# 3. ARMATURE VOLTAGE AND GOVERING EQUATIONS

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3. ARMATURE VOLTAGE AND GOVERING EQUATIONS
The induced voltage in a conductor is Then the induced voltage across armature terminal is Let Bp express average flux density under one pole.

From Eqs. (2) and (4), Let Substituting Eqs. (6) and (7) into Eqs. (5) and simplifying, where --- constant

Question : If saturation appears in magnetic field, will Ea become larger or smaller?
Note: Ea exists both in generator and in motor. In generator, Ea produces armature current. It is called generated emf or positive voltage. In motor, supply voltage U produces armature current and Ea opposes current to flow into armature winding. In this case it is called back emf or negative voltage.

2). Governing Equations of steady-state operation
Generator : Example: Separately excited generator Features : (1). (2). Ia is produced by Ea and has the same direction as Ea. Ea is positive voltage. (3). Torque T is called negative torque because it oppose n. Governing equation of armature circuit where --- resistant of armature winding --- voltage drop caused by brush contact resistant

The total equations are
Motor Example: Separately excited motor Features : (1). (2). Ia is produced by U and has the opposite direction from Ea. Ea is negative voltage. (3). Torque T is called positive torque because it produces n.

Governing equation of armature circuit
Total equations Summary Before writing governing equations, please draw schematic diagram according to excitation modes and then give correct reference direction in terms of operation modes. Example: Shunt generator and shunt motor

4. POWER FLOW AND EFFICIENCY
1). Losses Copper loss: armature circuit loss field circuit loss brush loss Core Loss Mechanical loss against windage and friction Additional loss (stray loss) --- cannot be computed accurately

2). Power Flowing diagram
Example: compound dc machine Compound generator Mechanical power electrical power Developed power (electromagnetic power): --- It is the total electrical power developed by or converted from the mechanical energy.

Power equations from power flow diagram:
electrical power Mechanical power Compound motor In motors, PM is the total mechanical power converted from input electrical power.

(3). Efficiency DEVELOPED TORQUE (ELECTROMAGNETIC TORQUE) The electromagnetic force acting on a conductor is where ic is the current of conductor. The force on the total conductors under each pole is

The force on all conductors of armature winding is
The developed torque produced by armature winding is where D is the outer diameter of rotor. Let

Substituting Eqs. (1), (3), (4), (5) and (6) into Eqs
Substituting Eqs. (1), (3), (4), (5) and (6) into Eqs. (2) and simplifying, we obtain Where is a constant. Questions 1. What is the affect of armature reaction to T if core involves in saturation ?

2. If load torque becomes larger, how about Ia under constant external voltage in shunt motor?
3. In shunt motor, if external voltage and load torque are all kept constant, when we put an external resistance into armature circuit, how will Ia change ? Useful Expression --- calculating T from power

Governing Equation for torque
In generator, --- torque under no-load operation --- input mechanical torque T is called an opposed torque. T1 has to overcome T to make prime mover deliver its mechanical energy into generator. In motor, where T is called an active torque. It drives motor and load to rotate.