Editing Torque converter (section)

===Theory of operation===
Torque converter equations of motion are governed by [[Leonhard Euler]]'s eighteenth century [[Euler's pump and turbine equation|turbomachine equation]]:
:<math>\tau = \sum \left [  r \times \frac{d}{dt} \left ( m \cdot v \right ) \right ]</math>

The equation expands to include the fifth power of radius; as a result, torque converter properties are very dependent on the size of the device.

Mathematical formulations for the torque converter are available from several authors.
<ref name="Bond Graph Modeling and Computer Simulation of Automotive Torque Converters" >
{{cite journal
	| last1 = Hrovat
	| first1 = D
	| last2 = Tobler
	| first2 = W
	| date = 1985
	| title = Bond Graph Modeling and Computer Simulation of Automotive Torque Converters
	| url = https://doi.org/10.1016/0016-0032(85)90067-5
	| journal = Journal of the Franklin Institute
	| volume = 319
	| pages = 93-114
	| doi = 10.1016/0016-0032(85)90067-5
	| access-date = 
| url-access = subscription
	}}
</ref>
<ref name="Dynamic Models for Torque Converter Equipped Vehicles">
{{cite journal
	| last1 = Kotwicki
	| first1 = A. J.
	| date = 1982
	| title = Dynamic Models for Torque Converter Equipped Vehicles
	| url = https://doi.org/10.4271/820393
	| journal = SAE Technical Paper Series
	| doi = 10.4271/820393
| url-access = subscription
	}}
</ref>

Hrovat derived the equations of the pump, turbine, stator, and conservation of energy. Four first-order differential equations can define the performance of the torque converter. 

<math>
I_i \dot{\omega_i} + \rho S_i \dot{Q} = -\rho (\omega_i R_i^2 + R_i \frac{Q}{A}\tan{\alpha_i} - \omega_\mathrm{s} R_\mathrm{s}^2 - R_\mathrm{s} \frac{Q}{A} \tan{\alpha_\mathrm{s}} ) Q + \tau_i
</math>

<math>
I_\mathrm{t} \dot{\omega_\mathrm{t}} + \rho S_\mathrm{t} \dot{Q} = -\rho (\omega_\mathrm{t} R_\mathrm{t}^2 + R_\mathrm{t} \frac{Q}{A}\tan{\alpha_\mathrm{t}} - \omega_i R_i^2 - R_i \frac{Q}{A} \tan{\alpha_i} ) Q + \tau_\mathrm{t}
</math>

<math>
I_\mathrm{s} \dot{\omega_\mathrm{s}} + \rho S_\mathrm{s} \dot{Q} = -\rho (\omega_\mathrm{s} R_\mathrm{s}^2 + R_\mathrm{s} \frac{Q}{A}\tan{\alpha_\mathrm{s}} - \omega_\mathrm{t} R_\mathrm{t}^2 - R_\mathrm{t} \frac{Q}{A} \tan{\alpha_\mathrm{t}} ) Q + \tau_\mathrm{s}
</math>

<math>
\rho (S_\mathrm{p} \dot{w_\mathrm{p}} + S_\mathrm{t} \dot{w_\mathrm{t}}+ S_\mathrm{s} \dot{w_\mathrm{s}}) + \rho \frac{L_\mathrm{f}}{A} \dot{Q} = 
\rho (R_\mathrm{p}^2 w_\mathrm{p}^2 + R_\mathrm{t}^2 w_\mathrm{t}^2 + R_\mathrm{s}^2 w_\mathrm{s}^2 - R_\mathrm{s}^2 w_\mathrm{p} w_\mathrm{s} - R_\mathrm{p}^2 w_\mathrm{t} w_\mathrm{p} - R_\mathrm{t}^2 w_\mathrm{s} w_\mathrm{t}) 
+ w_\mathrm{p} \frac{Q}{A} \rho (R_\mathrm{p} \tan{a_\mathrm{p}} - R_\mathrm{s} \tan{a_\mathrm{s}}) 
+ w_\mathrm{t} \frac{Q}{A} \rho (R_\mathrm{t} \tan{a_\mathrm{t}} - R_\mathrm{p} \tan{a_\mathrm{p}}) 
+ w_\mathrm{s} \frac{Q}{A} \rho (R_\mathrm{s} \tan{a_\mathrm{s}} - R_\mathrm{t} \tan{a_\mathrm{t}}) 
- P_L
</math>

where
* <math>\rho</math> is density
* <math>A</math> is flow area
* <math>R_\mathrm{p}</math> is pump radius
* <math>R_\mathrm{t}</math> is turbine radius
* <math>R_\mathrm{s}</math> is stator radius
* <math>a_\mathrm{p}</math> is pump exit angle
* <math>a_\mathrm{t}</math> is turbine exit angle
* <math>a_\mathrm{s}</math> is stator exit angle
* <math>I</math> is inertia
* <math>L_\mathrm{f}</math> is fluid inertia length


A simpler correlation is provided by Kotwicki.