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The transfer functionThe transfer functionThe transfer functionThe transfer function
• A linear and invariant system is described by the equation:
• where:
• y is the output
• x is ithe input
( ) ( 1) ( )
0 1 1
( ) ( 1) ( )
0 1 1
* * * *
* * * * , ( )
n n
n n
m m
m m
a y a y a y a y
b x b x b x b x n m
− •
−
− •
−
+ + + + =
= + + + + ≥
K
K
The transfer functionThe transfer functionThe transfer functionThe transfer function
• =L
L =
()
()=
⋯
⋯, ≥
( ) ( ) ( )zmzz
mmmssssssconstsbsbsb −−−=+++ −− *...***.... 2
2
2
1
10 1
( ) ( ) ( )n
nnnssssssconstsasasa −−−=+++ −− *...***.... 2
2
2
1
10 1
The transfer functionThe transfer functionThe transfer functionThe transfer function
• Zeros.
• Poles.
( ) ( ) ( )( ) ( ) ( )
)(*...**
*...***)(
2
2
1
1 mnssssss
ssssssctsG
n
zmzz≥
−−−
−−−=
∞→poles
sG )(
0)( →zeros
sG
)(*)(1
)(
)(
)(
sHsG
sG
sR
sC
+=
Block schematic
G(s)+
-
R(s) C(s)
G(s)+
-
R(s) C(s)
H(s)
Unity feed-back
Non-unity feed-back
)(1
)(
)(
)(
sG
sG
sR
sC
+=
V = −R
RV
V = 1 +R
RV
V = V
Circuits with OA: basic structures
Inverting
Non-inverting
Repeating
V = −R
RV +
R
RV +
R
R!V V =
R!
R
R + R
R + R!V −
R
RV
Sum
V = −1
sCRV
V = −R
R
1
sCR + 1V
V = −sCRV
Derivation and integration
Derivation
Derivation
Integration
Equivalent schematics
1
s+13
T ransfer Fcn
Subtract
Step
Scope2
Scope1
Scope
1
s
Integrator
13
Gain
G =1
+ 13=
1
1 + 13 ⋅1
Equivalent schematics
1
s+13
T ransfer Fcn
Subtract
Step
Scope2
Scope1
Scope
1
s
Integrator
13
Gain
G =1
+ 13=
1
1 + 13 ⋅1
G =1
+ 13=
1
1 + 13 ⋅1
Equivalent schematic with OA
1
s+13
T ransfer Fcn
Subtract
Step
Scope2
Scope1
Scope
1
s
In tegra tor
13
Gain
Equivalent schematic with OA SI
1
s+13
T ransfer Fcn
Subtract
Step
Scope2
Scope1
Scope
1
s
Integrator
13
Gain
-1
Equivalent schematic with OA SII
• =
'⋅!=
• =
(')'⋅(
!⋅
(')'⋅(
• =
'⋅=
⋅
=
• =
⋅
(
'
(
1
s +2s+42
T ransfer Fcn
Subtract
Step
Scope2
Scope1
Scope
1
s
Integrator1
1
s
Integrator
4
Gain1
2
Gain
Equivalent schematic with OA
• =
'⋅!=
• =
(')'⋅(
!⋅
(')'⋅(
• =
'⋅=
⋅
=
• =
⋅
(
'
(
1
s +2s+42
T ransfer Fcn
Subtract
Step
Scope2
Scope1
Scope
1
s
Integrator1
1
s
Integrator
4
Gain1
2
Gain
Equivalent schematic with OA
1
s +2s+42
T ransfer Fcn
Subtract
Step
Scope2
Scope1
Scope
1
s
Integrator1
1
s
Integrator
4
Gain1
2
Gain
Equivalent schematic with OA
1
s +2s+42
T ransfer Fcn
Subtract
Step
Scope2
Scope1
Scope
1
s
Integrator1
1
s
Integrator
4
Gain1
2
Gain
Equivalent schematic with OA
1
s +2s +4s+133 2
T ransfer Fcn
Subtract
Step
Scope2
Scope1
Scope
1
s
Integrator2
1
s
Integra tor1
1
s
Integrator
13
Gain2
4
Gain1
2
Gain
Equivalent schematic with OA
• =
*⋅'!⋅=
(*)'⋅(')+⋅(
⋅
(*)'⋅(')+⋅(
•
• =
*⋅'!⋅=
⋅
'⋅!=
⋅
(')'⋅(
!⋅
(')'⋅(
• =
'⋅=
⋅
=
⋅
(
'
(
Equivalent schematic with OA
First order system S-I
1
1
C s
R s Ts
( )
( )=
+)(*)(1
)(
)(
)(
sHsG
sG
sR
sC
+=
First order system S-I: step input
, - = 1 − ./
0
for t > 0
- = 1 →
, 1 = 1 − ./ =
= 0,632 = 63,20%
Second order system SSecond order system SSecond order system SSecond order system S----IIIIIIII
8()
9()=
:;
+ 2ζ:; + :;
8()
9()=
:;
+ 2ζ:;
ζ – factor de amortizare
:;- pulsatie naturala de oscilatie
)(*)(1
)(
)(
)(
sHsG
