ZF 9HP transmission

Improved fuel economy

The main objective in replacing the predecessor model was to improve vehicle fuel economy with extra speeds and a wider gear span to allow the engine speed level to be lowered (downspeeding), which is a decisive factor in improving energy efficiency and thus reducing fuel consumption. In addition, the lower engine speed level improves the noise-vibration-harshness comfort and the exterior noise is reduced. ZF claims that it is able to save an average of 16% in fuel compared with current 6-speed automatic transmissions.^[1]

Reduced manufacturing complexity

In order to avoid a further increase in manufacturing complexity while expanding the number of gear ratios, ZF switched from the conventional design method—in which the planetary gearset concept was limited to a purely serial or in-line power flow—to a more modern design method that utilizes a planetary gearset concept with combined parallel and serial power flow. This was only possible thanks to computer-aided design and has resulted in a globally patented gearset concept. The resulting progress is reflected in a better ratio of the number of gears to the number of components used compared to existing layouts. The 8HP has become the new reference standard (benchmark) for automatic transmissions.

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Planetary gearset concept: manufacturing complexity^[a]

With Assessment	Output: Gear Ratios	Innovation Elasticity[b] Δ Output : Δ Input	Input: Main Components
			Total	Gearsets	Brakes	Clutches
9HP Ref. Object	${\displaystyle n_{O1}}$ ${\displaystyle n_{O2}}$	Topic[b]	${\displaystyle n_{I}=n_{G}+}$ ${\displaystyle n_{B}+n_{C}}$	${\displaystyle n_{G1}}$ ${\displaystyle n_{G2}}$	${\displaystyle n_{B1}}$ ${\displaystyle n_{B2}}$	${\displaystyle n_{C1}}$ ${\displaystyle n_{C2}}$
Δ Number	${\displaystyle n_{O1}-n_{O2}}$		${\displaystyle n_{I1}-n_{I2}}$	${\displaystyle n_{G1}-n_{G2}}$	${\displaystyle n_{B1}-n_{B2}}$	${\displaystyle n_{C1}-n_{C2}}$
Relative Δ	Δ Output ${\displaystyle {\tfrac {n_{O1}-n_{O2}}{n_{O2}}}}$	${\displaystyle {\tfrac {n_{O1}-n_{O2}}{n_{O2}}}:{\tfrac {n_{I1}-n_{I2}}{n_{I2}}}}$ ${\displaystyle ={\tfrac {n_{O1}-n_{O2}}{n_{O2}}}\cdot {\tfrac {n_{I2}}{n_{I1}-n_{I2}}}}$	Δ Input ${\displaystyle {\tfrac {n_{I1}-n_{I2}}{n_{I2}}}}$	${\displaystyle {\tfrac {n_{G1}-n_{G2}}{n_{G2}}}}$	${\displaystyle {\tfrac {n_{B1}-n_{B2}}{n_{B2}}}}$	${\displaystyle {\tfrac {n_{C1}-n_{C2}}{n_{C2}}}}$
9HP 4HP[c]	9[d] 4[d]	Progress[b]	10 7	4 2[e]	3 2	3 3
Δ Number	5		3	2	1	0
Relative Δ	1.250 ${\displaystyle {\tfrac {5}{4}}}$	2.917[b] ${\displaystyle {\tfrac {5}{4}}:{\tfrac {3}{7}}={\tfrac {5}{4}}\cdot {\tfrac {7}{3}}={\tfrac {35}{12}}}$	0.429 ${\displaystyle {\tfrac {3}{7}}}$	1.000 ${\displaystyle {\tfrac {2}{2}}}$	0.500 ${\displaystyle {\tfrac {1}{2}}}$	0.000 ${\displaystyle {\tfrac {0}{3}}}$
9HP Aisin[f]	9[d] 4[d]	Progress[b]	10 8	4 3[g]	3 2	3 3
Δ Number	3		2	1	1	0
Relative Δ	0.500 ${\displaystyle {\tfrac {3}{6}}}$	2.000[b] ${\displaystyle {\tfrac {3}{6}}:{\tfrac {2}{8}}={\tfrac {1}{2}}\cdot {\tfrac {4}{1}}={\tfrac {2}{1}}}$	0.250 ${\displaystyle {\tfrac {2}{8}}}$	0.333 ${\displaystyle {\tfrac {1}{3}}}$	0.333 ${\displaystyle {\tfrac {-1}{3}}}$	0.000 ${\displaystyle {\tfrac {0}{3}}}$
9HP 8HP[h]	9[d] 8[d]	Current Market Position[b]	10 9	4 4	3 2	3 3
Δ Number	1		1	0	1	0
Relative Δ	0.125 ${\displaystyle {\tfrac {1}{8}}}$	1.125[b] ${\displaystyle {\tfrac {1}{8}}:{\tfrac {1}{9}}={\tfrac {1}{8}}\cdot {\tfrac {9}{1}}={\tfrac {9}{8}}}$	0.111 ${\displaystyle {\tfrac {1}{9}}}$	0.000 ${\displaystyle {\tfrac {0}{4}}}$	0.500 ${\displaystyle {\tfrac {1}{2}}}$	0.000 ${\displaystyle {\tfrac {0}{3}}}$
