dynamics of electric systems-spania · microsoft powerpoint - dynamics of electric...

Dynamics of electric systemsRobert Beloiu

University of Pitesti, Romania

[email protected]

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


• =L

L =

()

()=

⋯

⋯, ≥

( ) ( ) ( )zmzz

mmmssssssconstsbsbsb −−−=+++ −− *...***.... 2

2

2

1

10 1

( ) ( ) ( )n

nnnssssssconstsasasa −−−=+++ −− *...***.... 2

2

2

1

10 1


• 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


• =

'⋅!=

• =

(')'⋅(

!⋅

(')'⋅(

• =

'⋅=

⋅

=

• =

⋅

(

'

(

1

s +2s+42

T ransfer Fcn

Subtract

Step

Scope2

Scope1

Scope

1

s

Integrator1

1

s

Integrator

4

Gain1

2

Gain


1

s +2s+42

T ransfer Fcn

Subtract

Step

Scope2

Scope1

Scope

1

s

Integrator1

1

s

Integrator

4

Gain1

2

Gain


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


• =

*⋅'!⋅=

(*)'⋅(')+⋅(

⋅

(*)'⋅(')+⋅(

•

• =

*⋅'!⋅=

⋅

'⋅!=

⋅

(')'⋅(

!⋅

(')'⋅(

• =

'⋅=

⋅

=

⋅

(

'

(

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


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


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


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


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


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:;


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 block schematic - mechanic


+

+

+

− −

−

Experimental data

Experimental

data

DC motor transient behavior

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