modeling and simulation of dynamic systems

Modeling and simulation of dynamic systems is Pham Huy Hoang Electromotive force (emf) voltage (electromotance): is that which tends to cause current (actual electrons and ions) to flow, is the external work expended per unit of charge to produce an electric potential difference across two opencircuited terminals.

MODELING AND SIMULATION OF DYNAMIC SYSTEMS MIXED DISCIPLINE SYSTEMS PHAM HUY HOANG HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY INTRODUCTION MIXED DISCIPLINE SYSTEM: MIXED DISCIPLINE SYSTEM – COUPLING SYSTEM OF SINGLE-DISCIPLINE SYSTEMS Pham Huy Hoang ELECTROMECHANICAL SYSTEMS ARMATURE-CONTROLLED DC MOTOR Voltage is electric potential energy per unit charge (J/C = V) - referred to as "electric potential” Electromotive force (emf) voltage (electromotance): - is that which tends to cause current (actual electrons and ions) to flow; - is the external work expended per unit of charge to produce an electric potential difference across two open-circuited terminals; - is generated by a magnetic force (Faraday’s law) Pham Huy Hoang ELECTROMECHANICAL SYSTEMS Faraday's Law Any change in the magnetic environment* of a coil of wire will cause a voltage (emf) to be "induced" in the coil * The change of magnetic field strength, relative displacement between the magnet field and the coil Pham Huy Hoang Pham Huy Hoang ELECTROMECHANICAL SYSTEMS The back emf voltage across a DC motor: eb = K eω = K eθ& The torque developed by the motor: T = Kt i eb : back emf voltage θ : angular displacement of the rotor of the motor & θ = ω : angular velocity of the rotor T : torque applied to the rotor Ke : emf constant (Vs/rad) Ki : torque constant (Nm/A) Pham Huy Hoang ELECTROMECHANICAL SYSTEMS ia Ra La θ ,θ& = ω Jr eb Va TL Jd Bd vRa + vLa + eb − va = di Raia + La a + eb = va dt eb = K eω = K eθ& Raia + La dia + K eθ& = va dt (1) Pham Huy Hoang ELECTROMECHANICAL SYSTEMS ia Ra La θ ,θ& = ω eb Va Jr TL Jd Bd J = Jr + Jd T + TL − Bdθ& = Jθ&& T = Kt ia Kt ia + TL − Bdθ& = Jθ&& (2) Pham Huy Hoang ELECTROMECHANICAL SYSTEMS ia Ra La θ ,θ& = ω eb Va Jr TL Jd Bd Jθ&& + Bdθ& − Kt ia = TL di La a + Raia + K eθ& = va dt θ&& θ&  J 0    Bd    0 − Kt  θ  TL   0   +  K L    + 0 R  i  = v    i   e a  i   a  a   a   a a Pham Huy Hoang ELECTROMECHANICAL SYSTEMS ia Va Ra La θ ,θ& = ω eb Jr K, B TL Jd Bd Pham Huy Hoang ELECTROMECHANICAL SYSTEMS FIELD-CONTROLLED DC MOTOR Rf θ ,θ& = ω ia = const if Jr vf Jd Lf eb= Rf if + Lf Bd T = Kt i f di f dt = vf TL (1) Pham Huy Hoang ELECTROMECHANICAL SYSTEMS Rf vf ia = const if Jr Lf eb= θ ,θ& = ω T = Kt i f TL Jd Bd J = Jr + Jd T + TL − Bdθ& = Jθ&& T = Kt i f Kt i f + TL − Bdθ& = Jθ&& (2) Pham Huy Hoang ELECTROMECHANICAL SYSTEMS Rf vf if Jr Lf eb= θ ,θ& = ω ia = const TL Jd Bd T = Kt i f Jθ&& + Bd θ& − K t i f = TL di f Lf +Rf if =vf dt &&  θ   Bd − K t  θ&  TL  J =  L   +  R f  i f  v f  f  i     f Pham Huy Hoang ELECTROMECHANICAL SYSTEMS Rf vf ia = const if Jr Lf eb= θ ,θ& = ω T = Kt i f TL Jd Bd Pham Huy Hoang ELECTROMECHANICAL SYSTEMS MAGNETO-ELECTRO-MECHANICAL SYSTEMS Lenz’s law: increasing current in a coil will generate a counter emf which opposes the current (The emf always opposes the change in current) The relation of this counter emf to the current is the origin of the concept of inductance Pham Huy Hoang ELECTROMECHANICAL SYSTEMS Magnetic Force and Lorentz force law: - The force is perpendicular to both the velocity v of the charge q and the magnetic field B (N/A = Ns/C = Tesla) - The magnitude of the force is F = qvB sinθ (θ is the angle between the velocity and the magnetic field) Pham Huy Hoang ELECTROMECHANICAL SYSTEMS -The magnetic force on a stationary charge or a charge moving parallel to the magnetic field is zero - The direction of the force is given by the right hand rule Pham Huy Hoang ELECTROMECHANICAL SYSTEMS i1 v i2 R2 R1 L C k1 y b m1 a Magnetic force : K1i2 Electromotive force (emf ) : K y& c1 k2 x1 c2 m2 x2 Pham Huy Hoang ELECTROMECHANICAL SYSTEMS i1 v i2 R2 R1 L C y 1t 1t R1i1 + ∫ i1dt − ∫ i2dt = v C0 C0 a 1t di2 1t b − ∫ i1dt + L + R2i2 + ∫ i2dt − K x&1 = C0 dt C0 a Pham Huy Hoang ELECTROMECHANICAL SYSTEMS ∑ M O = J Oα &x& − K1i2a − (k1x1 + c1x&1)b + [k2 ( x2 − x1) + c2 ( x&2 − x&1)]b = m1b2   b m1&x&1 + (c1 + c2 )b2 x&1 − c2b2 x&2 + (k1 + k2 )b2 x1 − k2b2 x2 + K1abi2 = ∑ F = m2 &x&2 − k2 ( x2 − x1) − c2 ( x&2 − x&1) = m2 &x&2 k1 m2 &x&2 − c2 x&1 + c2 x&2 − k2 x1 + k2 x2 = b c1 m1 k2 x1 c2 m2 x2 Pham Huy Hoang 10 Pham Huy Hoang Pham Huy Hoang 11 Pham Huy Hoang Pham Huy Hoang 12 Pham Huy Hoang 13 ... change in current) The relation of this counter emf to the current is the origin of the concept of inductance Pham Huy Hoang ELECTROMECHANICAL SYSTEMS Magnetic Force and Lorentz force law: - The... displacement of the rotor of the motor & θ = ω : angular velocity of the rotor T : torque applied to the rotor Ke : emf constant (Vs/rad) Ki : torque constant (Nm/A) Pham Huy Hoang ELECTROMECHANICAL SYSTEMS. .. Hoang ELECTROMECHANICAL SYSTEMS Faraday's Law Any change in the magnetic environment* of a coil of wire will cause a voltage (emf) to be "induced" in the coil * The change of magnetic field strength,