LINEAR SYSTEM SIMULATOR

Aim:

To study the time response of a variety of simulated linear systems and to correlate the studies with theoretical

results.

Theory:

1) The block diagram of a general closed loop system is shown:

The closed loop transfer function in terms of the open loop transfer function G(s) is given by:

𝐶(𝑠)

𝑅(𝑠)=

𝐻(𝑠)

1 + 𝐻(𝑠)

2) The gain of an integrator (K) with respect to the pk to pk amplitude of the input square wave (Vin),

frequency of the input square wave, pk to pk amplitude of the output triangular wave is given by:

𝐾 =4𝑉𝑜𝑓

𝑉𝑖𝑛

3) The block diagram of a first order system of Type 0 is shown:

The closed loop transfer function is:

𝐶(𝑠)

𝑅(𝑠)=

𝐾𝐾3

𝐾𝐾3 + 1 + 𝑇𝑠

K is equal to the steady state output voltage when a unit step is applied to the system and T is the time

for the output to reach 63.2% of the final value.

4) The block diagram of a first order system of Type 1 is shown:

The closed loop transfer function is:

𝐶(𝑠)

𝑅(𝑠)=

𝐾𝐾1

𝐾𝐾1 + 𝑠

K is equal to the steady state output voltage when a unit step is applied to the system and T is the time

for the output to reach 63.2% of the final value.

5) The block diagram of a second order system of type 1 is given as:

The general form of transfer function of such a system is given as:

𝐶(𝑠)

𝑅(𝑠)=

𝜔2𝐾

𝑠2 + 2𝜁𝜔𝑛𝑠 + 𝜔2

Observations:

A) Determination of System Parameters for various components

i) Error Detector and Variable Gain

Input Signal = 204 mV (pk-to-pk)

Output Signal when input is given at Reference = -1900 mV (pk-to-pk)

Gain between reference input and output = -1900/204 = -9.31

Output Signal when input is given at Disturbance= -1920 mV (pk-to-pk)

Gain between disturbance input and output = -1920/204 = -9.41

Output Signal when input is given at feedback = -1920 mV (pk-to-pk)

Gain between feedback input and output = -1920/204 = -9.41

Output when input is given at 3 inputs = -5.60 V

Gain between output and 3 inputs = -5.6/0.204 = -27.45

This suggests that the output voltage of the Error detector unit (e0) when given an input of e1, e2, e3

is:

𝑒𝑜 = −9.4(𝑒1 + 𝑒2 + 𝑒3)

Reference (Input - 1) (Output - 2) Feedback (Input - 1) (Output - 2)

Disturbance (Input - 1) (Output - 2) Adder Output (Input - 1) (Output - 2)

ii) Disturbance Adder

Input voltage = 204 mV

Output when input is applied at terminal 1 only = -232 mV

Output when input is applied at terminal 2 only = -240 mV

Output when input is applied both to terminal 1 and 2 = -416 mv

Gain between terminal 1 and output = -232/204 = -1.137

Gain between terminal 2 and output = -240/204 = -1.176

Overall transfer function of the Disturbance adder with inputs e1, e2 and output e0 :

𝑒𝑜 = −(𝑒1 + 𝑒2)

Error 1 (Output - 2) Error 2 (Output - 2)

Error Sum (E1+E2) (Output - 2)

iii) Uncommitted Amplifier

Input voltage = 1.02 V

Output when input is applied at terminal 1 only = -1.02 V

Gain = -1

Unity Gain Amplifier (Output - 2)

iv) Integrator

Input voltage of the square wave (pk to pk) (Vin) = 1.02 V

Frequency of the square wave (f) = 20 Hz

Nature of the output wave form = Triangular

Peak to Peak Output Voltage (Vo) = 136 mV

Phase difference between input and output = 180 deg

The Gain of the integrator is given by:

𝐺 =4𝑉𝑜𝑓

𝑉𝑖𝑛 =

4 ∗ 0.136 ∗ 20

1.02= 10.67

Hence the transfer function of the integrator block is:

𝐻(𝑠) = −10.67

𝑠

Integrator Block (Output -2)

v) Time constant

For Block 1

Input Voltage (pk to pk) = 106 mV

Output Steady State voltage (pk to pk) = 880 mV

Time constant = 1.3 ms

Phase difference between input and output = 180 deg

Gain value (K) of the time constant block = 880/106 = 8.3

Transfer function of the time constant block:

𝐻(𝑠) = −8.3

0.0013𝑠 + 1

For Block 2

Input Voltage (pk to pk) = 106 mV

Output Steady State voltage (pk to pk) = 880 mV

Time constant = 1.3 ms

Phase difference between input and output = 180 deg

Gain value (K) of the time constant block = 880/106 = 8.3

Transfer function of the time constant block:

𝐻(𝑠) = −8.3

0.0013𝑠 + 1

Time Constant Block (Output - 2)

B) Study of First Order and Second Order Systems

i) First Order Type 0

S. No Input Type K Steady State Value (V) Time Constant (ms)

1 Square 4 1 Can’t be evaluated

First Order Type 0 (Output - 2) (Input - 1)

ii) First Order Type 1

S. No Input Type K Steady State Value

(V) Time Constant (ms)

1 Square 2 0.304 Can’t be evaluated

2 Square 10 0.900 14

3 Ramp 10 0.630 13

First Order Type 1 (Output - 2) (Input - 1) K=2 First Order Type 1 (Output - 2) (Input - 1) K=10

First Order Type 1 (Output - 2) ( Input - 1) K=10

iii) Second Order System Type 0

Applied input signal = 1.02 V pk to pk

S. No K Mp (V) t-peak (ms)

t-rise (ms) t-settling (ms) ζ ω

(rad/s)

1 4 0.3 2 1 12 0.3571 1681.67

2 10 0.48 1.2 0.606 12 0.2275 2688.49

Second Order Type 1 (Output - 2) (Input - 1) K=10 Second Order Type 1 (Output - 2) (Input - 1) K=4

From Theory:

Open Loop transfer Function of the System can be written as:

𝐻(𝑠) =5 ∗ 𝐾𝐾1𝐾3

(𝑠)(𝑠𝑇2 + 1)=

5 ∗ 10.67 ∗ 8.3 𝐾

𝑠(0.013𝑠 + 1)=

442.805𝐾

𝑠 + 0.013𝑠2

The theoretical closed loop transfer function is hence:

𝐺(𝑠) =𝐻(𝑠)

1 + 𝐻(𝑠)=

442.805𝐾

0.013𝑠2 + 𝑠 + 442.805𝐾=

34061.92𝐾

𝑠2 + 76.92𝑠 + 34061.92𝐾

Comparing from the standard form the following are derived:

𝜔 = √(34061.92𝐾)

𝜁 = 38.46

√(34061.92𝐾)

Theoretical Calculations:

S. No K ζ (theory) ω (theory)

1 1 0.208 184.559

2 4 0.104 369.12

3 10 0.066 583.626

Conclusion: For the first order type 1 systems, Low frequency input signal should be used so as to allow

enough time for the step response to reach the steady state value.

However, on using minimum frequency possible, steady state couldn’t be reached and a

finite steady state error was observed. Consequently time constant was immeasurable when

K was low.

For a first order system, speed of response increases as K increases.

Accurate values of Peak Time and Settling time couldn’t be recorded because of low

precision of the instrument and manual observation.

Since, 𝜔 depends on the inverse of peak time, and peak time is very small 𝜔 fluctuates

heavily on small changes in peak time. And since, peak time couldn’t be calculated

accurately a large deviation in theoretical and experimental values was recorded.