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# Basic dynamic elements

## 1. AUTOMATICS and AUTOMATIC CONTROL

LECTURE 4
Automation and Metrology Department
Room no 210A

## 2.

Basic dynamic elements
Any, more or less complex systems (objects) can be presented as a
connection of some appropriate basic dynamic elements
1. Proportional (noninertial):
a) Differential equation:
y(t) = k *u(t)
d) Frequency response:
P(ω) = k
b) Transfer function:
G(s) = k
Lm(ω)=20logk
e) Example:
c) Step response:
H(s) = k/s
h(t) = k1(t)
k=-R2/R1
Q(ω) = 0
φ(ω)=0

## 3.

Basic dynamic elements
2. Inertial (1st order):
a) Differential equation:
T y (t ) y (t ) k u (t )
c) Step response:
1
t
1 k
h(t ) k 1 e T
H ( s)
s Ts 1
d) Frequency response:
b) Transfer function:
Y ( s)
k
G( s)
U ( s) Ts 1
P ( )
k
1 2T 2
Q( )
k T
1 2T 2

## 4.

Basic dynamic elements
2. Inertial:
e) Example:
U 2 (s) Y (s)
1
G(s)
U1 ( s) U ( s) RCs 1

## 5.

Basic dynamic elements
c) Step response:
3. Oscilator (2nd order):
a) Differential equation:
h(t ) k 1
T 2 y 2d T y y k u
b) Transfer function:
1
e
1 d
1 d 2
arctg
,
d
Y (s)
k
G (s)
2 2
U ( s ) T s 2d T s 1
d
t
T
sin
1 d
t
Tn
2
Tn T * 2 undumped
d) Frequency response:
period

## 6.

Basic dynamic elements
3. Oscilator (2nd order):
e) Example1: damped harmonic oscillator
G(s)
1
k
1
2
ms bs k m s 2 b s 1
k
k

## 7.

Basic dynamic elements
3. Oscilator (2nd order):
e) Example 2: RLC circuit
di (t )
u (t ) i (t ) R L dt y (t )
dy (t )
i (t ) C
dt
dy 2 (t )
dy (t )
LC
RC
y (t ) u (t )
2
dt
dt
G( s)
1
LCs 2 RCs 1

## 8.

Basic dynamic elements
4. Integrator (ideal integrator):
a) Differential equation:
1
y (t )
Ti
t
u (t )dt
0
c) Step response:
1 1
H ( s)
s Ti s
h (t )
1
t
Ti
d) Frequency response:
b) Transfer function:
G (s)
Y (s)
1
U ( s ) Ti s
P( ) 0
Lm(ω)=-20logωTi
Q( )
φ(ω)=-π/2
e) Example: (water flow q to the tank with water level area Ch ):
1
h(t )
Ch
t
q(t )dt
0
1
T

## 9.

Basic dynamic elements
5. Real integrator (with inertia):
a) Differential equation:
dy (t )
1
T
y (t )
dt
Ti
t
u(t )dt
0
c) Step response:
1
1
1
1
1
t
H (s)
h(t ) t T 1 e T
s Ti s (Ts 1)
Ti
Ti
d) Frequency response:
b) Transfer function:
G(s)
Y (s)
1
U ( s ) Ti s (Ts 1)
P( )
T
Ti (1 2T 2 )
Q( )
1
Ti (1 2T 2 )

## 10.

Basic dynamic elements
5. Real integrator (with inertia):
e) Example: DC motor
M e ki i (t )
u (t ) i (t ) R ce (t )
d (t )
M (t ) M J d (t )
(t )
0
e
dt
dt
JR
d (t )
JR d 2 (t )
,
u (t ) Tm
ce
2
ki ce
dt
ki dt
d 2 (t ) d (t )
ku(t )
Tm
2
dt
dt
1
k
ce
d (t )
(t ) k u (t )dt
Tm
dt
0
k
G ( s)
s Tm s 1
t

## 11.

Basic dynamic elements
6. Differentiator (ideal):
a) Differential equation:
y (t ) Td
du (t )
dt
c) Step response:
H ( s) Td
h(t ) Td t
d) Frequency response:
b) Transfer function:
Y ( s)
G( s)
Td s
U ( s)
P( ) 0
Q( ) Td

## 12.

Basic dynamic elements
6. Differentiator (ideal)
e) Example: Ideal capasitor
du (t )
i (t ) C
dt
G( s)
I (s)
sC
U ( s)

## 13.

Basic dynamic elements
7. Real differentiator (with inertia): c) Step response:
a) Differential equation:
T
dy (t )
du (t )
y (t ) Td
dt
dt
Td
G ( s)
Ts 1
1
t
T
e
d) Frequency response:
b) Transfer function:
Td s
G ( s)
Ts 1
T
h(t ) d
T
P
kT 2
1 T 2
Q
k
1 T 2

## 14.

Basic dynamic elements
7. Real differentiator (with inertia):
e) Example: RC circuit
t
1
u (t ) i (t )dt Ri (t )
C0
y (t ) Ri (t )
t
1
y (t )dt y (t ) u (t )
RC 0
dy (t )
du (t )
RC
y (t ) RC
dt
dt
G ( s)
RCs
RCs 1

## 15.

Basic dynamic elements
8. Delay
c) Step response:
a) Differential equation:
1 sT0
H ( s ) ke
s
y(t ) k u(t To )
h(t ) k1 t T0
d) Frequency response:
b) Transfer function:
Y ( s)
G( s)
ke sTo
U ( s)
P( ) k cos( T0 )
Q( ) k sin( T0 )

## 16.

Basic dynamic elements
8. Delay
e) Example: conveyor (transporter)
T0
V
l