W9A 3-Speed[i]	9[d] 3[d]	Historical Market Position[b]	10 7	4 2	3 3	3 2
Δ Number	6		3	2	0	1
Relative Δ	2.000 ${\displaystyle {\tfrac {6}{3}}}$	4.667[b] ${\displaystyle {\tfrac {6}{3}}:{\tfrac {3}{7}}={\tfrac {2}{1}}\cdot {\tfrac {7}{3}}={\tfrac {14}{3}}}$	0.429 ${\displaystyle {\tfrac {3}{7}}}$	1.000 ${\displaystyle {\tfrac {1}{1}}}$	0.000 ${\displaystyle {\tfrac {0}{3}}}$	0.500 ${\displaystyle {\tfrac {1}{2}}}$
[a] Progress increases cost-effectiveness and is reflected in the ratio of forward gears to main components. It depends on the power flow: parallel: using the two degrees of freedom of planetary gearsets to increase the number of gears with unchanged number of components serial: in-line combined planetary gearsets without using the two degrees of freedom to increase the number of gears a corresponding increase in the number of components is unavoidable [b] Innovation elasticity classifies progress and market position Automobile manufacturers drive forward technical developments primarily in order to remain competitive or to achieve or defend technological leadership. This technical progress has therefore always been subject to economic constraints Only innovations whose relative additional benefit is greater than the relative additional resource input, i.e. whose economic elasticity is greater than 1, are considered for realization The required innovation elasticity of an automobile manufacturer depends on its expected return on investment. The basic assumption that the relative additional benefit must be at least twice as high as the relative additional resource input helps with orientation negative, if the output increases and the input decreases, is perfect 2 or above is good 1 or above is acceptable (red) below this is unsatisfactory (bold) [c] Direct predecessor To reflect the progress of the specific model change [d] plus 1 reverse gear [e] combined as a compound Ravigneaux gearset [f] Market predecessor As the reference standard for transverse engine vehicles, the Aisins AWTF-80 SC reflects the progress for car manufacturer and customer at that time [g] of which two gearstets are combined as a compound Ravigneaux gearset [h] Current reference standard (benchmark) The 8HP has become the new reference standard (benchmark) for automatic transmissions. Although designed for longitudinal installation, it is nevertheless the industry standard. [i] Historical reference standard (benchmark) 3-speed transmissions with torque converters have established the modern market for automatic transmissions and thus made it possible in the first place, as this design proved to be a particularly successful compromise between cost and performance It became the archetype and dominated the world market for around 3 decades, setting the standard for automatic transmissions. It was only when fuel consumption became the focus of interest that this design reached its limits, which is why it has now completely disappeared from the market What has remained is the orientation that it offers as a reference standard (point of reference, benchmark) for this market for determining progressiveness and thus the market position of all other, later designs All transmission variants consist of 7 main components Typical examples are Turbo-Hydramatic from GM Cruise-O-Matic from Ford TorqueFlite from Chrysler Detroit Gear from BorgWarner for Studebaker BW-35 from BorgWarner and as T35 from Aisin 3N 71 from Nissan/Jatco 3 HP from ZF Friedrichshafen W3A 040 and W3B 050 from Mercedes-Benz

