Gear steps should increase: Δ Step (first green highlighted line Δ Step) is always greater than 1 As progressive as possible: Δ Step is always greater than the previous step Not progressively increasing is acceptable (red) Not increasing is unsatisfactory (bold) Standard SPEED — from small to large gears: steady increase in shaft speed difference — Shaft speed differences should increase: Δ Shaft Speed (second line marked in green Δ (Shaft) Speed) is always greater than the previous one 1 difference smaller than the previous one is acceptable (red) 2 consecutive ones are a waste of possible ratios (bold) Blocks R1 (ring gear of the inner Ravigneaux gearset) and S2 (sun gear of the outer Ravigneaux gearset) Supports link with freewheel · blocks R1 (ring gear of the inner Ravigneaux gearset) and S2 (sun gear of the outer Ravigneaux gearset) in one direction Blocks C1 and C2 (the common carrier of the compound Ravigneaux gearset) Couples S1 (sun gear of the inner Ravigneaux gearset) with the input (turbine) Couples S2 (sun gear of the outer Ravigneaux gearset) with the input (turbine) Ordinary noted For direct determination of the ratio Elementary noted Alternative representation for determining the transmission ratio Contains only operands With simple fractions of both central gears of a planetary gearset Or with the value 1 As a basis For reliable And traceable Determination of specific torque and efficiency

History

The all new Simpson planetary gearset was introduced in 1973 and was produced through 1990 and has been used in a wide range of cars from Alfa Romeo, BMW,^[1] Citroën, Peugeot, and Fiat.^[2] There are versions for longitudinal and transverse engines.

Specifications

Planetary gearset concept

Planetary gearset concept: gear ratio quality^[a]

In-Depth Analysis[b] With Assessment And Torque Ratio[c] And Efficiency Calculation[d]		Planetary Gearset: Teeth[e]		Count	Nomi- nal[f] Effec- tive[g]	Cen- ter[h]
		Simpson				Avg.[i]
Model Type	Version First Delivery	S1[j] R1[k]	S2[l] R2[m]	Brakes Clutches	Ratio Span	Gear Step[n]
Gear		R		1	2	3
Gear Ratio[b]		i R {\displaystyle {i_{R}}} [b]		i 1 {\displaystyle {i_{1}}} [b]	i 2 {\displaystyle {i_{2}}} [b]	i 3 {\displaystyle {i_{3}}} [b]
Step[n]		− i R i 1 {\displaystyle -{\frac {i_{R}}{i_{1}}}} [o]		i 1 i 1 {\displaystyle {\frac {i_{1}}{i_{1}}}}	i 1 i 2 {\displaystyle {\frac {i_{1}}{i_{2}}}} [p]	i 2 i 3 {\displaystyle {\frac {i_{2}}{i_{3}}}}
Δ Step[q][r]					i 1 i 2 : i 2 i 3 {\displaystyle {\tfrac {i_{1}}{i_{2}}}:{\tfrac {i_{2}}{i_{3}}}}
Shaft Speed		i 1 i R {\displaystyle {\frac {i_{1}}{i_{R}}}}		i 1 i 1 {\displaystyle {\frac {i_{1}}{i_{1}}}}	i 1 i 2 {\displaystyle {\frac {i_{1}}{i_{2}}}}	i 1 i 3 {\displaystyle {\frac {i_{1}}{i_{3}}}}
Δ Shaft Speed[s]		0 − i 1 i R {\displaystyle 0-{\tfrac {i_{1}}{i_{R}}}}		i 1 i 1 − 0 {\displaystyle {\tfrac {i_{1}}{i_{1}}}-0}	i 1 i 2 − i 1 i 1 {\displaystyle {\tfrac {i_{1}}{i_{2}}}-{\tfrac {i_{1}}{i_{1}}}}	i 1 i 3 − i 1 i 2 {\displaystyle {\tfrac {i_{1}}{i_{3}}}-{\tfrac {i_{1}}{i_{2}}}}
Torque Ratio[c]		μ R {\displaystyle \mu _{R}} [c]		μ 1 {\displaystyle \mu _{1}} [c]	μ 2 {\displaystyle \mu _{2}} [c]	μ 3 {\displaystyle \mu _{3}} [c]
