TF Armature Controlled DC Motor

Circuit Diagram for determination of Kt

Tabulation for Determination of Kt

S.No

Armature Current (Ia) Amps

Armature voltage (Va) Amps

Speed (N) rpm

ω=2∏N/60 (rad/sec)

Eb=Va-IaRa (Volts)

EX.NO: DATE:

TRANSFER FUNCTION OF ARMATURE CONTROLLED DC MOTOR Aim To find the transfer function of armature controlled DC motor with control signal is applied to the armature. Apparatus Required Name of the Apparatus

S.No

Type

Range

Quantity

1

Ammeter

MC

(0-10) A

1

2

Ammeter

MI

(0-10) A

1

3

Voltmeter

MI

(0-300) V

1

4

Voltmeter

MC

(0-300) V

1

5

Voltmeter

MC

(0-30) V

1

6

Rheostat

Wire Wound

270Ω/1.5A

1

7

Rheostat

Wire Wound

50Ω/5A

1

8

DPST Switch

-

-

1

9

Tachometer

Digital

-

1

UPF

240V,10A,2.7KVA

1

10

Auto transformer

Formula Used Assume J = 0 .1 2 23 Kg-m2/rad & B= 1 N -m / (r a d/ s e c ) 1. Inductance of field winding (La) 2

L a= where,

Za-R 2π f

2 a

in ohms

Ra - Armature resistance in ohms Za – Armature impedance in ohms

Transfer function of Armature controlled DC shunt motor

Block Diagram

Model Graph

2. Torque constant of DC Motor (Kt)

ΔEb K t = Kb = Δω where,

in N-m/A

Kb = back emf constant

Δω

- Change in ω obtained from graph

Δ E b - Change in Eb obtained from graph Theory A DC machine can run as a motor, when a DC supply is given to its field winding to produce magnetic flux while the same DC source is used to supply current to the armature. Now the armature becomes a current carrying conductor and as it is kept in a magnetic field, it develops mechanical force. The direction of the force is given by Fleming’s Left Hand Rule. DC motor in control applications is used for delivering mechanical power to control elements while taking electrical control signal as input. Electrical input to DC motor is called control signal and that can be applied in two ways. In one method, the control signal is applied to the field winding while fixed voltage is applied to armature winding. This method is called Field controlled motor. In another method, control signal is applied to armature winding and constant voltage is applied to field winding. This method is called Armature controlled motor. Here the transfer function of armature controlled motor is to be found out. Derivation of Transfer Function Let Ra = Armature resistance in ohms La = Armature inductance in Henry Ia = Armature Current in Amps Va = Armature Voltage in Volts T = Torque developed by Motor in N-m Kt = Torque constant in N-m/A J = Moment of inertia of Motor in Kg-m2/ rad B = Frictional Coefficient of motor N-m/ (rad/sec) Kb = Back emf Constant in Volt / (rad/sec)

Circuit Diagram for determination of Ra

Tabulation to find Ra

S.No

Armature Voltage (V) Volts

Armature Current (I) Amps

Field Resistance (Ra=V/I) Ohms

Mean, Ra =_____ohms

According to Kirchhoff’s Voltage Law (KVL),

dI V a = IaR a + L a dta + e b -----------(1) On Taking Laplace of equ (1) I a ( s ) [ R a + S L a ] + E b ( s ) = V a ( s ) -----------(2)

Torque developed in a DC Motor is proportional to flux and armature current

T(t) ∝ i a ( t) .

..

T(t)= Ka ia (t) -----------(3)

On Taking Laplace of equ (3)

T(s) = K t I a (s) -----------(4) The differential equation governing the mechanical system of the motor is given by the expression

d 2θ dθ J +B = T (t) dt dt2

-----------(5)

On Taking Laplace of equ (5)

(Js 2 + Bs) θ (s) = T(s)

-----------(6)

The back emf of a DC machine is proportional to speed of the shaft

dθ

e b (t) α dt

e b(t) = K b

dθ dt

-----------(7)

On Taking Laplace of equ (7)

E b(s) = K bSθ(s) -----------(8)

Circuit diagram for determination of Za

Tabulation to find Za

S.No

Armature Voltage (V) Volts

Armature Current (I) Amps

Armature Impedance (Za) Ohms

Mean, Za=_____ohms

On equating expressions (4) & (6) and solving for Ia(s) gives

I a (s)=

(J S 2 +Bs) Kt

θ(s) -----------(9)

Substuting Equation (8) and (9) in (2) and rearranging gives

θ(s) Kt = 2 Va(s) (JS +BS)( R a +S L a )+ K t K b S

Precautions 1. While starting and stopping the machine the armature rheostat should be kept at maximum resistance position and the field rheostat should be kept at minimum resistance position. 2. DPST switch is kept open initially.

Procedure for determining Kt ¾ The connections are made as per the circuit diagram. ¾ By closing the DPST Switch, 220V DC Supply is given to the circuit and Start the motor using 3 point starter. ¾ The armature rheostat and field rheostat are varied to bring the speed to its rated value. ¾ The armature voltage is varied in steps by varying the armature rheostat and the corresponding speed is noted down in each step. The readings are taken for speeds below the rated value and tabulated. ¾ From the tabulated values, find the value of Kt

Procedure for determining Armature Resistance (Ra) ¾ The connections are made as per the circuit diagram. ¾ Supply is given by closing the DPST switch. ¾ Tabulate voltage and current readings for various load conditions. ¾ The effective resistance of motor armature is calculated from the tabulated readings.

Procedure for determining Armature Impedance (Za) ¾ The Connections are made as per the circuit diagram. ¾ The 230V, 50HZ, AC Supply is given to the circuit. ¾ By gradually varying the autotransformer, the different set of armature voltage (V) and armature current (I) readings are tabulated. ¾ From the tabulated values, the armature impedance (Za) value is calculated.

Result Thus the transfer function of armature controlled DC shunt motor was found.

θ(s) V a (s)

=