Lab:5

To plot and analyze the responses of first order systems

for various test signals.

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• Title:To plot and analyze the responses of first order systems forvarious test signals.

• Apparatus:Trainer kit of linear system simulator, CRO, connectingleads and multi-meter.

• Objectives:

1. To study the behavior of first order system for different kind ofexcitations.

2. To analyze the transient response of the system.

3. To analyze the steady state response of the system.

• Theory:

1. Description of Trainer Kit:

This setup has been designed to provide a convenient means

Figure 1: Trainer Kit of Linear System Simulator

of studying the transient response of linear systems. There arethree built in sources in the unit:

Square wave : Frequency 20− 80Hz.

: p-p amplitude 0− 2Volt

Triangular : Frequency 20− 80Hz.

: p-p amplitude 0− 2Volt

Trigger : Frequency 20− 80Hz.

: Amplitude ± 5Volt.

Following types of blocks have been included in the system:

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(a) Error Detector-Cum-Gain: The block has three inputs(e1,e2,e3)andone output (eo) which are related by the expression

eo = −K(e1 + e2 + e3)

where K is the gain. The value of K can be varied from 0to 10 by a ten-turn potentiometer.

(b) Integrater: The integrator block has an approximate trans-fer function of the form−

K1

s and is used in simulating type-1systems.

(c) Time constant: The two time constant blocks in the systemhave transfer functions of the form −

K2

sT+1 each. The secondblock has an ×5 option which results in a higher gain ifnecessary.

(d) Disturbance Adder: This is a two input (e1,e2), one output(eo) block having a defining equation of the form

eo = −(e1 + e2)

(e) Amplifier: It is used to provide negative feedback.

2. Transient Analysis of LTI system

First order systems:

These systems are characterized by one pole and/or a zero. hav-ing common transfer function of the form K/s and K/(sT +1).Unit step response of the systems are computed as follows and

Figure 2: Unit Step Response of First Order Transfer Functions

are shown in Figure 2:1. For C(s)/R(s) = G(s) = K/s and R(s) = 1/s⇒ C(s) = K/s2

On taking inverse laplace transform we get c(t) = Kt2. For G(s) = K/(sT + 1) and R(s) = 1/s

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Figure 3: Square Wave Response of First Order Transfer Functions

⇒ C(s) = K/s(sT + 1)On taking inverse laplace transform we get

c(t) = K(1− e−t/T ) (1)

Time constant of the system is defined from 1 as t=T whichgives

c(t) = K(1− e−1) = 0.632K

3. Steady State Analysis of LTI system

Steady state performance of a stable system is governed by itssteady state error to step,ramp and parabolic inputs. Consider

G(s)+R(s) E(s) C(s)

-B(s)

Figure 4: Unit Feedback System

the unity feedback system as shown in the Figure 4. Where R(s)is input, C(s) is output, B(s) is feedback signal and E(s) is errorsignal. The transfer function of the system is defined as:

C(s)

R(s)=

G(s)

1 +G(s)(2)

C(s) = E(s)G(s)

⇒ E(s) =1

1 +G(s)R(s)

The steady state error is calculated using final value theorem asfollows.

ess = limt→∞

e(t) = lims→0

sE(s) = lims→0

sR(s)

1 +G(s)(3)

The expression for steady state errors for various types of stan-dard test signals are derived below:

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(a) Unit step input : r(t) = u(t). Laplace Transform: R(S) =1/sFrom (3)

ess = lims→0

1

1 +G(s)=

1

1 +Kp(4)

where Kp is defined as the position error constant.

(b) Unit ramp input : r(t) = t. Laplace Transform: R(s) =1/s2

From (3)

ess = lims→0

1

s+ sG(s)= lim

s→0

1

sG(s)=

1

Kv(5)

where Kv is defined as the velocity error constant.

(c) Unit parabolic input : r(t) = t2/2. Laplace Transform:R(s) = 1/s3

From (3)

ess = lims→0

1

s2 + s2G(s)= lim

s→0

1

s2G(s)=

1

Ka(6)

where Ka is defined as the acceleration error constant.

• Pre lab assignments:

1. Obtain the characteristic equation of typical first order system.

2. Differentiate between linear and nonlinear system.

3. On what factors the time constant of first order system dependsand how it affects the system performance.

• Procedure:

1. Open loop response:In this section open loop transfer function of the integrator andtime constant blocks are determined.

(a) Integrator

i. Apply 1 V p-p square wave input of known frequency(f).

ii. Measure p-p output voltage of the integrator block.

iii. Calculate the gain constant K1 using the formula

K1 =4f(Output p-p)

Input p-p(7)

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iv. Write the transfer function of the block.

v. Simulate the system in MATLAB and compare simula-tion and experimental results.

(b) Time constant

i. Apply 1 V p-p square wave of known frequency (f) tothe input of time constant block.

ii. Determine the time constant i.e. the time time t = Tat which the response reaches 63.2%.

iii. Determine the steady state value of the response.

iv. Find the value of K2 using formula

K2 =p-p Steady state output

p-p Input amplitude(8)

v. Write the transfer function of the block.

vi. Repeat the procedure for second time constant block.

vii. Simulate the system in MATLAB and compare simula-tion and experimental results.

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2. Close loop response:

(a) First order system:There are two forms of first order system possible in thiskit. these are shown in Figure 5.

(a) Type 0 System (b) Type 1 System

Figure 5: Closed Loop First Order System

i. Make proper connections and apply 1 V p-p square waveinput.

ii. Calculate time constant and steady state error for dif-ferent values of K.

iii. To calculate steady state error for type-1 system use 1V p-p ramp input.

iv. Simulate the system in MATLAB and compare simula-tion and experimental results.

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• Observations:

1. Open Loop 1st order type-0 system:

S.No. K Time Constant T (msec) C(∞)p-p V ess = 1− C(∞)V

2. Closed Loop 1st order type-0 system:

S.No. K Time Constant T (msec) C(∞)p-p V ess = 1− C(∞)V

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3. Open Loop 1st order type-1 system:

S.No. K Time Constant T (msec) C(∞)p-p V ess = 1− C(∞)V

4. Closed Loop 1st order type-0 system:

S.No. K Time Constant T (msec) C(∞)p-p V ess = 1− C(∞)V

• Post Lab Assignment:

1. Describe how operational amplifier can be used to develop first,second and third order system.

2. List the references referred by you.

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