Efficiency η n {\displaystyle \eta _{n}} [d]		μ R i R {\displaystyle {\frac {\mu _{R}}{i_{R}}}} [d]		μ 1 i 1 {\displaystyle {\frac {\mu _{1}}{i_{1}}}} [d]	μ 2 i 2 {\displaystyle {\frac {\mu _{2}}{i_{2}}}} [d]	μ 3 i 3 {\displaystyle {\frac {\mu _{3}}{i_{3}}}} [d]
3HP 22	320 N⋅m (236 lb⋅ft) 1973	35 73	35 73	3 2	2.4795 2.0857[g][o]	1.5746
						1.5746[n]
Gear		R		1	2	3
Gear Ratio[b]		−2.0857[o][g] − 2 1 {\displaystyle -{\tfrac {2}{1}}}		2.4795 181 73 {\displaystyle {\tfrac {181}{73}}}	1.4795[p] 108 73 {\displaystyle {\tfrac {108}{73}}}	1.0000 1 1 {\displaystyle {\tfrac {1}{1}}}
Step		0.8412[o]		1.0000	1.6759[p]	1.4795
Δ Step[q]					1.1328
Speed		-1.1888		1.0000	1.6759	2.4795
Δ Speed		1.1888		1.0000	0.6759	0.8035
Torque Ratio[c]		–2.0440 –2.0231		2.4303 2.4060	1.4699 1.4651	1.0000
Efficiency η n {\displaystyle \eta _{n}} [d]		0.9800 0.9700		0.9802 0.9704	0.9935 0.9903	1.0000
3HP 22	Small Engines 1973	35 73	41 73	3 2	2.7331 2.0857[g][o]	1.6532
						1.6532[n]
Gear		R		1	2	3
Gear Ratio[b]		−2.0857[o][g] − 73 35 {\displaystyle -{\tfrac {73}{35}}}		2.7331 6 , 983 2 , 555 {\displaystyle {\tfrac {6,983}{2,555}}}	1.5616[p] 114 73 {\displaystyle {\tfrac {114}{73}}}	1.0000 1 1 {\displaystyle {\tfrac {1}{1}}}
Step		0.7631[o]		1.0000	1.7501[p]	1.5616
Δ Step[q]					1.1207
Speed		-1.3103		1.0000	1.7501	2.7331
Δ Speed		1.3103		1.0000	0.7501	0.9829
Torque Ratio[c]		–2.0440 –2.0231		2.6755 2.6470	1.5504 1.5448	1.0000
Efficiency η n {\displaystyle \eta _{n}} [d]		0.9800 0.9700		0.9789 0.9685	0.9928 0.9892	1.0000
3HP 22	Porsche 944 1981	28 68	32 64	3 2	2.7143 2.4286[g][o]	1.6475
						1.6475[n]
Gear		R		1	2	3
Gear Ratio[b]		−2.4286[o][g] − 17 7 {\displaystyle -{\tfrac {17}{7}}}		2.7143 19 7 {\displaystyle {\tfrac {19}{7}}}	1.5000[p] 3 2 {\displaystyle {\tfrac {3}{2}}}	1.0000 1 1 {\displaystyle {\tfrac {1}{1}}}
Step		0.8947[o]		1.0000	1.8095[p]	1.5000
Δ Step[q]					1.2063
Speed		-1.1176		1.0000	1.8095	2.7143
Δ Speed		1.1176		1.0000	0.8095	0.9048
Torque Ratio[c]		–2.3800 –2.3557		2.6562 2.6275	1.4900 1.4850	1.0000
Efficiency η n {\displaystyle \eta _{n}} [d]		0.9800 0.9700		0.9786 0.9680	0.9933 0.9900	1.0000
Actuated shift elements
Brake A[t]					❶
Brake B[u]					❶	❶
Brake C[v]		❶		❶
Clutch D[w]				❶	❶	❶
Clutch E[x]		❶				❶
Geometric ratios: speed conversion
Gear Ratio[b] R & 2 Ordinary[y] Elementary Noted[z]	i R = − R 1 S 1 {\displaystyle i_{R}=-{\frac {R_{1}}{S_{1}}}}		i 2 = S 2 + R 2 R 2 {\displaystyle i_{2}={\frac {S_{2}+R_{2}}{R_{2}}}}
	i R = − R 1 S 1 {\displaystyle i_{R}=-{\tfrac {R_{1}}{S_{1}}}}		i 2 = 1 + S 2 R 2 {\displaystyle i_{2}=1+{\tfrac {S_{2}}{R_{2}}}}
Gear Ratio[b] 1 & 3 Ordinary[y] Elementary Noted[z]	i 1 = S 1 ( S 2 + R 2 ) + R 1 S 2 S 1 R 2 {\displaystyle i_{1}={\frac {S_{1}(S_{2}+R_{2})+R_{1}S_{2}}{S_{1}R_{2}}}}				i 3 = 1 1 {\displaystyle i_{3}={\frac {1}{1}}}
	i 1 = 1 + S 2 R 2 ( 1 + R 1 S 1 ) {\displaystyle i_{1}=1+{\tfrac {S_{2}}{R_{2}}}\left(1+{\tfrac {R_{1}}{S_{1}}}\right)}
Kinetic ratios: torque conversion
Torque Ratio[c] R & 2	μ R = − R 1 S 1 η 0 {\displaystyle \mu _{R}=-{\tfrac {R_{1}}{S_{1}}}\eta _{0}}		μ 2 = 1 + S 2 R 2 η 0 {\displaystyle \mu _{2}=1+{\tfrac {S_{2}}{R_{2}}}\eta _{0}}
