Articles in this blog's slice are, for the time being, ony in italian.
Le equazioni differenziali ordinarie del I ordine hanno la forma:
d x ( t ) d t + x ( t ) Ο = x p Ο \frac{dx(t)}{dt} + \frac{x(t)}{\tau} = \frac{x_p}{\tau} d t d x ( t ) β + Ο x ( t ) β = Ο x p β β La soluzione generale Γ¨:
x ( t ) = k e β t / Ο + h x(t) = k e^{-t/\tau}+h x ( t ) = k e β t / Ο + h Con h soluzione particolare, che si determina grazie alla condizione iniziale:
h = x ( 0 ) β x p β
β βΉ β
β x ( t ) [ x ( 0 ) β x p ] e β t / Ο + x p h = x(0)-x_p \implies x(t)\big[x(0)-x_p\big]e^{-t/\tau}+x_p h = x ( 0 ) β x p β βΉ x ( t ) [ x ( 0 ) β x p β ] e β t / Ο + x p β Le equazioni differenziali ordinarie del II ordine hanno la forma:
d 2 x ( t ) d t 2 + 2 Ξ± d x ( t ) d t + Ο 0 2 x ( t ) = 0 \frac{d^2 x(t)}{dt^2} + 2\alpha \frac{d x(t)}{dt} +\omega_0^2 x(t)= 0 d t 2 d 2 x ( t ) β + 2 Ξ± d t d x ( t ) β + Ο 0 2 β x ( t ) = 0 Se Ξ± β Ο 0 \alpha \neq \omega_0 Ξ± ξ = Ο 0 β
x ( t ) = A e s t β
β βΉ β
β s 1 , 2 = β Ξ± Β± Ξ± 2 β Ο 0 2 β
β βΉ β
β x ( t ) = A 1 e s 1 t + A 2 e s 2 t x(t) = A e^{st}
\implies s_{1,2} = -\alpha \pm \sqrt{\alpha^2-\omega_0^2}
\implies x(t)=A_1 e^{s_1 t} + A_2 e^{s_2 t} x ( t ) = A e s t βΉ s 1 , 2 β = β Ξ± Β± Ξ± 2 β Ο 0 2 β β βΉ x ( t ) = A 1 β e s 1 β t + A 2 β e s 2 β t Si determinano A 1 A_1 A 1 β A 2 A_2 A 2 β x ( 0 ) x(0) x ( 0 ) d x d t β£ t = 0 \frac{dx}{dt}\Big|_{t=0} d t d x β β t = 0 β
Lβordine di complessitΓ di un circuito Γ¨ ricavato calcolando il numero di condizioni iniziali indipendenti:
n D n_D n D β n C n_C n C β n L n_L n L β
ordine circuito = n D β n C β n L \text{ordine circuito} = n_D - n_C - n_L ordine circuito = n D β β n C β β n L β Un circuito Γ¨ detto autonomo quando i generatori sono spenti. Lβenergia immagazzinata dai componenti dinamici (condensatori e induttori) Γ¨ rilasciata ai resistori.
