31.8
Crescita e decadimento della corrente nei circuiti RL
La crescita e il decadimento della corrente nei circuiti RL possono essere compresi considerando un circuito RL in serie composto da un resistore, un…
Si consideri un circuito RL costituito da un resistore, un induttore, una sorgente costante di campi elettromagnetici e gli interruttori S1 e S2.
Quando l'interruttore S1 è chiuso, la corrente nel circuito aumenta, generando campi elettromagnetici attraverso il resistore e l'induttore. Questi campi elettromagnetici vengono utilizzati nella regola del ciclo di Kirchhoff per trovare il tasso di crescita corrente.
Riorganizzando e integrando l'equazione, si ottiene la corrente nel circuito RL con emf.
Dopo aver raggiunto lo stato stazionario, l'interruttore S2 viene chiuso mentre S1 viene aperto, formando un singolo anello che bypassa la sorgente emf. Ciò si traduce in un decadimento della corrente attraverso il resistore e l'induttore. La corrente decrestata si ottiene utilizzando la regola di Kirchhoff.
L'induttanza della quantità rispetto alla resistenza è chiamata costante di tempo induttiva.
Il grafico corrente rispetto al tempo mostra che, quando il tempo è uguale alla costante di tempo, la corrente cresce fino al 63% del suo valore finale, mentre durante il decadimento, allo stesso valore della costante di tempo, la corrente decade al 37% del suo valore originale.
Quindi, la corrente aumenta gradualmente da zero a uno stato stazionario, ma decade esponenzialmente con il tempo.
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Q1: Why does current not rise immediately to its final value when an RL circuit is first closed?
When the circuit closes, the increasing current produces an induced emf across the inductor that opposes the applied emf, following Lenz's law. This opposing emf counteracts the current increase, forcing the current to start at zero and rise gradually toward its steady-state value ε/R rather than jumping instantly.
Q2: What is the inductive time constant and what does it tell us about circuit behavior?
The inductive time constant is the ratio of inductance to resistance (L/R). It measures how quickly current builds toward its final value. At time equal to one time constant, current grows to 63% of its final value during growth, or decays to 37% of its original value during decay.
Q3: How does the energy stored in an inductor change as current grows in an RL circuit?
As current increases from zero toward its steady-state value ε/R, the energy stored in the inductor increases asymptotically from zero to a maximum value. This stored energy in the magnetic field grows proportionally with the increasing current until the circuit reaches steady state.
Q4: What happens to current when the emf source is disconnected from an RL circuit?
When the emf source is disconnected and the circuit forms a single loop with only the resistor and inductor, the initial current ε/R decreases exponentially with time. The energy stored in the inductor is gradually depleted through the resistor until the current reaches zero.
Q5: How do you apply Kirchhoff's loop rule to find current growth in an RL circuit?
Kirchhoff's loop rule states that the sum of emfs around a closed loop equals zero. In an RL circuit with applied emf, the applied emf equals the sum of the emf across the resistor and the induced emf across the inductor. Rearranging and integrating this equation yields the current growth equation.
Q6: Why does current decay exponentially rather than linearly when an RL circuit is disconnected from its emf source?
During decay, the inductor's induced emf is proportional to the rate of current change. As current decreases, the induced emf also decreases, resulting in a slower decay rate at each moment. This proportional relationship produces exponential decay rather than a constant linear decrease.
Q7: How can you compare the behavior of RL circuits to RC circuits in terms of current response?
Both RL and RC circuits exhibit exponential responses to switching, but with different time constants. RL circuits have time constant L/R, while RC circuits have time constant RC. Both show asymptotic approach to steady state, though the physical mechanisms differ—inductors oppose current change while capacitors oppose voltage change.