mak-411e experimental methods in m.e. fall 2007...

1İstanbul Technical UniversityDr. Erdinç Altuğ

MAK-411EEXPERIMENTAL METHODS IN M.E.

Fall 2007-2008

LAB: Experimental Analysis of System Response

Dr. Erdinç Altuğ

November 6, 2007


The Purpose of the Experiment:

The responses of a 2nd order system (a RLC circuit) to various inputs will be analyzed and compared with the theoretical results. System response is used to compare performance of various control systems.

In this experiment, a RLC circuit (resistance, inductance and capacitance elements in series) is used as a second order system.


The Series RLC Circuit and Mechanical Analog


• A transient response or natural response is the response of a system to a change from equilibrium. Transient response is from an initial state to a final state.

In a mechanical system a simple example is a mass/spring/damper system. The transient response is the position of the mass as the system returns to equilibrium after an initial force or a non zero initial condition.

In electrical engineering a simple example would be the output of a 5 volt DCpower supply when it is turned on: the transient response is from the time the switch is turned on and the output is a steady 5 volts. At this point the power supply reaches its steady-state response of a constant 5 volts.

• The steady-state response follows the transient response. Steady State response is the behavior of the system as time goes to infinity.

In electrical engineering, a steady-state response is the electrical response of a system at equilibrium.

Total Response = Transient Response + Steady State Response

The System Response


System Response in our experiment

• For Transient Response Analysis, a step input and for Steady State Response Analysis, a sinusoidal input will be applied to the RLC circuit.

• If mathematical system model known => System response can be obtained using Transfer Functions by applying Laplace Transformation

• If no model exists => input is applied and system response analyzed experimentally

You will do the both. (Why?)


Page 96a

Continued



Pages 91-92 Modeling: Electric Circuits


Page 92 (1)


Pages 92-93a

Continued


Pages 92-93b

I: current

Q: charge


Page 93

Solving the ODE (1) for the Current Discussion of Solutions


Page 94


Page 94


Pages 94-95


n

….(1)

Natural frequency


n….(2)


n

n

The Damping Ratio

n

Roots of the

Equation 2

(zeta)


Solution Analysis

n


n


nd

n

1 2( ) cos( ) sin( )t tc n nx t K e t K e tα αω ω− −= +Solution in the form:


Summary: Equations for RLC Circuit (Also in lab sheet)

2

2 2( )2

n

n n

G ss s

ωζω ω

=+ +

2 1n LC

ω =

2R C

Lζ =

2

11LCs RCs+ +

Transfer Function:

( )E s ( )C s

The natural frequency

The damping Ratio


A) The Step Response

Step response is one way of observing the effect of circuit self-behavior, which is called the transient response or natural response.


Step Response of 2nd order System


Equations for The Step Response (Also in lab sheet)

2

2

10

n

x ex

πζ

ζ−=

2

d

T πω

=21d nω ω ζ= −

0 21n

t π

ω ζ=

−2

( )1

0x eζπ

ζ

−

−=

rd

t π βω−

=2

1 1tan

ζβ

ζ− −

=

Logarithmic Decrement

Period

Peak Time

Rise Time

Maximum Overshoot

Damped Natural

Frequency


B) Frequency-Response Analysis

• The frequency response is the system's response to sinusoidal inputs.

• The output of a linear time-invariant system to a sinusoidal input is a sinusoid of the same frequency but with a different magnitude and phase. The magnitude and phase differences between the input and output sinusoids are functions of frequency.

Frequency-Response: Steady-State response of a system to sinusoidal input

• A representation of the system's response to sinusoidal inputs at varying frequencies.

0( ) sinx t X tω= ⇒ 0( ) sin( )y t Y tω θ= +


• The most common ways to represent frequency response of a systemare Bode plot and Nyquist plot. A Bode plot shows the relationship between the magnitude and phase values of the open-loop transfer function G(j*w) and input frequency (when the input frequency w goes from zero to infinity). A Nyquist plot is a representation of open-loop transfer function G(j*w) in s plane. The Nyquist plot allows us to predict the stability performance of a closed-loop system by observing its open-loop behavior.

• Both amplitude (magnitude) and phase are affected by frequency and characterize the sinusoidal response. Its phase versus frequency is also significant. When plotted, they are called Bode or frequency response plots.

• After enough time, the transient response decays away and the steady-state response alone is left.


Advantages of controller design based on frequency response analysis1. A powerful tool both for analyzing dynamic systems and for designing

controllers.2. It is applicable to dynamic models of any order.3. The designer can specify the desirable closed-loop response characteristics.4. Information on stability margins and sensitivity characteristics is provided.

Disadvantage1. It is generally iterative in nature hence it can be time-consuming.


• After enough time, the transient response decays away and the steady-state response alone is left.

It can be understood in terms of |G(jw)| - magnitude of the Fourier transform of the impulse response (transfer function)∠G(jw) – phase of the Fourier transform of the impulse response (transfer function)

Bode plots are plots of log magnitude and phase against log frequency.

Used to plot a greater range of frequenciesUsed to plot decibel-type informationTransfer function is now “additive”


Bode Plot 2nd Order System

The LTI 2nd order differential equation

ωn is the natural frequencyζ is the damping ratio

Bandwidth is defined as the frequency at which the closed-loop system gain drops 3 dB below peak.

)()()(2)( 222

2

txtydt

tdydt

tydnnn ωωζω =++

ωn=1ζ=[0.01 0.1 0.4 1 1.5]

Bode plots are plots of log magnitude and phase against log frequency.

Used to plot a greater range of frequenciesUsed to plot decibel-type informationTransfer function is now “additive”


Equations for Frequency Response (Also in lab sheet)

21 2p nω ω ζ= −

2

1( )2 1

pG ωζ ζ

=−

1( )2nG ωζ

=

( ) out

in

VGV

ω =

(sec) 360(sec)T

φφ =

0 0.707ζ≤ ≤For the range:

Phase Shift

Resonant Frequency

Magnitude of Resonant Peak

Amplitude Ratio


Modeling in MATLAB% Simulate a series RLC circuit.

% The output is the voltage across the capacitor. % Set the R, L, and C values. R = 2; L = e-2; C = 100E-4;

% Set the system matrices. % xdot = Ax + Bu, y = Cx + Du. A = [0 1/C; -1/L -R/L]; B = [0; 1/L]; C = [1 0]; D = [0];

% Create a Matlab state-space model. sys = ss(A, B, C, D);

% Compute and plot the step response.

[y, t] = step(sys);

plot(t, y);

title('Step Response');

% Compute and plot the zero-state response to a sinusoidal input.

dt = 0.003;

tf = 5;

t = 0 : dt : tf;

u = sin(t);

[ysine, t] = lsim(sys, u, t);

plot(t, ysine);

title('Sine Response');

%To plot the bode diagram

Bode(sys);


Lab Procedure

1. Set-up the circuit (Oscilloscope, Signal Generator, Resistance, Inductance, Capacitance, Connection Cables)

2. Apply Step input to the RLC circuit3. Change R, L, and C values and observe effect of each element on the

system (oscillations, steady-state error, etc.) Pick a good response.4. Calculate theoretical values for and . Determine the

experimental values.5. Switch input to Sin wave. (Frequency Response Analysis)6. Increase the value of the Sine input, and read the output and phase

difference. Fill the table.7. Calculate the amplitude ratio, phase difference. Plot Bode diagrams using

Table.8. Plot Bode diagrams using R, L, and C values. (Theoretical) 9. Compare the theoretical and the experimental results, explain the reasons

for the differences.

nω ζ

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