Final notu – elektromekanik dönüşüm, DC ve asenkron makineler

externally force move the conductor opposite to the magnetic force, mech. energy converted into electric energy

In a conservative system:

0.9The energy supplied by the source stored in the system as mag. energy ( W_i=W_m )

In a linear system the energy stored in the system

3.3 Magnetic Field as a Medium

Case 1. Constant Flux

A magnetic circuit is moved by a distance dx in time dt in a region where the flux is held constant. The energy supplied by the source must be zero because the initial and final values of the flux are the same.

For a conservative system

The mechanical force acting on the magnetic circuit.

Case 2. Constant Current

For a change in flux dφ when I is held constant. the energy input:

For conservative system F_m dx+dW_m=2 dW_m F_m=dW_mdx

When the flux is constant

The rate of change of stored magnetic energy (constant flux)

total airgap inductance

Magnetic Circuit with Air Gap

so the energy in each air gap

When the Flux constant =BA

The rotor of dc machine called armature.

Wound machine (mach. with electromagnets)

mechanical Angle θ_m=θ (Electrical Angle)

The arc by the coil =180^° ⇒ full pitch coil

less than 180^° fractional pitch coil

If φ_p is the flux per pole: φ=φ_pθ

θ : the angular position of the coil in electrical degrees

If there are P poles in a machine θ=P2θ_m

angular f. of the induced emf ω=P2 ω_m → angular f. of the rotor

The max value of the induced emf in a single turn coil

In a dc machine, the average value of the induced voltage in a single turn coil is

Force on a Conductor

For linear current-carrying conductor:

Motor action: the force on the upper conductors of the coil

Revolving Field of a Three Phase Motor:

In a 2 pole motor, the time taken by the flux to complete one revolution along the periphery of the air gap is the same as input. T

the speed of the revolving field in revolutions per second n_s=1T_s=2fP (dönme frekansı)

Sabit frekanslı bir kaynak için hız sabit.

Slip speed: (the difference between synchronous and rotor speed)

Per unit slip:

Revolving Field of two Phase Motor

power angle δ : rotor lags, revolving field

The power is proportional δ

Under no load, δ is almost zero

The torque developed max @ δ=90^°

Torque developed by rotating the system T_e=12 i^2dLdθ rotor speedω_m=2Pω

Torque exerted on the rotor T_e=-12 i^2(L_d-L_q)2θ

Direct current Generators

Induced emf Equation:

Average value of the induced emf total number of coil (slots for a 2 layer winding) and a the number of parallel paths ( a=2 for wave, a=P for lap winding)

The total turns per parallel path: turns between negative and positive brushes

The electrical power developed by the generator:

The average torque developed by a dc machine:

The torque developed by the dc machine:

Power Flow diagram: V_a=V_t+I_aR_a

A seperately Exited DC Generator

Maximum Efficiency Criterion

For a PM dc generator, the power output is

The max power developed when

As R_a→0 , the speed of shunt motor ω_m→ω_mnL . In other words, a shunt motor is basically a constant speed motor.

Methods of Speed Control

Armature Resistance Control

For series or a compound motor:

for a shunt motor

Field Control method

Inserting an external resistor in series with the field winding.

Braking or Reversing DC motors

Plugging: Reverse Supply connections to the armature terminals. At any time during the plugging,

So the braking torque:

The armature disconnected from S. connected across a variable resistor R. The back emf produce a reverse current.

Polyphase Induction Motors

The Synchronous Speed of the revolving field is

(slip) revolving field relative rotor speed

f_r=sf At stand still, the slip is 1 and f_r=f

Per phase eq. circuit

per phase emf in rotor cct.

per phase induced emf in the stator winding.

per phase induced emf in the rotor winding under blocked-rotor condition.

Torque developed by the motor

Speed Torque Characteristic

When the slip falls (below breakdown slip s_b ), the hypothetical resistance becomes the dominating factor.

Y bağlantıdaki hat gerilimine V_L dersek faz gerilim V_p=V_L3 olur.

The stator current

An approximate Eq. circuit

Maximum Power Criterion

P_d is max when the equivalent ^s_p load resistance is equal to the magnitude of the standstill impedance of the motor.

Maximum Torque Criterion

If the stator impedance is so small

When the core loss is considered a part of the rotational loss the power input to the motor using the approximate circuit.

The motor efficiency is

3I_2^2(R_1+R_2)=P_r as the criterion for the maximum efficiency of an induction motor. (stator and rotor copper losses.)

Equivalent Circuit Parameters

The Stator resistance test:

The blocked-Rotor Test:

The No-load test

W_oc = power input I_oc input Current V_oc rated applied voltage

Let P_fw be the friction and windage loss on a per-phase basis.

Then, the power loss in R_c is P_oc=W_oc-P_fwφ