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Quality

The 9HP is only 0.24 inches (6 mm) longer than, and weighs 16.5 lbs (7.5 kg) less than, the outgoing six-speed transmission. The compact packaging is achieved by using a number of innovative design features: a new compact hydraulic vane-type pump, two patented dog clutches,^[5] which replace bulkier conventional clutch packs, and a nested gear set.^[2]

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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]
										Avg.[i]
Model Type	Version First Delivery · Weight			S4[j] R4[k]	S3[l] R3[m]	S2[n] R2[o]	S1[p] R1[q]	Brakes Clutches	Ratio Span	Gear Step[r]
Gear	R	1	2	3	4	5	6	7	8	9
Gear Ratio[b]	${\displaystyle {i_{R}}}$[b]	${\displaystyle {i_{1}}}$[b]	${\displaystyle {i_{2}}}$[b]	${\displaystyle {i_{3}}}$[b]	${\displaystyle {i_{4}}}$[b]	${\displaystyle {i_{5}}}$[b]	${\displaystyle {i_{6}}}$[b]	${\displaystyle {i_{7}}}$[b]	${\displaystyle {i_{8}}}$[b]	${\displaystyle {i_{9}}}$[b]
Step[r]	${\displaystyle -{\frac {i_{R}}{i_{1}}}}$[s]	${\displaystyle {\frac {i_{1}}{i_{1}}}}$	${\displaystyle {\frac {i_{1}}{i_{2}}}}$[t]	${\displaystyle {\frac {i_{2}}{i_{3}}}}$	${\displaystyle {\frac {i_{3}}{i_{4}}}}$	${\displaystyle {\frac {i_{4}}{i_{5}}}}$	${\displaystyle {\frac {i_{5}}{i_{6}}}}$	${\displaystyle {\frac {i_{6}}{i_{7}}}}$	${\displaystyle {\frac {i_{7}}{i_{8}}}}$	${\displaystyle {\frac {i_{8}}{i_{9}}}}$
Δ Step[u][v]			${\displaystyle {\tfrac {i_{1}}{i_{2}}}:{\tfrac {i_{2}}{i_{3}}}}$	${\displaystyle {\tfrac {i_{2}}{i_{3}}}:{\tfrac {i_{3}}{i_{4}}}}$	${\displaystyle {\tfrac {i_{3}}{i_{4}}}:{\tfrac {i_{4}}{i_{5}}}}$	${\displaystyle {\tfrac {i_{4}}{i_{5}}}:{\tfrac {i_{5}}{i_{6}}}}$	${\displaystyle {\tfrac {i_{5}}{i_{6}}}:{\tfrac {i_{6}}{i_{7}}}}$	${\displaystyle {\tfrac {i_{6}}{i_{7}}}:{\tfrac {i_{7}}{i_{8}}}}$	${\displaystyle {\tfrac {i_{7}}{i_{8}}}:{\tfrac {i_{8}}{i_{9}}}}$
Shaft Speed	${\displaystyle {\frac {i_{1}}{i_{R}}}}$	${\displaystyle {\frac {i_{1}}{i_{1}}}}$	${\displaystyle {\frac {i_{1}}{i_{2}}}}$	${\displaystyle {\frac {i_{1}}{i_{3}}}}$	${\displaystyle {\frac {i_{1}}{i_{4}}}}$	${\displaystyle {\frac {i_{1}}{i_{5}}}}$	${\displaystyle {\frac {i_{1}}{i_{6}}}}$	${\displaystyle {\frac {i_{1}}{i_{7}}}}$	${\displaystyle {\frac {i_{1}}{i_{8}}}}$	${\displaystyle {\frac {i_{1}}{i_{9}}}}$
Δ Shaft Speed[w]	${\displaystyle 0-{\tfrac {i_{1}}{i_{R}}}}$	${\displaystyle {\tfrac {i_{1}}{i_{1}}}-0}$	${\displaystyle {\tfrac {i_{1}}{i_{2}}}-{\tfrac {i_{1}}{i_{1}}}}$	${\displaystyle {\tfrac {i_{1}}{i_{3}}}-{\tfrac {i_{1}}{i_{2}}}}$	${\displaystyle {\tfrac {i_{1}}{i_{4}}}-{\tfrac {i_{1}}{i_{3}}}}$	${\displaystyle {\tfrac {i_{1}}{i_{5}}}-{\tfrac {i_{1}}{i_{4}}}}$	${\displaystyle {\tfrac {i_{1}}{i_{6}}}-{\tfrac {i_{1}}{i_{5}}}}$	${\displaystyle {\tfrac {i_{1}}{i_{7}}}-{\tfrac {i_{1}}{i_{6}}}}$	${\displaystyle {\tfrac {i_{1}}{i_{8}}}-{\tfrac {i_{1}}{i_{7}}}}$	${\displaystyle {\tfrac {i_{1}}{i_{9}}}-{\tfrac {i_{1}}{i_{8}}}}$
