### Formulas

*Shunt Generator*

For a shunt generator with armature induced voltage **E _{a}**, armature current

**I**and armature resistance

_{a}**R**, the terminal voltage

_{a}**V**is:

**V = E**

_{a}– I_{a}R_{a}The field current

**I**for a field resistance

_{ f}**R**is:

_{ f}**I**

_{ f}= V / R_{ f}The armature induced voltage **E _{a}** and torque

**T**with magnetic flux

**F**at angular speed

**w**are:

**E**

_{a}= k_{ f}Fw = k_{m}w**T = k**

_{ f}FI_{a}= k_{m}I_{a}where

**k**and

_{ f}**k**are design coefficients of the machine.

_{m}Note that for a shunt generator:

– induced voltage is proportional to speed,

– torque is proportional to armature current.

The airgap power **P _{e}** for a shunt generator is:

**P**

_{e}= wT = E_{a}I_{a}= k_{m}w I_{a}*Shunt Motor*

For a shunt motor with armature induced voltage **E _{a}**, armature current

**I**and armature resistance

_{a}**R**, the terminal voltage

_{a}**V**is:

**V = E**

_{a}+ I_{a}R_{a}The field current

**I**for a field resistance

_{ f}**R**is:

_{ f}**I**

_{ f}= V / R_{ f}The armature induced voltage **E _{a}** and torque

**T**with magnetic flux

**F**at angular speed

**w**are:

**E**

_{a}= k_{ f}Fw = k_{m}w**T = k**

_{ f}FI_{a}= k_{m}I_{a}where

**k**and

_{ f}**k**are design coefficients of the machine.

_{m}Note that for a shunt motor:

– induced voltage is proportional to speed,

– torque is proportional to armature current.

The airgap power **P _{e}** for a shunt motor is:

**P**

_{e}= wT = E_{a}I_{a}= k_{m}w I_{a}*Series Motor*

For a series motor with armature induced voltage **E _{a}**, armature current

**I**, armature resistance

_{a}**R**and field resistance

_{a}**R**, the terminal voltage

_{ f}**V**is:

**V = E**

_{a}+ I_{a}R_{a}+ I_{a}R_{ f}= E_{a}+ I_{a}(R_{a}+ R_{ f})The field current is equal to the armature current.

The armature induced voltage **E _{a}** and torque

**T**with magnetic flux

**F**at angular speed

**w**are:

**E**

_{a}= k_{ f}Fw I_{a}= k_{m}w I_{a}**T = k**

_{ f}FI_{a}^{2}= k_{m}I_{a}^{2}where

**k**and

_{ f}**k**are design coefficients of the machine.

_{m}Note that for a series motor:

– induced voltage is proportional to both speed and armature current,

– torque is proportional to the square of armature current,

– armature current is inversely proportional to speed for a constant induced voltage.

The airgap power **P _{e}** for a series motor is:

**P**

_{e}= wT = E_{a}I_{a}= k_{m}w I_{a}^{2}NOTATION | ||||||

The symbol font is used for some notation and formulae. If the Greek symbols for alpha beta delta do not appear here [ a b d ] the symbol font needs to be installed for correct display of notation and formulae. | ||||||

BE f G I j k m N n P p R | susceptance induced voltage frequency conductance current j-operator coefficient number of phases number of turns rotational speed power pole pairs resistance | [siemens, S] [volts, V] [hertz, Hz] [siemens, S] [amps, A] [1Ð90°] [number] [number] [number] [revs/min] [watts, W] [number] [ohms, W] | SsTVXYZdFfhqw | voltamperes slip torque terminal voltage reactance admittance impedance loss angle magnetic flux phase angle efficiency temperature angular speed | [volt-amps, VA] [per-unit] [newton-metres, Nm] [volts, V] [ohms, W] [siemens, S] [ohms, W] [degrees, °] [webers, Wb] [degrees, °] [per-unit] [centigrade, °C] [radians/sec] |