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Linear Systems
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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.