Torque Ratio[c]	${\displaystyle \mu _{R}}$[c]	${\displaystyle \mu _{1}}$[c]	${\displaystyle \mu _{2}}$[c]	${\displaystyle \mu _{3}}$[c]	${\displaystyle \mu _{4}}$[c]	${\displaystyle \mu _{5}}$[c]	${\displaystyle \mu _{6}}$[c]	${\displaystyle \mu _{7}}$[c]	${\displaystyle \mu _{8}}$[c]	${\displaystyle \mu _{9}}$[c]
Efficiency ${\displaystyle \eta _{n}}$[d]	${\displaystyle {\frac {\mu _{R}}{i_{R}}}}$[d]	${\displaystyle {\frac {\mu _{1}}{i_{1}}}}$[d]	${\displaystyle {\frac {\mu _{2}}{i_{2}}}}$[d]	${\displaystyle {\frac {\mu _{3}}{i_{3}}}}$[d]	${\displaystyle {\frac {\mu _{4}}{i_{4}}}}$[d]	${\displaystyle {\frac {\mu _{5}}{i_{5}}}}$[d]	${\displaystyle {\frac {\mu _{6}}{i_{6}}}}$[d]	${\displaystyle {\frac {\mu _{7}}{i_{7}}}}$[d]	${\displaystyle {\frac {\mu _{8}}{i_{8}}}}$[d]	${\displaystyle {\frac {\mu _{9}}{i_{9}}}}$[d]
9HP 28 9HP 48	280 Nm[x] · 2013 · 78 kg (172 lb) 450 Nm[y] · 2013 · 86 kg (190 lb) 480 Nm[z] · 2013 · 86 kg (190 lb)			42 110	42 110	91[aa] 133	42 86	3[ab] 3[ac]	9.8085 7.9402 [g][s]	1.5007
										1.3303[r]
Gear	R	1	2	3	4	5	6	7	8	9
Gear Ratio[b]	−3.8049 [s][g] ${\displaystyle -{\tfrac {3,142,144}{825,825}}}$	4.7001 ${\displaystyle {\tfrac {184,832}{39,325}}}$	2.8419 ${\displaystyle {\tfrac {369,664}{130,075}}}$	1.9094 ${\displaystyle {\tfrac {5,776}{3,025}}}$	1.3818 [v] ${\displaystyle {\tfrac {76}{55}}}$	1.0000 ${\displaystyle {\tfrac {1}{1}}}$	0.8081 [w] ${\displaystyle {\tfrac {34,048}{42,133}}}$	0.6995 [v][w] ${\displaystyle {\tfrac {6,272}{8,967}}}$	0.5802 [v] ${\displaystyle {\tfrac {76}{131}}}$	0.4792 ${\displaystyle {\tfrac {2,176}{4,541}}}$
Step	0.8095[s]	1.0000	1.6538	1.4884	1.3818	1.3818	1.2375	1.1553	1.2056	1.2107
Δ Step[u]			1.1112	1.0771	1.0000[v]	1.1167	1.0711	0.9583[v]	0.9958[v]
Speed	-1.2353	1.0000	1.6538	2.4615	3.4014	4.7001	5.5816	6.7197	8.1015	9.8085
Δ Speed	1.2353	1.0000	0.6538	0.8077	0.9399	1.2987	1.1161[w]	0.9035[w]	1.3818	1.7066
Torque Ratio[c]	-3.5391 –3.4099	4.5931 4.5402	2.7922 2.7675	1.8884 1.8779	1.3742 1.3704	1.0000	0.8005 0.7966	0.6904 0.6857	0.5717 0.5673	0.4653 0.4582
Efficiency ${\displaystyle \eta _{n}}$[d]	0.9302 0.8962	0.9772 0.9660	0.9825 0.9738	0.9890 0.9835	0.9945 0.9917	1.0000	0.9906 0.9857	0.9870 0.9803	0.9854 0.9779	0.9710 0.9561
Actuated shift elements[ad]
Brake A[ae]	❶	❶	❶	❶	❶
Brake C[af]			❶				❶		❶
Brake D[ag]	❶	❶						❶	❶	❶
Clutch B[ah]	❶			❶		❶				❶
Clutch E[ai]					❶	❶	❶	❶	❶	❶
Clutch F[aj]		❶	❶	❶	❶	❶	❶	❶
Geometric ratios: speed conversion
Gear Ratio[b] R & 1 Ordinary[ak] Elementary Noted[al]	${\displaystyle i_{R}={\frac {(S_{1}S_{2}-R_{1}R_{2})(S_{3}+R_{3})(S_{4}+R_{4})}{S_{1}S_{2}R_{3}R_{4}}}}$					${\displaystyle i_{1}={\frac {(S_{2}+R_{2})(S_{3}+R_{3})(S_{4}+R_{4})}{S_{2}R_{3}R_{4}}}}$
	${\displaystyle i_{R}=\left(1-{\tfrac {R_{1}R_{2}}{S_{1}S_{2}}}\right)\left(1+{\tfrac {S_{3}}{R_{3}}}\right)\left(1+{\tfrac {S_{4}}{R_{4}}}\right)}$					${\displaystyle i_{1}=\left(1+{\tfrac {R_{2}}{S_{2}}}\right)\left(1+{\tfrac {S_{3}}{R_{3}}}\right)\left(1+{\tfrac {S_{4}}{R_{4}}}\right)}$
Gear Ratio[b] 2 & 3 Ordinary[ak] Elementary Noted[al]	${\displaystyle i_{2}={\frac {(S_{1}+R_{1})(S_{3}+R_{3})(S_{4}+R_{4})}{R_{1}R_{3}R_{4}}}}$					${\displaystyle i_{3}={\frac {(S_{3}+R_{3})(S_{4}+R_{4})}{R_{3}R_{4}}}}$
