Question list

LABORATORY WORK #1.5-1.6

The purpose of this work is to study:

- Nyquist diagrams

- Gain frequency characteristics

- Phase frequency characteristics

- Bandwidth function.

Task 1

Your task is to obtain Nyquist diagram using the following Matlab script:

H = tf([100],[0.1 1.1 1])

nyquist(H)

Using this example your task is to obtain Nyquist diagrams for the following transfer functions:

-

-

-

Each student has to choose its values for within the interval (0;+10).

Task 2

Using the following Matlab scripts define gain frequency characteristics of the 1^st and 2^nd order links:

1^st order links:

w=0:0.05:3;

T=[1 2 3 4];

for k=1:4

Gnum=[0 1]; Gden=[T(k) 1];

Gjomega=freqs(Gnum,Gden,w);

Gmag=abs(Gjomega);

plot(w, Gmag);

title('Gain Frequency Response of G(s)')

xlabel('Omega')

ylabel('G(j*omega)')

grid on

hold on

end

hold off

Underdamped links:

w=0:0.05:3;

z=[0.25 0.5 0.707 0.9];

for k=1:4

Gnum=[0 0 1]; Gden=[1 2*z(k) 1];

Gjomega=freqs(Gnum, Gden, w);

Gmag=abs(Gjomega);

plot(w, Gmag);

title('Gain Frequency Response of G(s)')

xlabel('Omega')

ylabel('G(j*omega)')

grid on

hold on

end

hold off

Overdamped links:

w=0:0.05:3;

z=[1 1.25 1.5 1.75 2];

for k=1:4

Gnum=[0 0 1]; Gden=[1 2*z(k) 1];

Gjomega=freqs(Gnum, Gden, w);

Gmag=abs(Gjomega);

plot(w, Gmag);

title('Gain Frequency Response of G(s)')

xlabel('Omega')

ylabel('G(j*omega)')

grid on

hold on

end

hold off

Using these examples your task is to obtain Matlab scripts and gain frequency characteristics for the following transfer functions:

-

-

-

Each student has to choose its values for within the interval (0;+10).

Task 3

Using the following Matlab scripts define phase frequency characteristics of the 1^st and 2^nd order links:

1^st order links:

Matlab script:

w=0:0.1:3;

t=[1 2 3 4];

for k=1:4

x=w*t(k);

phi=-atan(x)

y=phi*180/pi;

title('Phase Frequency Response')

xlabel('Omega')

ylabel('Phase angle')

plot(w, y);

grid on

hold on

end

hold off

Overdamped links:

w=0:0.1:6;

for k=1:4

t3=[2 3 4 5];

t4=[1 2 3 4];

x1=w*t3(k);

x2=w*t4(k);

phi=-atan(x1)-atan(x2)

y=phi*180/pi;

plot(w,y)

title('Phase Frequency Response')

xlabel('Omega')

ylabel('Phase angle')

grid on

hold on

end

hold off

Underdamped links:

Y = atan(X) returns the inverse tangent (arctangent) for each element of X. For real elements

of X, atan(X) is in the range

The atan function operates element-wise on arrays. The function's domains and ranges include complex values.

All angles are in radians.

Matlab script:

x = -20:0.01:20;

y=atan(x)*180/pi;

plot(x,y)

title('Arctangent Function')

xlabel('X')

ylabel('Angle (degrees)')

grid on

The singular point is defined by the point where .

