Modelizacion

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ANÁLISIS DE SEÑALES Y SISTEMAS. Facultad Regional Buenos Aires. Departamento de Electrónica Modelización de Sistemas Simples

Ecuaciones de elementos de Sistemas Físicos Sistema Eléctrico.

Corriente: i(t ) [A]

Tensión: V (t ) [V ]

Inductancia: L [Hy ]

Resistencia: R [Ω ] V (t ) = R ⋅ i (t ) i(t ) =

di (t ) dt

V (t ) = L ⋅

i (t ) =

1 ⋅ V (t ) R

Capacitor: C [F ] V (t ) =

1 t V (t )·dt L −∫∞

1 C

i(t ) = C ⋅

t

∫ i(t )·dt

−∞

dV (t ) dt

Sistema Mecánico Translacional ⎡m⎤ ⎡ Kg ·m ⎤ Fuerza: F (t ) [N ] = ⎢ 2 ⎥ Velocidad: v(t ) ⎢ ⎥ ⎣s⎦ ⎣ s ⎦

N ·s Amortiguador: B ⎡⎢ ⎤⎥ ⎣ m ⎦

F (t ) = B ⋅ v (t ) v(t ) =

Masa: M [Kg ]

N Resorte: K ⎡⎢ ⎤⎥ ⎣m⎦

dv(t ) dt

F (t ) = K ∫ v(t )·dt

F (t ) = M ⋅ v(t ) =

1 ⋅ F (t ) B

1 M

t

−∞

t

∫ F (t )·dt

1 dF (t ) ⋅ K dt

v(t ) =

−∞

Sistema Mecánico Rotacional ⎡ Kg ·m 2 ⎤ Momento: T (t ) [N ·m] = ⎢ ⎥ 2 ⎣ s ⎦

⎡1⎤ Velocidad angular: ω (t ) ⎢ ⎥ ⎣s⎦

J [Kg·m 2 ]

Amortiguamiento Momento de BR [N ·m·s ] Viscoso: Inercia: T (t ) = B R ⋅ ω (t ) ω (t ) =

T (t ) = J ⋅

1 ⋅ T (t ) BR

ω (t ) =

1 J

KR [N ⋅ m]

Resorte Torsional:

dω (t ) dt

t

T (t ) = K R ∫ ω (t )·dt −∞

ω (t ) =

t

∫ T (t )·dt

−∞

1 dT (t ) ⋅ K R dt

Sistema Mecánico de Fluidos ⎡ m3 ⎤ Flujo: q(t ) ⎢ ⎥ ⎣ s ⎦

⎡ N ⎤ Presión: P(t ) ⎢ 2 ⎥ ⎣m ⎦

Resistencia Hidráulica:

N ·s RH ⎡⎢ 5 ⎤⎥

P (t ) = R H ⋅ q (t ) q(t ) =

1 ⋅ P(t ) RH

⎣m ⎦

⎡ N ·s 2 ⎤ 5 ⎥ ⎣ m ⎦

Inertancia Hidráulica:

LH ⎢

P (t ) = L H ⋅ q(t ) =

1 LH

⎡ m5 ⎤

Compliance Hidráulica:

dq (t ) dt

t

∫ P(t )·dt

−∞

Msc. Ing. Franco Martin Pessana [email protected] FRBA. Universidad Tecnológica Nacional

P(t ) =

1 CH

q (t ) = C H ⋅

CH ⎢ ⎥ ⎣N ⎦ t

∫ q(t )·dt

−∞

dP(t ) dt

ANÁLISIS DE SEÑALES Y SISTEMAS. Facultad Regional Buenos Aires. Departamento de Electrónica Modelización de Sistemas Simples Sistema Físico Calórico ⎡J ⎤ Flujo Calórico: q(t ) ⎢ ⎥ ⎣s⎦

Temperatura (*): θ (t ) [º C ]

Resistencia Térmica:

º C·s ⎤ RT ⎡⎢ ⎥ ⎣ J ⎦

θ (t ) = RT ⋅ q (t ) q(t ) =

°C ⋅ s ⎤ LT ⎡⎢ ⎥

Inertancia Térmica:

⎣ J ⎦

θ (t ) = LT ⋅

1 ⋅ θ (t ) RT

1 q(t ) = LT

J CT ⎡⎢ ⎤⎥ ⎣º C ⎦

Capacidad Térmica:

dq (t ) dt

θ (t ) =

1 CT

t

∫ q(t )·dt

−∞

t

∫ θ (t )·dt

q(t ) = C T ⋅

−∞

dθ (t ) dt

Analogías entre Sistemas Físicos Sistemas

Modelización Serie

Modelización Paralelo

V (t ) ≡ F (t ) ≡ T (t ) ≡ P(t ) ≡ θ (t ) i(t ) ≡ v(t ) ≡ ω (t ) ≡ q(t ) ≡ q(t )

V (t ) ≡ v(t ) ≡ ω (t ) ≡ q(t ) i(t ) ≡ F (t ) ≡ T (t ) ≡ P(t )

Eléctrico

R

L

C

R

L

C

Traslacional

B

M

1 K

1 K

M

Rotacional

BR

J

1 KR

1 KR

J

Fluidos

RH

LH

CH

1 B 1 BR 1 RH

CH

LH

Calórico

RT

LT

CT

Transformador Ideal

Engranajes Ideales

V P (t ) i S (t ) N P = = V S (t ) i P (t ) N S

T1 (t ) ω 2 (t ) N 1 = = T2 (t ) ω 1 (t ) N 2

Reflexiones de un Circuito de Secundario a Primario de Transformador Relación de Transformación: η =

Modelización Serie Resistencia Inductancia Capacitor

V P (t ) = η 2 ·R ⋅ i P (t ) V P (t ) = η 2 ·L ⋅

V P (t ) =

η2 C

t

di P (t ) dt

∫ i P (t )·dt

NP NS

Modelización Paralelo 1 ⋅ V P (t ) η ·R

i P (t ) =·

i P (t ) =

2

1 t V P (t )·dt η 2 ·L −∫∞

i P (t ) =

−∞

Msc. Ing. Franco Martin Pessana [email protected] FRBA. Universidad Tecnológica Nacional

C dV P (t ) ⋅ dt η2

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