	${\displaystyle i_{2}=\left(1+{\tfrac {S_{1}}{R_{1}}}\right)\left(1+{\tfrac {S_{3}}{R_{3}}}\right)\left(1+{\tfrac {S_{4}}{R_{4}}}\right)}$					${\displaystyle i_{3}=\left(1+{\tfrac {S_{3}}{R_{3}}}\right)\left(1+{\tfrac {S_{4}}{R_{4}}}\right)}$
Gear Ratio[b] 5–7 Ordinary[ak] Elementary Noted[al]	${\displaystyle i_{5}={\frac {1}{1}}}$	${\displaystyle i_{6}={\frac {S_{3}(S_{1}+R_{1})(S_{4}+R_{4})}{S_{3}(S_{1}+R_{1})(S_{4}+R_{4})+S_{1}R_{3}S_{4}}}}$					${\displaystyle i_{7}={\frac {S_{3}(S_{2}+R_{2})(S_{4}+R_{4})}{S_{3}(S_{2}+R_{2})(S_{4}+R_{4})+R_{2}R_{3}S_{4}}}}$
		${\displaystyle i_{6}={\tfrac {1}{1+{\tfrac {\tfrac {R_{3}}{S_{3}}}{\left(1+{\tfrac {R_{1}}{S_{1}}}\right)\left(1+{\tfrac {R_{4}}{S_{4}}}\right)}}}}}$					${\displaystyle i_{7}={\tfrac {1}{1+{\tfrac {\tfrac {R_{3}}{S_{3}}}{\left(1+{\tfrac {S_{2}}{R_{2}}}\right)\left(1+{\tfrac {R_{4}}{S_{4}}}\right)}}}}}$
Gear Ratio[b] 4 & 8 & 9 Ordinary[ak] Elementary Noted[al]	${\displaystyle i_{4}={\frac {S_{4}+R_{4}}{R_{4}}}}$		${\displaystyle i_{8}={\frac {S_{3}(S_{4}+R_{4})}{S_{4}(S_{3}+R_{3})+S_{3}R_{4}}}}$			${\displaystyle i_{9}={\frac {S_{3}(R_{1}R_{2}-S_{1}S_{2})(S4+R_{4})}{S_{3}(R_{1}R_{2}-S_{1}S_{2})(S_{4}+R_{4})+R_{1}R_{2}R_{3}S_{4}}}}$
	${\displaystyle i_{4}=1+{\tfrac {S_{4}}{R_{4}}}}$		${\displaystyle i_{8}={\tfrac {1}{1+{\tfrac {\tfrac {R_{3}}{S_{3}}}{1+{\tfrac {R_{4}}{S_{4}}}}}}}}$			${\displaystyle i_{9}={\tfrac {1}{1+{\tfrac {\tfrac {R_{3}}{S_{3}}}{\left(1-{\tfrac {S_{1}S_{2}}{R_{1}R_{2}}}\right)\left(1+{\tfrac {R_{4}}{S_{4}}}\right)}}}}}$
Kinetic ratios: torque conversion
Torque Ratio[c] R & 1	${\displaystyle \mu _{R}=\left(1-{\tfrac {R_{1}R_{2}}{S_{1}S_{2}}}{\eta _{0}}^{2}\right)\left(1+{\tfrac {S_{3}}{R_{3}}}\eta _{0}\right)\left(1+{\tfrac {S_{4}}{R_{4}}}\eta _{0}\right)}$					${\displaystyle \mu _{1}=\left(1+{\tfrac {R_{2}}{S_{2}}}\eta _{0}\right)\left(1+{\tfrac {S_{3}}{R_{3}}}\eta _{0}\right)\left(1+{\tfrac {S_{4}}{R_{4}}}\eta _{0}\right)}$
Torque Ratio[c] 2 & 3	${\displaystyle \mu _{2}=\left(1+{\tfrac {S_{1}}{R_{1}}}\eta _{0}\right)\left(1+{\tfrac {S_{3}}{R_{3}}}\eta _{0}\right)\left(1+{\tfrac {S_{4}}{R_{4}}}\eta _{0}\right)}$					${\displaystyle \mu _{3}=\left(1+{\tfrac {S_{3}}{R_{3}}}\eta _{0}\right)\left(1+{\tfrac {S_{4}}{R_{4}}}\eta _{0}\right)}$
Torque Ratio[c] 5–7	${\displaystyle \mu _{5}={\tfrac {1}{1}}}$	${\displaystyle \mu _{6}={\tfrac {1}{1+{\tfrac {{\tfrac {R_{3}}{S_{3}}}\cdot {\tfrac {1}{\eta _{0}}}}{\left(1+{\tfrac {R_{1}}{S_{1}}}\eta _{0}\right)\left(1+{\tfrac {R_{4}}{S_{4}}}\eta _{0}\right)}}}}}$					${\displaystyle \mu _{7}={\tfrac {1}{1+{\tfrac {{\tfrac {R_{3}}{S_{3}}}\cdot {\tfrac {1}{\eta _{0}}}}{\left(1+{\tfrac {S_{2}}{R_{2}}}\eta _{0}\right)\left(1+{\tfrac {R_{4}}{S_{4}}}\eta _{0}\right)}}}}}$
Torque Ratio[c] 4 & 8 & 9	${\displaystyle \mu _{4}=1+{\tfrac {S_{4}}{R_{4}}}\eta _{0}}$		${\displaystyle \mu _{8}={\tfrac {1}{1+{\tfrac {{\tfrac {R_{3}}{S_{3}}}\cdot {\tfrac {1}{\eta _{0}}}}{1+{\tfrac {R_{4}}{S_{4}}}\eta _{0}}}}}}$			${\displaystyle \mu _{9}={\tfrac {1}{1+{\tfrac {{\tfrac {R_{3}}{S_{3}}}\cdot {\tfrac {1}{\eta _{0}}}}{\left(1-{\tfrac {S_{1}S_{2}}{R_{1}R_{2}}}\cdot {\tfrac {1}{{\eta _{0}}^{2}}}\right)\left(1+{\tfrac {R_{4}}{S_{4}}}\eta _{0}\right)}}}}}$