C d v ( t ) d t + v ( t ) R = 0 C \frac{d v(t)}{dt}+\frac{v(t)}{R}=0 C d t d v ( t ) β + R v ( t ) β = 0 Condizione iniziale:
V 0 = ( t = t 0 = 0 s ) V_0=(t=t_0=0s) V 0 β = ( t = t 0 β = 0 s ) Energia iniziale immagazzinata:
E ( t = 0 ) = 1 2 C V 0 2 E(t=0)=\frac{1}{2} C V_0^2 E ( t = 0 ) = 2 1 β C V 0 2 β Altre formule importanti sono:
d v ( t ) d t + v ( t ) Ο = 0 con R C = Ο \frac{d v(t)}{dt}+\frac{v(t)}{\tau}=0\quad \text{con } RC = \tau d t d v ( t ) β + Ο v ( t ) β = 0 con RC = Ο v ( t ) = V 0 e β t / Ο v(t) = V_0 e^{-t/\tau} v ( t ) = V 0 β e β t / Ο I dati relativi alla resistenza sono:
i R ( t ) = v ( t ) R = V 0 R e β t / Ο p R ( t ) = v ( t ) i R ( t ) = V 0 2 R e β ( 2 t ) / Ο E R ( t ) = β« 0 t p ( t ~ ) d t ~ = 1 2 C V 0 2 ( 1 β e β ( 2 t ) / Ο ) \eq{
i_R(t) &= \frac{v(t)}{R}= \frac{V_0}{R} e^{-t/\tau}\\
p_R(t) &= v(t)i_R(t)= \frac{V_0^2}{R} e^{-(2t)/\tau}\\
E_R(t) &= \int_0^t p(\widetilde{t})d\widetilde{t}
= \frac{1}{2} C V_0^2 \bigg(1-e^{-(2t)/\tau} \bigg) \\
} i R β ( t ) p R β ( t ) E R β ( t ) β = R v ( t ) β = R V 0 β β e β t / Ο = v ( t ) i R β ( t ) = R V 0 2 β β e β ( 2 t ) / Ο = β« 0 t β p ( t ) d t = 2 1 β C V 0 2 β ( 1 β e β ( 2 t ) / Ο ) β β
L d i ( t ) d t + R i ( t ) = 0 L \frac{d i(t)}{dt}+Ri(t)=0 L d t d i ( t ) β + R i ( t ) = 0 Condizione iniziale:
I 0 = i ( t = t 0 = 0 s ) I_0=i(t=t_0=0s) I 0 β = i ( t = t 0 β = 0 s ) Energia iniziale immagazzinata:
E ( t = 0 ) = 1 2 L I 0 2 E(t=0)=\frac{1}{2} L I_0^2 E ( t = 0 ) = 2 1 β L I 0 2 β Altre formule:
d i ( t ) d t + R i ( t ) L = 0 con L R = Ο d i ( t ) d t + i ( t ) Ο β 1 = 0 i ( t ) = I 0 e β t / Ο \eq{
& \frac{d i(t)}{dt}+\frac{Ri(t)}{L}=0\quad \text{con } \frac{L}{R} = \tau \\
& \frac{d i(t)}{dt}+i(t)\tau^{-1}=0 \\
& i(t) = I_0 e^{-t/\tau}
} β d t d i ( t ) β + L R i ( t ) β = 0 con R L β = Ο d t d i ( t ) β + i ( t ) Ο β 1 = 0 i ( t ) = I 0 β e β t / Ο β β I dati relativi alla resistenza sono:
v R ( t ) = R i ( t ) = R I 0 e β t / Ο p R ( t ) = v R ( t ) i ( t ) = R I 0 2 e β ( 2 t ) / Ο E R ( t ) = 1 2 L I 0 2 ( 1 β e β ( 2 t ) / Ο ) \eq{
v_R(t) &= Ri(t)= R I_0 e^{-t/\tau}\\
p_R(t) &= v_R(t)i(t)= R I_0^2 e^{-(2t)/\tau}\\
E_R(t) &= \frac{1}{2} L I_0^2 \bigg(1-e^{-(2t)/\tau} \bigg) \\
} v R β ( t ) p R β ( t ) E R β ( t ) β = R i ( t ) = R I 0 β e β t / Ο = v R β ( t ) i ( t ) = R I 0 2 β e β ( 2 t ) / Ο = 2 1 β L I 0 2 β ( 1 β e β ( 2 t ) / Ο ) β β La costante di tempo Ο β₯ 0 \tau \geq 0 Ο β₯ 0 1 / e 1/e 1/ e 36.8 % 36.8\% 36.8%