Torque Ratio[c] 1 & 3	μ 1 = 1 + S 2 R 2 η 0 ( 1 + R 1 S 1 η 0 ) {\displaystyle \mu _{1}=1+{\tfrac {S_{2}}{R_{2}}}\eta _{0}\left(1+{\tfrac {R_{1}}{S_{1}}}\eta _{0}\right)}				μ 3 = 1 1 {\displaystyle \mu _{3}={\tfrac {1}{1}}}
Revised 14 January 2026 Nomenclature S n = {\displaystyle S_{n}=} sun gear: number of teeth R n = {\displaystyle R_{n}=} ring gear: number of teeth C n = {\displaystyle \color {gray}{C_{n}=}} carrier or planetary gear carrier (not needed) s n = {\displaystyle s_{n}=} sun gear: shaft speed r n = {\displaystyle r_{n}=} ring gear: shaft speed c n = {\displaystyle c_{n}=} carrier or planetary gear carrier: shaft speed With n = {\displaystyle n=} gear is i n = {\displaystyle i_{n}=} gear ratio or transmission ratio ω 1 ; n = ω t = {\displaystyle \omega _{1;n}=\omega _{t}=} shaft speed shaft 1: input (turbine) shaft ω 2 ; n = {\displaystyle \omega _{2;n}=} shaft speed shaft 2: output shaft T 1 ; n = T t = {\displaystyle T_{1;n}=T_{t}=} torque shaft 1: input (turbine) shaft T 2 ; n = {\displaystyle T_{2;n}=} torque shaft 2: output shaft μ n = {\displaystyle \mu _{n}=} torque ratio or torque conversion ratio η n = {\displaystyle \eta _{n}=} efficiency i 0 = {\displaystyle i_{0}=} stationary gear ratio η 0 = {\displaystyle \eta _{0}=} (assumed) stationary gear efficiency Gear ratio (transmission ratio) i n {\displaystyle i_{n}} — speed conversion — The gear ratio i n {\displaystyle i_{n}} is the ratio of input shaft speed ω 1 ; n {\displaystyle \omega _{1;n}} to output shaft speed ω 2 ; n {\displaystyle \omega _{2;n}} and therefore corresponds to the reciprocal of the shaft speeds i n = 1 ω 2 ; n ω 1 ; n = ω 1 ; n ω 2 ; n = ω t ω 2 ; n {\displaystyle i_{n}={\frac {1}{\frac {\omega _{2;n}}{\omega _{1;n}}}}={\frac {\omega _{1;n}}{\omega _{2;n}}}={\frac {\omega _{t}}{\omega _{2;n}}}} Torque ratio (torque conversion ratio) μ n {\displaystyle \mu _{n}} — torque conversion — The torque ratio μ n {\displaystyle \mu _{n}} is the ratio of output torque T 2 ; n {\displaystyle T_{2;n}} to input torque T 1 ; n {\displaystyle T_{1;n}} minus efficiency losses and therefore corresponds (apart from the efficiency losses) to the reciprocal of the shaft speeds too μ n = i n η n ; η 0 = ω 1 ; n η n ; η 0 ω 2 ; n = T 2 ; n η n ; η 0 T 1 ; n {\displaystyle \mu _{n}=i_{n}\eta _{n;\eta _{0}}={\frac {\omega _{1;n}\eta _{n;\eta _{0}}}{\omega _{2;n}}}={\frac {T_{2;n}\eta _{n;\eta _{0}}}{T_{1;n}}}} whereby η n ; η 0 {\displaystyle \eta _{n;\eta _{0}}} may vary from gear to gear according to the formulas listed in this table and 0 ≤ η n ; η 0 ≤ 1 {\displaystyle 0\leq \eta _{n;\eta _{0}}\leq 1} Efficiency The efficiency η n {\displaystyle \eta _{n}} is calculated from the torque ratio in relation to the gear ratio (transmission ratio) η n = μ n i n {\displaystyle \eta _{n}={\frac {\mu _{n}}{i_{n}}}} Power loss for single meshing gears is in the range of 1 % to 1.5 % helical gear pairs, which are used to reduce noise in passenger cars, are in the upper part of the loss range spur gear pairs, which are limited to commercial vehicles due to their poorer noise comfort, are in the lower part of the loss range Corridor for torque ratio