sG
sR
sC
+=
Second order system response: step inputSecond order system response: step inputSecond order system response: step inputSecond order system response: step input
0,-1
arctgsin1
1
sin1
cos1)(
2
2
2
≥
+
−−=
=
−+−= −
tt
ttetc
d
dd
tn
ξ
ξω
ξ
ξ
ωξ
ξωξω
Second order system response: step inputSecond order system response: step inputSecond order system response: step inputSecond order system response: step input
Transitory response: Transitory response: Transitory response: Transitory response:
step inputstep inputstep inputstep input
-=- delay time
->- rise time
-?- peak time
-- settling time
σ?- peak value
Determination of transient parametersDetermination of transient parametersDetermination of transient parametersDetermination of transient parameters
Rising timettttrrrr :
0,-1
arctgsin1
1
sin1
cos1)(
2
2
2
≥
+
−−=
=
−+−= −
tt
ttetc
d
dd
tn
ξ
ξω
ξ
ξ
ωξ
ξωξω
1( )r
c t =
21
1( ) cos sinn rt
r d r d rc t e t t
−ξω ξ = − ω + ω − ξ
21 ξωω −= nd
Determination of transient parametersDetermination of transient parametersDetermination of transient parametersDetermination of transient parameters
Rising timettttrrrr :
0n rte−ξω ≠
20
1cos sin
d r d rt t
ξω + ω =
− ξ
σ
ω
ξ
ξω d
rt −=−
−=2
d
1tg
dd
rtω
βπ
σ
ω
ω
−=
= darctg
1
21 ξωω −= nd
Determination of transient parametersDetermination of transient parametersDetermination of transient parametersDetermination of transient parameters
Overshooting time ttttpppp :
0== pttdt
dc
0,-1
arctgsin1
1
sin1
cos1)(
2
2
2
≥
+
−−=
=
−+−= −
tt
ttetc
d
dd
tn
ξ
ξω
ξ
ξ
ωξ
ξωξω
Determination of transient parametersDetermination of transient parametersDetermination of transient parametersDetermination of transient parameters
Overshooting time ttttpppp :
20 0
1sin sinn
p
tn
d p d p
t t
dce t t
dt
−ξω
=
ω= ω = => ω =
− ξ
0 2 3, , , , ...d ptω = π π π
d ptω = π
p
d
tπ
=ω
Determination of transient parametersDetermination of transient parametersDetermination of transient parametersDetermination of transient parameters
Overshooting value σσσσpppp :
p
d
tπ
=ω
0,-1
arctgsin1
1
sin1
cos1)(
2
2
2
≥
+
−−=
=
−+−= −
tt
ttetc
d
dd
tn
ξ
ξω
ξ
ξ
ωξ
ξωξω
Determination of transient parametersDetermination of transient parametersDetermination of transient parametersDetermination of transient parameters
Overshooting value σσσσpppp :
σ? = , -? − 1
= ./AB
CDE cos H +
ξ
1 − ξJsin H − 1
= ./
A
/A'J C
>> cos(pi)
ans =
-1
>> sin(pi)
ans =
1.2246e-016[ ]100* %d
pe
σ− π
ωσ =
21 ξωω −= nd
M = ζ:;
Determination of transient parametersDetermination of transient parametersDetermination of transient parametersDetermination of transient parameters
Settling time ttttssss :
4 44
s
n
t T= = =σ ξω
3 33
s
n
t T= = =σ ξω
- ± 2%
- ± 5%
DC motor model
−−=Ω
++=
sf
AAAAA
mmmdt
tdJ
tedt
tdiLtiRU
)(
)()(
)(
60
2 n
AnAAn
n
AnAAnn n
IRUIRUK
⋅
−=
Ω
−=Φ
π
−Ω⋅−⋅Φ⋅=Ω⋅⋅
Ω⋅Φ⋅+⋅⋅+⋅=
sA
AAAAA
msfsIKssJ
sKsIsLsIRU
)()()(
)()()(
DC motor model
[ ]
[ ]
+⋅⋅−⋅Φ⋅=Ω
⋅+⋅Ω⋅Φ⋅−=
fJsmsIKs
LsRsKUsI
sA
AA
AA
1)()(
1)()(
1
sLA+ RA
KΦ1
sJ + f
KΦ
U A Ω
ms
+ +
-
-I A(s)
Ω
Ω (s)
DC motor transfer function
( )( )( ) fsJ
m
sLRfsJ
sKsUK
fsJ
m
sLR
sKsU
fsJ
K
fsJ
msI
fsJ
Ks
s
AA
A
s
AA
A
sA
+−
++
ΩΦ−Φ=
=+
−+
ΦΩ−
+
Φ=
=+
−+
Φ=Ω
)()(...
