Feedback Control Systems FCS Lecture7 Mathematical Modeling of

Feedback Control Systems (FCS) Lecture-7 Mathematical Modeling of Real World Systems Dr. Imtiaz Hussain email: imtiaz. hussain@faculty. muet. edu. pk URL : http: //imtiazhussainkalwar. weebly. com/ 1

Modelling of Mechanical Systems • Automatic cruise control • The purpose of the cruise control system is to maintain a constant vehicle speed despite external disturbances, such as changes in wind or road grade. • This is accomplished by measuring the vehicle speed, comparing it to the desired speed, and automatically adjusting the throttle. • The resistive forces, bv, due to rolling resistance and wind drag act in the direction opposite to the vehicle's motion. 2

Modelling of Mechanical Systems • The transfer function of the systems would be 3

Electromechanical Systems • Electromechanics combines electrical and mechanical processes. • Devices which carry out electrical operations by using moving parts are known as electromechanical. – – Relays Solenoids Electric Motors Switches and e. t. c 4

Potentiometer 5

Potentiometer • R 1 and R 2 vary linearly with θ between the two extremes: 6

Potentiometer • Potentiometer can be used to sense angular position, consider the circuit of figure-1. Figure-1 • Using the voltage divider principle we can write: 7

D. C Drives • Speed control can be achieved using DC drives in a number of ways. • Variable Voltage can be applied to the armature terminals of the DC motor. • Another method is to vary the flux per pole of the motor. • The first method involve adjusting the motor’s armature while the latter method involves adjusting the motor field. These methods are referred to as “armature control” and “field control. ” 8

Example-2: Armature Controlled D. C Motor Ra Input: voltage u Output: Angular velocity u La ia B eb T J Electrical Subsystem (loop method): nt sta n =co Vf Mechanical Subsystem

Example-2: Armature Controlled D. C Motor Ra Power Transformation: u La ia B eb T J Torque-Current: Voltage-Speed: where Kt: torque constant, Kb: velocity constant For an ideal motor Combing previous equations results in the following mathematical model: t an t s n =co Vf

Example-2: Armature Controlled D. C Motor Taking Laplace transform of the system’s differential equations with zero initial conditions gives: Eliminating Ia yields the input-output transfer function

Example-2: Armature Controlled D. C Motor Reduced Order Model Assuming small inductance, La 0

Example-3: Armature Controlled D. C Motor If output of the D. C motor is angular position θ then we know Ra u La ia B eb J T θ t an t s n =co Which yields following transfer function Vf

Example-3: Field Controlled D. C Motor Ra Rf ef if Applying KVL at field circuit Mechanical Subsystem Lf Tm B ω La J ea

Example-3: Field Controlled D. C Motor Power Transformation: Torque-Current: where Kf: torque constant Combing previous equations and taking Laplace transform (considering initial conditions to zero) results in the following mathematical model:

Example-3: Field Controlled D. C Motor Eliminating If(S) yields If angular position θ is output of the motor Ra Rf ef if Lf Tm La J B θ ea

Example-4 An armature controlled D. C motor runs at 5000 rpm when 15 v applied at the armature circuit. Armature resistance of the motor is 0. 2 Ω, armature inductance is negligible, back emf constant is 5. 5 x 10 -2 v sec/rad, motor torque constant is 6 x 10 -5, moment of inertia of motor 10 -5, viscous friction coefficient is negligible, moment of inertia of load is 4. 4 x 10 -3, viscous friction coefficient of load is 4 x 10 -2. La Ra ea 15 v ia N 1 Bm eb T Jm BL t =co an t s n Vf JL N 2 L 1. Drive the overall transfer function of the system i. e. ΩL(s)/ Ea(s) 2. Determine the gear ratio such that the rotational speed of the load is reduced to half and torque is doubled.

System constants ea = armature voltage eb = back emf Ra = armature winding resistance = 0. 2 Ω La = armature winding inductance = negligible ia = armature winding current Kb = back emf constant = 5. 5 x 10 -2 volt-sec/rad Kt = motor torque constant = 6 x 10 -5 N-m/ampere Jm = moment of inertia of the motor = 1 x 10 -5 kg-m 2 Bm=viscous-friction coefficients of the motor = negligible JL = moment of inertia of the load = 4. 4 x 10 -3 kgm 2 BL = viscous friction coefficient of the load = 4 x 10 -2 N-m/rad/sec gear ratio = N 1/N 2

Example-4 Since armature inductance is negligible therefore reduced order transfer function of the motor is used. La Ra ea 15 v ia N 1 Bm eb T Jm BL t =co an t s n Vf JL N 2 L

Example-5 A field controlled D. C motor runs at 10000 rpm when 15 v applied at the field circuit. Filed resistance of the motor is 0. 25 Ω, Filed inductance is 0. 1 H, motor torque constant is 1 x 10 -4, moment of inertia of motor 10 -5, viscous friction coefficient is 0. 003, moment of inertia of load is 4. 4 x 10 -3, viscous friction coefficient of load is 4 x 10 -2. Ra Rf ef if Lf Tm La ea B m ωm N 1 Jm BL JL N 2 L 1. Drive the overall transfer function of the system i. e. ΩL(s)/ Ef(s) 2. Determine the gear ratio such that the rotational speed of the load is reduced to 500 rpm.

Example-5 + + r e _ kp - La Ra + ea _ N 1 + ia JM BM T e b _ BL θ JL N 2 if = Constant c

Numerical Values for System constants r = angular displacement of the reference input shaft c = angular displacement of the output shaft θ = angular displacement of the motor shaft K 1 = gain of the potentiometer shaft = 24/π Kp = amplifier gain = 10 ea = armature voltage eb = back emf Ra = armature winding resistance = 0. 2 Ω La = armature winding inductance = negligible ia = armature winding current Kb = back emf constant = 5. 5 x 10 -2 volt-sec/rad K = motor torque constant = 6 x 10 -5 N-m/ampere Jm = moment of inertia of the motor = 1 x 10 -5 kg-m 2 Bm=viscous-friction coefficients of the motor = negligible JL = moment of inertia of the load = 4. 4 x 10 -3 kgm 2 BL = viscous friction coefficient of the load = 4 x 10 -2 N-m/rad/sec n= gear ratio = N 1/N 2 = 1/10

System Equations e(t)=K 1[ r(t) - c(t) ] or E(S)=K 1 [ R(S) - C(S) ] Ea(s)=Kp E(S) (1) (2) Transfer function of the armature controlled D. C motor Is given by Km = Ea(S) S(Tm. S+1) θ(S)