So in order to construct a phase frequency characteristic it’s necessary to create Matlab scripts before and after the singular point:

Before the singular point:

for w=0:0.1:1

t=1;

zeta=0.9;

a=(2*zeta*w*t)/(1-w.^2*t.^2);

alpha=-atan(a)

y=alpha*180/pi;

a,alpha,y

plot(w,y,'-bs')

title('Phase Frequency Response, zeta=0.9')

xlabel('Omega')

ylabel('Phase angle')

grid on

hold on

end

hold off

After the singular point:

for w=1.1:0.2:6

t=1;

zeta=0.9;

a=(2*zeta*w*t)/(1-w.^2*t.^2);

alpha=-atan(a)

y=-alpha*180/pi;

z=-y-180;

plot(w,z,'-bs')

title('Phase Frequency Response, zeta=0.9')

xlabel('Omega')

ylabel('Phase angle')

grid on

hold on

end

hold off

for w=0:0.1:1

t=1;

zeta=0.1;

a=(2*zeta*w*t)/(1-w.^2*t.^2);

alpha=-atan(a)

y=alpha*180/pi;

a,alpha,y

plot(w,y,'-rs')

title('Phase Frequency Response, zeta=0.1')

xlabel('Omega')

ylabel('Phase angle')

grid on

hold on

end

hold off

for w=0:0.1:1

t=1;

zeta=0.5;

a=(2*zeta*w*t)/(1-w.^2*t.^2);

alpha=-atan(a)

y=alpha*180/pi;

a,alpha,y

plot(w,y,'-gs')

title('Phase Frequency Response, zeta=0.5')

xlabel('Omega')

ylabel('Phase angle')

grid on

hold on

end

hold off

After the singular point:

for w=1.1:0.2:6

t=1;

zeta=0.9;

a=(2*zeta*w*t)/(1-w.^2*t.^2);

alpha=-atan(a)

y=-alpha*180/pi;

a,alpha,y

plot(w,y,'-bs')

title('Phase Frequency Response, zeta=0.9')

xlabel('Omega')

ylabel('Phase angle')

grid on

hold on

end

hold off

for w=1.1:0.2:6

t=1;

zeta=0.1;

a=(2*zeta*w*t)/(1-w.^2*t.^2);

alpha=-atan(a)

y=alpha*180/pi;

a,alpha,y

plot(w,y,'--rs')

title('Phase Frequency Response, zeta=0.1')

xlabel('Omega')

ylabel('Phase angle')

grid on

hold on

end

hold off

for w=1.1:0.2:6

t=1;

zeta=0.9;

a=(2*zeta*w*t)/(1-w.^2*t.^2);

alpha=-atan(a)

y=-alpha*180/pi;

z=-90+y;

plot(w,z,'-bs')

title('Phase Frequency Response, zeta=0.9')

xlabel('Omega')

ylabel('Phase angle')

grid on

hold on

end

hold off

for w=1.1:0.2:6

t=1;

zeta=0.9;

a=(2*zeta*w*t)/(1-w.^2*t.^2);

alpha=-atan(a)

y=-alpha*180/pi;

z=90+y;

plot(w,z,'-bs')

title('Phase Frequency Response, zeta=0.9')

xlabel('Omega')

ylabel('Phase angle')

grid on

hold on

end

hold off

Using these examples your task is to obtain Matlab scripts and phase frequency characteristics for the following transfer functions:

-

-

-

Each student has to choose its values for within the interval (0;+10).

Task 4

Define frequency bandwidth characteristic for the following transfer function:

num=[0 0 1]; den=[1 1 1];

sys = tf(num,den)

fb = bandwidth(sys)

Transfer function:

-----------

s^2 + s + 1

fb =

1.2711

Using this example your task is to obtain Matlab scripts and define frequency bandwidth characteristics for the following transfer functions:

-

-

-

Each student has to choose its values for within the interval (0;+10).

Question list

1. Mass-Spring-Damper System

2. Block diagram models

3. Frequency Response Methods

4. The connection between the Laplace and Fourier transforms

5. The particular solution – the system forced movement component

6. Frequency characteristics of control systems

7. The rules to construct the Nyquist diagram

8. Frequency bandwidth characteristics of control systems

9. Gain-frequency characteristics of control systems

10. Phase-frequency characteristics of control systems

11. The notion of a link

12. Frequency characteristics of typical dynamic links

- amplification links

- the 1^st order links

- overdamped links

- underdamped links

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