and efficiency in planetary gearsets, the stationary gear ratio i 0 {\displaystyle i_{0}} is formed via the planetary gears and thus by two meshes for reasons of simplification, the efficiency for both meshes together is commonly specified there the efficiencies η 0 {\displaystyle \eta _{0}} specified here are based on assumed efficiencies for the stationary ratio i 0 {\displaystyle i_{0}} of η 0 = 0.9800 {\displaystyle \eta _{0}=0.9800} (upper value) and η 0 = 0.9700 {\displaystyle \eta _{0}=0.9700} (lower value) for both interventions together The corresponding efficiency for single-meshing gear pairs is η 0 1 2 {\displaystyle {\eta _{0}}^{\tfrac {1}{2}}} at 0.9800 1 2 = 0.98995 {\displaystyle 0.9800^{\tfrac {1}{2}}=0.98995} (upper value) and 0.9700 1 2 = 0.98489 {\displaystyle 0.9700^{\tfrac {1}{2}}=0.98489} (lower value) Layout Input and output are on opposite sides Planetary gearset 1 is on the input (turbine) side Input (turbine) shaft is, if actuated, S1 or R2 Output shaft is R1 Total ratio span (total gear ratio/total transmission ratio) nominal ω 2 ; n ω 2 ; 1 = ω 2 ; n ω 2 ; 1 ω 2 ; n ω 2 ; 1 ω 2 ; 1 ω 2 ; n = 1 ω 2 ; 1 1 ω 2 ; n = ω t ω 2 ; 1 ω t ω 2 ; n = i 1 i n {\displaystyle {\frac {\omega _{2;n}}{\omega _{2;1}}}={\frac {\frac {\omega _{2;n}}{\omega _{2;1}\omega _{2;n}}}{\frac {\omega _{2;1}}{\omega _{2;1}\omega _{2;n}}}}={\frac {\frac {1}{\omega _{2;1}}}{\frac {1}{\omega _{2;n}}}}={\frac {\frac {\omega _{t}}{\omega _{2;1}}}{\frac {\omega _{t}}{\omega _{2;n}}}}={\frac {i_{1}}{i_{n}}}} A wider span enables the downspeeding when driving outside the city limits increase the climbing ability when driving over mountain passes or off-road or when towing a trailer Total ratio span (total gear ratio/total transmission ratio) effective ω 2 ; n m a x ( ω 2 ; 1 ; | ω 2 ; R | ) = m i n ( i 1 ; | i R | ) i n {\displaystyle {\frac {\omega _{2;n}}{max(\omega _{2;1};|\omega _{2;R}|)}}={\frac {min(i_{1};|i_{R}|)}{i_{n}}}} The span is only effective to the extent that the reverse gear ratio matches that of 1st gear see also Standard R:1 Digression Reverse gear is usually longer than 1st gear the effective span is therefore of central importance for describing the suitability of a transmission because in these cases, the nominal spread conveys a misleading picture which is only unproblematic for vehicles with high specific power Market participants Manufacturers naturally have no interest in specifying the effective span Users have not yet formulated the practical benefits that the effective span has for them The effective span has not yet played a role in research and teaching Contrary to its significance the effective span has therefore not yet been able to establish itself either in theory or in practice. End of digression Ratio span's center ( i 1 i n ) 1 2 {\displaystyle (i_{1}i_{n})^{\frac {1}{2}}} The center indicates the speed level of the transmission Together with the final drive ratio it gives the shaft speed level of the vehicle Average gear step ( ω 2 ; n ω 2 ; 1 ) 1 n − 1 = ( i 1 i n ) 1 n − 1 {\displaystyle \left({\frac {\omega _{2;n}}{\omega _{2;1}}}\right)^{\frac {1}{n-1}}=\left({\frac {i_{1}}{i_{n}}}\right)^{\frac {1}{n-1}}} There are n − 1 {\displaystyle n-1} gear steps between n {\displaystyle n} gears with decreasing step width the gears connect better to each other shifting comfort increases Sun 1: sun gear of gearset 1: inner Ravigneaux gearset Ring 1: ring gear of gearset 1: inner Ravigneaux gearset Sun 2: sun gear of gearset 2: outer Ravigneaux gearset Ring 2: ring gear of gearset 2: outer Ravigneaux gearset Standard 50:50 — 50 % is above and 50 % is below the average gear step — With steadily decreasing gear steps (yellow highlighted line Step) and a particularly large step from 1st to 2nd gear the lower half of the gear steps (between the small gears; rounded down, here the first 1) is always larger and the upper half of the gear steps (between the large gears; rounded up, here the last 1) is always smaller than the average gear step (cell highlighted yellow two rows above on the far right) lower half: smaller gear steps are a waste of possible ratios (red bold) upper half: larger gear steps are unsatisfactory (red bold) Standard R:1 — reverse and 1st gear have the same ratio — The ideal reverse gear has the same transmission ratio as 1st gear no impairment when maneuvering especially when towing a trailer a torque converter can only partially compensate for this deficiency Plus 11.11 % minus 10 % compared to 1st gear is good Plus 25 % minus 20 % is acceptable (red) Above this is unsatisfactory (bold) see also Total ratio span (total gear ratio/total transmission ratio) effective Standard 1:2 — gear step 1st to 2nd gear as small as possible — With continuously decreasing gear steps (yellow marked line Step) the largest gear step is the one from 1st to 2nd gear, which for a good speed connection and a smooth gear shift must be as small as possible A gear ratio of up to 1.6667 : 1 (5 : 3) is good Up to 1.7500 : 1 (7 : 4) is acceptable (red) Above is unsatisfactory (bold) From large to small gears (from right to left) Standard STEP — from large to small gears: steady and progressive increase in gear steps — Gear steps should increase: Δ Step (first green highlighted line Δ Step) is always greater than 1 As progressive as possible: Δ Step is always greater than the previous step Not progressively increasing is acceptable (red) Not increasing is unsatisfactory (bold) Standard SPEED — from small to large gears: steady increase in shaft speed difference — Shaft speed differences should increase: Δ Shaft Speed (second line marked in green Δ (Shaft) Speed) is always greater than the previous one 1 difference smaller than the previous one is acceptable (red) 2 consecutive ones are a waste of possible ratios (bold) Blocks S1 Supports link with freewheel · blocks S1 in one direction Blocks C1 Couples S1 with the input (turbine) Couples R2 with the input (turbine) Ordinary noted For direct determination of the gear ratio Elementary noted Alternative representation for determining the transmission ratio Contains only operands With simple fractions of both central gears of a planetary gearset Or with the value 1 As a basis For reliable And traceable Determination of the torque conversion ratio and efficiency