[a] Revised 14 January 2026 Nomenclature ${\displaystyle S_{n}=}$ sun gear: number of teeth ${\displaystyle R_{n}=}$ ring gear: number of teeth ${\displaystyle \color {gray}{C_{n}=}}$ carrier or planetary gear carrier (not needed) ${\displaystyle s_{n}=}$ sun gear: shaft speed ${\displaystyle r_{n}=}$ ring gear: shaft speed ${\displaystyle c_{n}=}$ carrier or planetary gear carrier: shaft speed With ${\displaystyle n=}$ gear is ${\displaystyle i_{n}=}$ gear ratio or transmission ratio ${\displaystyle \omega _{1;n}=\omega _{t}=}$ shaft speed shaft 1: input (turbine) shaft ${\displaystyle \omega _{2;n}=}$ shaft speed shaft 2: output shaft ${\displaystyle T_{1;n}=T_{t}=}$ torque shaft 1: input (turbine) shaft ${\displaystyle T_{2;n}=}$ torque shaft 2: output shaft ${\displaystyle \mu _{n}=}$ torque ratio or torque conversion ratio ${\displaystyle \eta _{n}=}$ efficiency ${\displaystyle i_{0}=}$ stationary gear ratio ${\displaystyle \eta _{0}=}$ (assumed) stationary gear efficiency [b] Gear ratio (transmission ratio) ${\displaystyle i_{n}}$ — speed conversion — The gear ratio ${\displaystyle i_{n}}$ is the ratio of input shaft speed ${\displaystyle \omega _{1;n}}$ to output shaft speed ${\displaystyle \omega _{2;n}}$ and therefore corresponds to the reciprocal of the shaft speeds ${\displaystyle i_{n}={\frac {1}{\frac {\omega _{2;n}}{\omega _{1;n}}}}={\frac {\omega _{1;n}}{\omega _{2;n}}}={\frac {\omega _{t}}{\omega _{2;n}}}}$ [c] Torque ratio (torque conversion ratio) ${\displaystyle \mu _{n}}$ — torque conversion — The torque ratio ${\displaystyle \mu _{n}}$ is the ratio of output torque ${\displaystyle T_{2;n}}$ to input torque ${\displaystyle T_{1;n}}$ minus efficiency losses and therefore corresponds (apart from the efficiency losses) to the reciprocal of the shaft speeds too ${\displaystyle \mu _{n}=i_{n}\eta _{n;\eta _{0}}={\frac {\omega _{1;n}\eta _{n;\eta _{0}}}{\omega _{2;n}}}={\frac {T_{2;n}\eta _{n;\eta _{0}}}{T_{1;n}}}}$ whereby ${\displaystyle \eta _{n;\eta _{0}}}$ may vary from gear to gear according to the formulas listed in this table and ${\displaystyle 0\leq \eta _{n;\eta _{0}}\leq 1}$ [d] Efficiency The efficiency ${\displaystyle \eta _{n}}$ is calculated from the torque ratio in relation to the gear ratio (transmission ratio) ${\displaystyle \eta _{n}={\frac {\mu _{n}}{i_{n}}}}$ 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 ${\displaystyle i_{0}}$ 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 ${\displaystyle \eta _{0}}$ specified here are based on assumed efficiencies for the stationary ratio ${\displaystyle i_{0}}$ of ${\displaystyle \eta _{0}=0.9800}$ (upper value) and ${\displaystyle \eta _{0}=0.9700}$ (lower value) for both interventions together The corresponding efficiency for single-meshing gear pairs is ${\displaystyle {\eta _{0}}^{\tfrac {1}{2}}}$ at ${\displaystyle 