9 Basic Characteristics of Lead, Lag and Lag-Lead Compensation

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Control Engineering (ME55)

Sess Se ssio ion n IV

SYSTEM COMPENSATION Basic Characteristics Of Lead, Lag And Lag-Lead Compensation:

R (S )

C (S ) COMPENSATOR

PLANT

FEEDBACK ELELMENT

Lead compensation compensation essenti essentially ally yields yields an appropri appropriate ate improvemen improvementt in transient resonse and a small imrovement in steady state accuracy. Lag compensation on the other hand, yields an appreciable improvement in steady state accuracy ast the expense of increasing theh transient responsetime. Lag-lead compensation combines the characterisitcs of both lead lead compensation compensation and and lag compensatio compensation. n. The use of a lead or lag lag 1 Srin rinidhi.R i.R, Fa Facul culty, ty, Me Mechan chanic icaal En Engine ineering ing, SJ SJCE, My Mysore 570006

Session IV: IV: 22 22/11/20 /2006



compenastion saises the order of the system by one. The use of a laglead compensator raises the order of the system by two (unless cancellation occurs between the zeroes of the lag-lead network and the poles of the uncompensated open-loop transfer function), which means that the system becomes more complex and it is more difficult to control the transient response behavior. The particular situation determines the type of the compensation to be used.

Lead Compensation

Procdure for the design of suitable compensator starts from the type of network adopted in the process. For example a Mechanical Network or an electrical network and derive the transfer function for the type of network chosen.Once the transfer fucntion is obtained, then the procedure for designing lead compensator based on the root-locus aproach or frequency response approach.

Lead Network. A schematic diagram of an electrial lead network is shown . Mechanical Network comprises of SPRINGS and DAMPERS and

Electrical

Network

comprises

of

RESISTORS

AND

CAPACITORS. The name lead network comes from the fact that for a sinusoidal input, ei, the output e0 of the network is also sinusoidal with phase lead. The phase lead angle is a function of the input frequency.

2 Srinidhi.R, Faculty, Mechanical Engineering, SJCE, Mysore 570006

Session IV: 22/11/2006



The transfer fucntion of the network is derived assunming the sourec network to be ZERO and the output load impedance is infinite.

• Basic Step involves finding out the Transfer Function (TF) of the network

• Next step involves procedures for designing lead compensator based on the root locus approach and /or Frequency Response approach 1. TRANSFER FUNCTION • Consider a system represented by an ELECTRICAL & a

MECHANICAL network

• Lead Network comes from the fact that for a sinusoidal input ei, the output eo is also sinusoidal with phase lead 3 Srinidhi.R, Faculty, Mechanical Engineering, SJCE, Mysore 570006




• Phase lead angle is a function of input frequency • In order to derive the TF, the complex impedances Z1 and Z2 Z1=[ R1/(R1Cs+1)] and Z2=R2

• Assumption made is that the source impedance is ZERO and that of the output load is INFINITE

• XO(S)/XI(S) = (S+1/T)/(S+1/ αT) • This has a ZERO at s= -(1/T) • This has a POLE at s = -(1/ αT) • Zero is always located far to the left • Minimum value of α is limited by the physical construction of the lead network

• If α is very very small then it is necessary to cascade an amplifier in order to compensate for the attenuation of the lead network .





• In the polar plot, for a given value of α, the angle between the positive real axis and the tangent line drawn from the origin to the semicircle gives the MAXIMUM PHASE ANGLE

Φm

• Frequency at the tangent point is ωn • Phase angle at ω = ωn is given by • Sin Φm = (1-α /2)/(1+α /2)= (1-α)/(1+α) • Shows relationship of LEAD ANGLE &Value α 5 Srinidhi.R, Faculty, Mechanical Engineering, SJCE, Mysore 570006




• In Bode Plot of lead network for α=0.1 shows that corner frequency as

ω=1/T and ω=1/ αT , ωm is the geometric mean of

two corner frequencies

• Log ωm = ½ { log(1/T) + log(1/ αT)}, which is ωm = 1/ √ αT • Lead Network is a high pass filter ,hence low frequencies are attenuated

• Additional GAIN is required elsewhere to increase low frequency gain Consider the elctrical lead network as shown. Using defined symbols, Z1={R1/(R2Cs+1)} and Z2= R2 The transfer function between the output Eo(s) and input Ei(s) is [Eo(s)/Ei(s) ] = [Z2/(Z1+Z2)] = [R2/(R1+R2)] [(R1Cs+1)/((R1R2/R1+R2)Cs+1)] Defining R1C=T & [R2/(R1+R2)]= α<1

and substituting in the above,

then the transfer fucntion becomes

[Eo(s)/Ei(s) ] = α{Ts+1/ (αTs+1)} =[( s+(1/T))/ (s+(1/ αT))] [Eo(s)/Ei(s)] = [(s+(1/T))/ (s+(1/ αT))] On the similar lines, the transfer fucntion for a mecanical lead network may also be obtained which is the same as that of an electrical lead network.





[Eo(s)/Ei(s)] = [(s+(1/T))/ (s+(1/ αT))]

A lead network has the following transfer fucntion.

α{Ts+1/ (αTs+1)} =[( s+(1/T))/ (s+(1/ αT))] (for α <1) This has a ZERO at s= -1/T and a pole at s= -

[1/(α T)]. Since α <1,

ZERO is always located to the right of the pole in the complex plane. It may also be noted tat for a small value of α, the pole is located far to the left. The minimum value of α is limited by the physical construction of the lead network. The minimum value of α is usually taken to be 0,.07. If the value of α is small, it is necessary to cascade an amplifier in order to compensate for the attenaution of the lead network. In the polar plot of the lead network replacing

s by jω

α [(jωT+1)/( jω αT+1)] (0< α<1) For a given value of α, the angle between the positive real axis and the tangent line drawn from the origin to the semicircle gives the

maximum phase angle Φ m, calling frequency at the tangent point to be ωm, the phase angle at ω = ωm, is

Sin Φm = [((1- α)/2)/ ((1+ α)/2)] = [(1- α)/ (1+ α)]

This relates the maximum phase angle and the value of α. In the Bode diagram of the lead network (when α= 0.1), the corner frequencies for the lead network are ω = 1/T and ω = 1/( αT). From





the figure it is clear that ωm is the geometric mean of the two corner frequencies or , Log ωm = ½[ log (1/T) + log 1/( αT)]

ω = ωm = 1/ √ αT As seen from the figure, the lead network is basically a high passs filter. (The high frequencies are passed but the low frequencies are attenuated). Therefore an additional gain elsewhere is needd to increase the low frequency gain.

LAG COMPENSATION: The Procedural aspects of designing a suitable lag compensator is exactly the same as given in the lead compensator design.



Mass F= m F x :Mass

Tra



• In Bode Plot of lag network for β=10 shows that corner frequency as ω=1/T and ω=1/ βT frequencies

• Lag Network is a low pass filter, hence high frequencies are attenuated



9 Basic Characteristics of Lead, Lag and Lag-Lead Compensation

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