Il circuito raggiunge la condizione di regime (transitorio esaurito) dopo un tempo di 5 Ο 5\tau 5 Ο
La velocitΓ di risposta Γ¨ inversamente proporzionale a Ο \tau Ο
L d i ( t ) d t + R i ( t ) + V 0 + 1 C β« 0 t i ( t ~ ) d t ~ = 0 L \frac{d i(t)}{dt}+ R i(t)+V_0 + \frac{1}{C} \int_0^t i(\widetilde{t})d\widetilde{t}=0 L d t d i ( t ) β + R i ( t ) + V 0 β + C 1 β β« 0 t β i ( t ) d t = 0 d 2 i ( t ) d t 2 + R L d i ( t ) d t + i ( t ) L C = 0 \frac{d^2 i(t)}{dt^2}+\frac{R}{L}\frac{d i(t)}{dt}+\frac{i(t)}{LC}=0 d t 2 d 2 i ( t ) β + L R β d t d i ( t ) β + L C i ( t ) β = 0 La costante di smorzamento Ξ± \alpha Ξ±
Ξ± = R 2 L β
β [ 1 s e c . ] \alpha = \frac{R}{2L} \; \bigg[\frac{1}{\rm sec.} \bigg] Ξ± = 2 L R β [ sec. 1 β ] La pulsazione di risonanza Ο 0 \omega_0 Ο 0 β
Ο 0 = 1 L C \omega_0 = \frac{1}{\sqrt{LC}} Ο 0 β = L C β 1 β Condizioni iniziali con i ( 0 ) = I 0 i(0)=I_0 i ( 0 ) = I 0 β
d i d t β£ t = 0 = β 1 L [ R I 0 + V 0 ] \frac{d i}{dt}\bigg|_{t=0} = -\frac{1}{L}\bigg[R I_0 + V_0 \bigg] d t d i β β t = 0 β = β L 1 β [ R I 0 β + V 0 β ]
C d v ( t ) d t + v ( t ) R + I 0 + 1 L β« 0 t v ( t ~ ) d t ~ = 0 C \frac{d v(t)}{dt} + \frac{v(t)}{R}+I_0 + \frac{1}{L} \int_0^t v(\widetilde{t})d\widetilde{t}=0 C d t d v ( t ) β + R v ( t ) β + I 0 β + L 1 β β« 0 t β v ( t ) d t = 0 d 2 v ( t ) d t 2 + 1 R C d v ( t ) d t + v ( t ) L C = 0 \frac{d^2 v(t)}{dt^2}+\frac{1}{RC}\frac{d v(t)}{dt}+\frac{v(t)}{LC}=0 d t 2 d 2 v ( t ) β + RC 1 β d t d v ( t ) β + L C v ( t ) β = 0 La costante di smorzamento Ξ± \alpha Ξ±
Ξ± = 1 2 R C β
β [ 1 s e c . ] \alpha = \frac{1}{2RC} \; \bigg[\frac{1}{\rm sec.} \bigg] Ξ± = 2 RC 1 β [ sec. 1 β ] La pulsazione di risonanza Ο 0 \omega_0 Ο 0 β
Ο 0 = 1 L C \omega_0 = \frac{1}{\sqrt{LC}} Ο 0 β = L C β 1 β Condizioni iniziali con v ( 0 ) = V 0 v(0)=V_0 v ( 0 ) = V 0 β
d v d t β£ t = 0 = β 1 R C [ R I 0 + V 0 ] \frac{dv}{dt}\bigg|_{t=0} = -\frac{1}{RC}\bigg[R I_0 + V_0 \bigg] d t d v β β t = 0 β = β RC 1 β [ R I 0 β + V 0 β ] Si confrontano i circuiti RLC serie ed RLC parallelo nelle diverse tipologie di smorzamento:
