Concept Of Transfer function:

Selection Of System Parameters

Selected Input Output

• In any system ,first the system parameters are designed and their values are selected as per the requirements.

• The input is selected next to see the performance of the design system.

• Effect of system parameters = output Input

• Mathematically such a function explaining the effect of system parameters on input to produce output is called Transfer function.

Definition of Transfer Function

T(s)R(s) C(s)

• Mathematically it defined as the ratio of Laplace transform of output of the system to the Laplace transform of input under the assumption that all initial condition are zero.

• T(s)= Laplace transform of the output = C(s) Laplace transform of the input R(s)

Advantages:

• Individual analysis of various components is possible by the transfer function approach.• As it uses a Laplace's approach, it converts time domain equations to simple algebraic equation.•It suggests operational method of expressing equations which relate output to input.•Once transfer function is known, output response for any type of reference input can be calculated.

Disadvantages: •Only applicable to Linear Time Invariant system.•It does not provide any information concerning the physical structure of the system. i.e. whether it is mechanical or electrical cannot be judged.•Effects arising due to initial conditions are totally neglected.

Procedure to determine the transfer function of a system:

•Write down the time domain equations for the system by introducing different variables in the system.

•Take the Laplace transform of the system equation assuming all initial conditions to be zero.

•Identify the system input and output variables.

•Eliminating the introduced variables, get the resultant equation in terms of input and output variables.

•Take the ratio of Laplace transform of output variable to Laplace transform of input variable to get the transfer function model of the system.

Transfer function of separately excited D.C.Motor

Electro Mechanical System

R a = Armature Resistance ,ΩL a = Armature inductance, HI a = Armature current, AVa =Armature Voltage, VR f = Field Resistance ,ΩL f = Field inductance, HI f = Field current, AVf = Field Voltage, Veb = Back E.M.F, VK t = Torque constant, N-m/AT = Torque Developed by motor, N-Mω = Angular displacement of shaft, radiansJ = Moment of inertia, Kg-m²/radB= Frictional Co-Efficient of motor and load, N-M (rad/Sec)K b = Back E.M.F constant, V (rad/sec)

Equivalent circuit of armature ( field excitation kept constant)

By kvl, we can write :

ia Ra + La d(Ia ) + eb = Va …….(1)

dt

Torque of d.c motor is proportional to the product of flux and current . Since the flux is constant in this system , the torque is proportional to ia alone.

T is proportional to Ф ia

Torque (T) =Kt ia……(2)

The differential equation governing the mechanical system is given by

Jd2ω/d2t+ B dω / dt…….(3)

The back emf of d.c.motor

eb is proportional to dω/dt

eb=kb dω….(4)

dt

By taking laplace transform of equation (1),(2),(3)&(4)

ia(s)Ra+La(s) + eb(s) =Va(s)……….(5)

T(s) = kt . ia(s)…………………………..(6)

Js2 ω(s) + B s ω(s) = T(s) )………….(7)

eb (s) = Kb s ω(s)………………….....(8)

La

Ra= ĩa = Electrical time constant

JB = ĩm = mechanical time constant

Vds

Vqr

=Rds + Lds

Md ω r Rqr + Lqr P

ids

iqr

Generalized theory system

QR= Armature coil = A , rqs = ra , Lqs = La

DS= Field coil = F , rds = rf , Lds = Lf

Md = Mutual inductance between coil DS & QR

ωr = Speed in radians/sec

Transfer function of separately excited D.C.Motor

The generalized mathematical model is as shown in fig above. The voltage equations for this model can be written as

dsqr

ds qr

General Torque ConventionAfter substituting the equivalent parameters we getVf= (rf+Lfp)if (2)Vt=Mdωr if+ (ra+Lap)ia (2)

Tm=mechanical torque of the machine results in the rotation of shaft in the reference positive direction for ωr.Te=electromagnetic torque of the machine results in the rotation of shaft in the reference positive direction ωr.

These torques have to overcome inertia and damping torques as shown in figure Te+Tm=Jpωr+D ωr

TL= load torque, causes the torque to flow out of the machine at positive speed.TL= -Tm

Equation Te=Jp ωr+D ωr+TL

J=polar moment of inertia of motot and load in kgm2

D=viscous co-efficient of friction for moving parts in Nm/radian/sec.Mdifia=Jp ωr+D ωr+TL (3)

Torque Equations

For obtaining Transfer function

Here the armature voltage control with constant field excitation is considered. Since the field current is constant if is replaced by If .The transfer function relating the speed ωr & armature terminal voltage vt can be obtained from equations 2&3. From equation 3&2 friction torque Dωr , Load torque TL & Armature inductance La are neglected for simplicity. So equations reduces to:

Vt=Md ωrIf+ra Ia

MdIfia=Jp ωr

Vt=km ωr+raia (4)

Kmia=Jpωr (5)

elimination of ia from the equation we get

The transfer function in s-domian

Finally block diagram of d.c. separately excited motor

Electro Mechanical System

Transfer function of separately excited D.C.Generator

Consider a separately excited generator

Rf = field resistane in ,ohmsLf = field inductance in H.Vf =applied voltage in V.Ea = generated voltage in V. For a generator

Ea = ФZNP 60A

Ea proportional to Ф

Ea proportional to if

Let Ka be the generator constant so,

Ea = Ka if --------(1)

By KVL to field circuit

Vf=if.Rf+Lf d(if/dt)-------- (2)

Taking Laplace Transform both equations (1) & ( 2)

Ea(s) = Ka.If (s)

Vf(s)= RfIf(s)+Lf(s). If(s)

If(s)= vf(s) Rf+sLf

Ea(s)=Ka.Vf(s) Rf+sLf

Ea(s) = Ka

Vf(s) (Rf+sLf)

Generalized theory system

Transfer function of separately excited D.C.Generator

Vds

Vqr

=Rds +Lds

Md ω r Rqr + Lqr P

Ids

iqr

The generalized mathematical model is as shown in fig above. The voltage equations for this model can be written as

dsqr

ds qr

QR= Armature coil = A , rqs = ra , Lqs = La

DS= Field coil = F , rds = rf , Lds = Lf

Md = Mutual inductance between coil DS & QR

ωr = Speed in radians/sec(1)

No load operation: ia = 0

Transfer function in s-domain is

On load operation: iqr = -ia

For a load impedance(RL+Lfp) of armature

The armature current in s- domain

Block diagram of no-load operation

Block diagram of on-load operation