1 Il circuito RLC e la notazione complessa

(1)

CIRCUITI IN CORRENTE ALTERNATA

In questo capitolo indicheremo in grassetto variabili a valori complessi e con Re() la parte reale di un numero complesso.

La motivazione matematica per l’uso di quantit`a complesse nello studio dei circuiti in corrente alternata sta nella grande semplificazione che l’uso della formula di Eulero consente nella trattazione di espressioni contenenti funzioni trigonometriche.

Ci`o porta all’introduzione dell’impedenza complessa di un elemento circuitale, una quantit`a fisica di cui analizzeremo il significato.

1 Il circuito RLC e la notazione complessa

Consideriamo il circuito RLC in serie

V0cos(ωt)

L C

R

L’equazione del circuito si scrive:

V = L ¨˙ I + R ˙I + 1

CI (1)

L’equazione omogenea associata `e dello stesso tipo di quella risolta nel capitolo sulle correnti continue, e le soluzioni per I hanno in funzione del tempo lo stesso andamento di quelle trovate per Q. Ora dovremo cercare una soluzione particolare della equazione completa.

Dalla teoria delle equazioni differenziali lineari sappiamo che tale soluzione ha la forma

I = I0cos(ωt + ϕ) (2)

dove I0 e ϕ sono costanti che dipendono da V0, L, R, C ed ω. Notate che la soluzione generale della omogenea, quali che siano le condizioni iniziali, decresce esponenzialmente nel tempo; a tempi sufficientemente grandi la soluzione sar`a dunque all’incirca uguale alla (2). Diciamo che il circuito `e nel

• regime transitorio, nel periodo iniziale in cui i termini esponenziali non sono trascurabili rispetto alla (2);

• regime stazionario nel periodo successivo, in cui la corrente `e uguale all’incirca a (2).

Le costanti d’integrazione stanno solo nella soluzione generale della omogenea, quindi le condizioni iniziali hanno effetti solo nel regime transitorio, mentre nel regime stazionario l’andamento della corrente `e determinato solo dalle costanti del circuito.

Il metodo delle impedenze complesse si applica al regime stazionario e d’ora in avanti ci occuperemo solo di questo.

Poich`e cercheremo le fasi di tensioni e corrente nel circuito relativamente alla fase di V (t), assegneremo fase zero a V (t):

V = V0cos(ωt) = Re Ve^iωt

; V= V0 (3)

Cerchiamo la soluzione particolare nella forma:

I = I0cos(ωt + ϕ) = Re Ie^iωt

; I= I0e^iϕ (4)

(2)

Ora il compito `e trovare I0 e ϕ. Procederemo sostituendo le espressioni complesse della tensione e della corrente:

Ie^iωt ; Ve^iωt (5)

nella equazione (1):

iωVe^iωt=

−ω²LI + iωRI + 1 CI

e^iωt (6)

questa uguaglianza deve valere per ogni t; quindi:

iωV = −ω²LI + iωRI + 1

CI (7)

V=

iωL + R + 1 iωC

I≡ ZI ; Z= R + i

ωL − 1 ωC

(8)

Otteniamo dunque un’equazione algebrica per I; la risolviamo e calcoliamo la corrente fisica reale:

I(t) = Re Ie^iωt = Re V Ze^iωt

= Re |V|

|Z|e^i(ωt−ϕ^Z⁾

= |V|

|Z|cos(ωt − ϕZ) (9)

ϕZ `e la fase di Z:

ϕZ = arctanωL − _ωC¹

R (10)

Se avessimo introdotto anche la fase di V (t), ϕZ sarebbe la fase di Z meno quella di V.

L’ampiezza della corrente `e dunque data da:

I0=|V|

|Z| = V0

q

R²+ ωL −_ωC¹ 2 (11)

Provate ad ottenere gli stessi risultati senza utilizzare le espressioni complesse.

Dalla (10) deduciamo che se nel circuito abbiamo:

• Solo R: ϕ^Z = 0 ⇒ corrente e tensione sono in fase.

• Solo L: ϕ^Z = ^π₂ ⇒ I = I⁰cos(ωt −^π₂) ⇒ la corrente ’segue’ la tensione di 90^o.

• Solo C: ϕ^Z = −^π₂ ⇒ I = I0cos(ωt +^π₂) ⇒ la corrente ’precede’ la tensione di 90^o.

• Solo L e C: la corrente precede o segue la tensione di 90^o a seconda del valore di ωL relativamente a quello di _ωC¹ .

Inoltre:

• Per ω = ^√_LC¹ tensione e corrente sono in fase qualunque sia il valore di R.

Dalla (11) vediamo che, al variare di ω, la corrente ha un massimo (risonanza) per:

ω = 1

√LC ≡ ω0 (12)

Come abbiamo gi`a visto, alla risonanza la fase si annulla.

Introducendo il fattore di merito alla risonanza Q0 (adimensionale):

Q0= ω0L R = 1

R r L

C (13)

riscriviamo I0:

I0= V0

R r

1 + Q²₀

ω

ω0−^ω_ω⁰2 (14)

Come potete vedere nelle figure che seguono, a parità di L e di C, quindi di ω0, il circuito è tanto più selettivo alla risonanza quanto più è grande Q0, cioè più piccola è R rispetto a ^q^L_C. Cosa succede per R = 0 ?.

(3)

0

V0 R

I0(ω)

ω

··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ···

^Q⁰^{= 1}

·

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···^Q⁰^{= 3}

ω₀ ⁻

π 2

0

π 2

ϕ(ω)

ω

··· ··· ··· ··· ··· ··· ···

·

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···

ω₀

2 L’impedenza complessa

La (7) e la (8) ci mostrano che, introducendo le impedenze complesse dei tre elementi circuitali fonda- mentali:

Z_R= R ; Z_L= iωL ; Z_C= 1

iωC (15)

ed utilizzando la notazione complessa possiamo scrivere una generalizzazione della legge di Ohm tra le ampiezze complessein cui la resistenza `e sostituita dall’impedenza di ciascun elemento:

V= Z · I (16)

Notate che in questa equazione compaiono solo le ampiezze indipendenti dal tempo: la dipendenza dal tempo compare nel fattore e^iωt che abbiamo ’semplificato’.

Con le stesse equazioni abbiamo già visto che le impedenze in serie si sommano come le resistenze, possiamo facilmente predire che ciò è vero anche per le impedenze in parallelo. Verifichiamolo in un semplice circuito:

V

I(t)

C L ^I^L^(t) ^I^C^(t)

Per le correnti nei due elementi e per la corrente totale abbiamo:

I_L= V

Z_L (17)

I_C= V

Z_C (18)

I= IL+ IC=

1 Z_L + 1

Z_C

V= V

Z_tot ; Z_tot= Z_LZ_C Z_L+ ZC

=

L C

i ωL − ωC¹

(19)

I= i ωL −_ωC¹

L C

V (20)

I(t) = I0cos(ωt + ϕ) (21)