Tipologia Soluzioni RLC Serie RLC Parallelo Sovrasmorzamento: Ξ± > Ο 0 \alpha > \omega_0 Ξ± > Ο 0 β s 1 , 2 = β Ξ± Β± Ξ± 2 β Ο 0 2 β R β s_{1,2}=-\alpha \pm \sqrt{\alpha^2 -\omega_0^2} \in \R^- s 1 , 2 β = β Ξ± Β± Ξ± 2 β Ο 0 2 β β β R β C > ( 4 L ) / R 2 C > (4L)/ R^2 C > ( 4 L ) / R 2 L > 4 R 2 C L > 4 R^2 C L > 4 R 2 C Sottosmorzamento: Ξ± < Ο 0 \alpha < \omega_0 Ξ± < Ο 0 β s 1 , 2 = β Ξ± Β± j Ξ± 2 β Ο 0 2 β C s_{1,2}=-\alpha \pm j\sqrt{\alpha^2 -\omega_0^2} \in \C s 1 , 2 β = β Ξ± Β± j Ξ± 2 β Ο 0 2 β β β C C < ( 4 L ) / R 2 C < (4L)/ R^2 C < ( 4 L ) / R 2 L < 4 R 2 C L < 4 R^2 C L < 4 R 2 C Senza smorzamento: Ξ± = 0 < Ο 0 \alpha = 0 < \omega_0 Ξ± = 0 < Ο 0 β s 1 , 2 = Β± j Ξ± 2 β Ο 0 2 β C s_{1,2}=\pm j\sqrt{\alpha^2 -\omega_0^2} \in \C s 1 , 2 β = Β± j Ξ± 2 β Ο 0 2 β β β C R = 0 R=0 R = 0 1 / R = 0 1/R = 0 1/ R = 0 Smorzamento critico: Ξ± = Ο 0 \alpha = \omega_0 Ξ± = Ο 0 β s 1 , 2 = β Ξ± β R s_{1,2}=-\alpha \in \R s 1 , 2 β = β Ξ± β R
N.B. Nei casi del sottosmorzamento e dellβassenza dello smorzamento, le soluzioni sono numeri complessi coniugati: s 1 = s 2 βΎ s_1 = \overline{s_2} s 1 β = s 2 β β
Nellβambito dei circuiti elettrici, in particolare per gli argomenti trattati, la funzione gradino unitario si definisce come:
u ( t ) = { 0 t < 0 1 t > 0 β t = 0 u(t) =
\begin{cases}
0 \quad t < 0\\
1 \quad t > 0\\
\nexists \quad t = 0
\end{cases} u ( t ) = β© β¨ β§ β 0 t < 0 1 t > 0 β t = 0 β Il gradino modella rapide variazioni di corrente e tensione.
Mentre per la teoria dei segnali Γ¨ utile definire u ( 0 ) = 1 / 2 u(0)=1/2 u ( 0 ) = 1/2 t = 0 t=0 t = 0
Il circuito RC autonomo si presenta nella forma:
Il circuito RL autonomo si presenta nella forma:
Formula RC Autonomo RL Autonomo Condizione iniziale v ( t = 0 ) = V 0 v(t=0)=V_0 v ( t = 0 ) = V 0 β i ( t = 0 ) = I 0 i(t=0)=I_0 i ( t = 0 ) = I 0 β Energia iniziale immagazzinata E ( t = 0 ) = 1 2 C V 0 2 E(t=0)=\frac{1}{2} C V_0^2 E ( t = 0 ) = 2 1 β C V 0 2 β E ( t = 0 ) = 1 2 L I 0 2 E(t=0)=\frac{1}{2} L I_0^2 E ( t = 0 ) = 2 1 β L I 0 2 β Integrazione nel tempo C d v ( t ) d t + v ( t ) β V s u ( t ) R = 0 C \frac{d v(t)}{dt}+\frac{v(t)-V_s u(t)}{R}=0 C d t d v ( t ) β + R v ( t ) β V s β u ( t ) β = 0 L d i ( t ) d t + R i ( t ) β V s u ( t ) = 0 L \frac{d i(t)}{dt}+R i(t)-V_s u(t)=0 L d t d i ( t ) β + R i ( t ) β V s β u ( t ) = 0 Risposta del circuito Ο = R C \tau = RC Ο = RC d v ( t ) d t + v ( t ) Ο = V s u ( t ) Ο = 0 \frac{d v(t)}{dt}+\frac{v(t)}{\tau}=\frac{V_s u(t)}{\tau}=0 d t d v ( t ) β + Ο v ( t ) β = Ο V s β u ( t ) β = 0 Ο = L / R \tau = L/R Ο = L / R d i ( t ) d t + i ( t ) Ο = I s u ( t ) R Ο \frac{di(t)}{dt}+\frac{i(t)}{\tau} = \frac{I_s u(t)}{R\tau} d t d i ( t ) β + Ο i ( t ) β = R Ο I s β u ( t ) β Risposta completa v ( t ) = [ V 0 β V s ] e β t / Ο + V s v(t) = [V_0 - V_s] e ^{-t/\tau}+V_s v ( t ) = [ V 0 β β V s β ] e β t / Ο + V s β i ( t ) = [ I 0 β I s ] e β t / Ο + I s i(t) = [I_0 - I_s] e ^{-t/\tau}+I_s i ( t ) = [ I 0 β β I s β ] e β t / Ο + I s β Risposta transitoria v T = [ V 0 β V s ] e β t / Ο v_T=[V_0 - V_s] e^{-t/\tau} v T β = [ V 0 β β V s β ] e β t / Ο i T = [ I 0 β I s ] e β t / Ο i_T=[I_0 - I_s] e^{-t/\tau} i T β = [ I 0 β β I s β ] e β t / Ο Risposta a regime v R = V s v_R = V_s v R β = V s β i R = I s i_R = I_s i R β = I s β Risposta al gradino v ( t ) = V 0 e β t / Ο + V s ( 1 β e β t / Ο ) v(t)=V_0 e^{-t/\tau}+V_s (1-e^{-t/\tau}) v ( t ) = V 0 β e β t / Ο + V s β ( 1 β e β t / Ο ) i ( t ) = I 0 e β t / Ο + I s ( 1 β e β t / Ο ) i(t)=I_0 e^{-t/\tau}+I_s (1-e^{-t/\tau}) i ( t ) = I 0 β e β t / Ο + I s β ( 1 β e β t / Ο ) Risposta naturale v N = V 0 e β t / Ο v_N = V_0 e^{-t/\tau} v N β = V 0 β e β t / Ο i N = I 0 e β t / Ο i_N = I_0 e^{-t/\tau} i N β = I 0 β e β t / Ο Risposta forzata v F = V s ( 1 β e β t / Ο ) v_F=V_s(1-e^{-t/\tau}) v F β = V s β ( 1 β e β t / Ο ) i F = I s ( 1 β e β t / Ο ) i_F=I_s(1-e^{-t/\tau}) i F β = I s β ( 1 β e β t / Ο )
RLC Serie RLC Parallelo v < 0 β i ( t ) = 0 v<0 \to i(t)=0 v < 0 β i ( t ) = 0 i ( 0 ) = I 0 = 0 i(0)=I_0 = 0 i ( 0 ) = I 0 β = 0 v < 0 β v ( t ) = 0 v<0 \to v(t)=0 v < 0 β v ( t ) = 0 v ( 0 ) = V 0 = 0 v(0)=V_0 = 0 v ( 0 ) = V 0 β = 0 d i d t t = 0 = 1 L [ V s β V 0 ] \frac{d i}{dt}_{t=0}=\frac{1}{L}[V_s - V_0] d t d i β t = 0 β = L 1 β [ V s β β V 0 β ] d v d t t = 0 = 1 C [ I s β I 0 ] \frac{dv}{dt}_{t=0} =\frac{1}{C}[I_s - I_0 ] d t d v β t = 0 β = C 1 β [ I s β β I 0 β ] v L ( t ) = L d i d t v_L(t)=L \frac{d i}{dt} v L β ( t ) = L d t d i β i C ( t ) = C d v d t i_C(t)=C\frac{dv}{dt} i C β ( t ) = C d t d v β v R ( t ) = R i ( t ) v_R (t)=R i(t) v R β ( t ) = R i ( t ) i R ( t ) = v ( t ) / R i_R (t)=v(t)/R i R β ( t ) = v ( t ) / R v C ( t ) = V s β v R ( t ) β v L ( t ) v_C (t)=V_s-v_R (t)-v_L (t) v C β ( t ) = V s β β v R β ( t ) β v L β ( t ) i L ( t ) = I s β i R ( t ) β i C ( t ) i_L (t)=I_s-i_R (t)-i_C (t) i L β ( t ) = I s β β i R β ( t ) β i C β ( t )
