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# Direct Current Machines

### Formulas

Shunt Generator
For a shunt generator with armature induced voltage Ea, armature current Ia and armature resistance Ra, the terminal voltage V is:
V = Ea – IaRa
The field current I f for a field resistance R f is:
I f = V / R f

The armature induced voltage Ea and torque T with magnetic flux F at angular speed w are:
Ea = k fFw = kmw
T = k fFIa = kmIa
where k f and km are design coefficients of the machine.

Note that for a shunt generator:
– induced voltage is proportional to speed,
– torque is proportional to armature current.

The airgap power Pe for a shunt generator is:
Pe = wT = EaIa = kmw Ia

Shunt Motor
For a shunt motor with armature induced voltage Ea, armature current Ia and armature resistance Ra, the terminal voltage V is:
V = Ea + IaRa
The field current I f for a field resistance R f is:
I f = V / R f

The armature induced voltage Ea and torque T with magnetic flux F at angular speed w are:
Ea = k fFw = kmw
T = k fFIa = kmIa
where k f and km are design coefficients of the machine.

Note that for a shunt motor:
– induced voltage is proportional to speed,
– torque is proportional to armature current.

The airgap power Pe for a shunt motor is:
Pe = wT = EaIa = kmw Ia

Series Motor
For a series motor with armature induced voltage Ea, armature current Ia, armature resistance Ra and field resistance R f, the terminal voltage V is:
V = Ea + IaRa + IaR f = Ea + Ia(Ra + R f)
The field current is equal to the armature current.

The armature induced voltage Ea and torque T with magnetic flux F at angular speed w are:
Ea = k fFw Ia = kmw Ia
T = k fFIa2 = kmIa2
where k f and km are design coefficients of the machine.

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 Pe for a series motor is:
Pe = wT = EaIa = kmw Ia2

 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. B E 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] S s T V X Y Z d F f h q w 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]

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