...)()(
...
...)()(
2
( )( )( )AA
A
sLRfsJ
sKsUKs
++
ΩΦ−Φ=Ω
)()()(
2
( )( ) ( )
( ) ( )......
...)(
)(
22
2
=Φ++++
Φ=
=Φ+++
Φ=
Ω
KfRJRfLsJLs
K
KsLRfsJ
K
sU
s
AAAA
AAA
DC motor transfer function
( )( ) ( )
( ) ( )......
...)(
)(
22
2
=Φ++++
Φ=
=Φ+++
Φ=
Ω
KfRJRfLsJLs
K
KsLRfsJ
K
sU
s
AAAA
AAA
( )
( )
( )
( )JL
KfR
JL
JRfLss
JL
KfR
JL
KfR
JL
K
JL
KfR
JL
JRfLss
JL
K
A
A
A
AA
A
A
A
A
A
A
A
A
AA
A
2
2
2
2
2
2
...
Φ++
++
Φ+
Φ+
Φ
=Φ+
++
+
Φ
=
2212
)(nn
n
ssKsG
ωζω
ω
++=
( )
( )
JL
JRfL
JL
KfR
JL
KfR
JL
K
K
A
AAn
A
An
A
A
A
+=
Φ+=
Φ+
Φ
=
ζω
ω
2
;
;
2
21
DC motor dynamic parameters
( )
( )( )
−+
Φ−=
Φ+=
AAnAnA
AAn
nA
A
RLRL
KRLf
L
KfRJ
ξωω
ξω
ω
2
2222
2
2
2
( )
JL
JRfL
JL
KfR
A
AAn
A
An
+=
Φ+=
ζω
ω
2
;
2
R-L circuit
• ω = 2πP
• QR = ωS = 9 tg φ
• WX = YX 9 + QRJ
• φ = ωΔ-[\
]
• Δ- – voltage and current delay
Δ-
9
S
W
Y
Example:
DC motor nominal data
DC nominal data
Data Value []
1 UA 200 V
2 IA 2,0 A
3 nn 1500 rot/min
Measured data
Data Value []
1 RA 15,8 Ω2 LA 0,41 H
Transient behavior of the DC motor
-? = 0,064
X = 1600,50 `a-/c
Xd= = 1371,01 `a-/c
-? = 0,064
σ? = 0,1674
J and f determination
=−
=
=
+
=
9776,641
4945,0ln
1
ln
2
2
2
ζ
πω
π
σ
π
σ
ζ
p
n
p
p
t
( )
( )( )
=−+
Φ−=
=Φ+
=
0279,02
2
0011,0
222
2
2
2
AAnAnA
AAn
nA
A
RLRL
KRLf
L
KfRJ
ξωω
ξω
ω
-? = 0,064
σ? = 0,1674
Experimental data
DC motor block schematic
DC motor block schematic - electric
DC motor equivalent schematic
+−
−
−
DC motor equivalent schematic
DC motor block schematic - mechanic
DC motor equivalent schematic
+
+
+
− −
−
DC motor equivalent schematic
DC motor equivalent schematic
Experimental data
Experimental
data
DC motor transient behavior