0.9800^{\tfrac {1}{2}}=0.98995}$ (upper value) and ${\displaystyle 0.9700^{\tfrac {1}{2}}=0.98489}$ (lower value) [e] Layout Input and output are on the same side Planetary gearset 4 is on the input (turbine) side Input shafts are, if actuated, S1, R1 + S3, and C3 + R4 Output shaft is C4 [f] Total ratio span (total gear ratio/total transmission ratio) nominal ${\displaystyle {\frac {\omega _{2;n}}{\omega _{2;1}}}={\frac {\frac {\omega _{2;n}}{\omega _{2;1}\omega _{2;n}}}{\frac {\omega _{2;1}}{\omega _{2;1}\omega _{2;n}}}}={\frac {\frac {1}{\omega _{2;1}}}{\frac {1}{\omega _{2;n}}}}={\frac {\frac {\omega _{t}}{\omega _{2;1}}}{\frac {\omega _{t}}{\omega _{2;n}}}}={\frac {i_{1}}{i_{n}}}}$ 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 [g] Total ratio span (total gear ratio/total transmission ratio) effective ${\displaystyle {\frac {\omega _{2;n}}{max(\omega _{2;1};|\omega _{2;R}|)}}={\frac {min(i_{1};|i_{R}|)}{i_{n}}}}$ 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 [h] Ratio span's center ${\displaystyle (i_{1}i_{n})^{\frac {1}{2}}}$ The center indicates the speed level of the transmission Together with the final drive ratio it gives the shaft speed level of the vehicle [i] Average gear step ${\displaystyle \left({\frac {\omega _{2;n}}{\omega _{2;1}}}\right)^{\frac {1}{n-1}}=\left({\frac {i_{1}}{i_{n}}}\right)^{\frac {1}{n-1}}}$ There are ${\displaystyle n-1}$ gear steps between ${\displaystyle n}$ gears with decreasing step width the gears connect better to each other shifting comfort increases [j] Sun 4: sun gear of gearset 4 [k] Ring 4: ring gear of gearset 4 [l] Sun 3: sun gear of gearset 3 [m] Ring 3: ring gear of gearset 3 [n] Sun 2: sun gear of gearset 2 [o] Ring 2: ring gear of gearset 2 [p] Sun 1: sun gear of gearset 1 [q] Ring 1: ring gear of gearset 1 [r] 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 4) is always larger and the upper half of the gear steps (between the large gears; rounded up, here the last 4) 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) [s] 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 [t] 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) [u] From large to small gears (from right to left) [v] 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) [w] 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) [x] 280 N⋅m (207 lb⋅ft) for both gasoline and diesel[1] [y] 450 N⋅m (332 lb⋅ft) for gasoline[1] [z] 480 N⋅m (354 lb⋅ft) for diesel[1] [aa] 91/133 or 104/152 [ab] Thereof 1 dog break[5] [ac] Thereof 1 dog clutch[5] [ad] Permanently coupled elements C1, C2, and R3 S3 and S4 C3 and R4 [ae] Dog brake blocks S3 and S4 [af] Blocks S1 [ag] Blocks R2 [ah] Couples S1 with input (turbine) [ai] Couples C3 and R4 with input (turbine) [aj] Dog clutch couples R1 and S2 with input (turbine) [ak] Ordinary noted For direct determination of the gear ratio [al] Elementary noted Alternative representation for determining the transmission ratio Contains only operands With ordinary 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 rate and efficiency

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How it works

An animated drive line schematic & a rotational speeds nomogram

These ordinates are positioned on the abscissa in strict accordance with the proportions of the sun gears' teeth numbers relative to those of their rings. Consequently, the output ratios on the ordinate C₄ (carrier of planetary gearset 4) follows closely to those of the actual transmission. Note that elements A and F are labelled swapped (cf. legend below).

Nomogram

▶️ Interactive nomogram Archived 2017-02-02 at the Wayback Machine

This interactive nomogram is a real geometric calculator exactly representing the rotational speeds of the transmission's 3x4 = 12 internal shafts for each of its 9 ratios (+ reverse), grouped according to their 5 permanent coupling on 4 joint ordinates and 3 independent ordinates. These ordinates are positioned on the abscissa in strict accordance with the proportions of the sun gears' teeth numbers relative to those of their rings. Consequently, the output ratios on the 6th ordinate (carrier of the fourth planetary gearset) follows closely those of the actual transmission. This advantageous geometric construction sets us free from Robert Willis' famous and tedious formula,^[6] because all calculations are exclusively determined by lengths ratios, respectively teeth numbers on the abscissa for the 4 epicyclic ratios, and of rotational speeds on the 6th ordinate for the 10 gear ratios.

Legend

A: Dog brake (blocks S₃ and S₄)
C: Brake (blocks S₁)
D: Brake (blocks R₂)
B: Clutch (couples S₁ with input shaft)
E: Clutch (couples C₃ and R₄ with input shaft)
F: Dog clutch (couples R₁ and S₂ with input shaft)