Transient Analysis of RL Circuits
RL circuits—circuits containing a resistor (R) and an inductor (L) in series—are fundamental building blocks in electrical engineering. When a step voltage is suddenly applied to such a circuit, the current does not instantly reach its maximum value. Instead, the inductor initially resists changes in current, causing a gradual increase until a steady state is reached. This chapter explores how to model and simulate the current response of RL circuits to a step voltage input, focusing on the transient (time-dependent) behavior that occurs immediately after the voltage is applied.
1234567891011121314151617181920212223import numpy as np def rl_step_response(V, R, L, t): """ Calculate current through an inductor in an RL circuit after a step input. Parameters: V : float Step input voltage (volts) R : float Resistance (ohms) L : float Inductance (henries) t : array_like Time points (seconds) Returns: I : ndarray Current at each time point (amperes) """ tau = L / R # Time constant return (V / R) * (1 - np.exp(-t / tau))
The time-dependent behavior of an RL circuit during a step response is governed by the inductor's opposition to changes in current. When the voltage is first applied, the inductor generates a back electromotive force (EMF) that resists the increase in current. The rate at which current rises is determined by the time constant, given by tau = L / R. A larger inductance or a smaller resistance leads to a slower current increase, while a smaller inductance or larger resistance allows the current to rise more quickly. Over time, the current approaches its steady-state value of V / R as the effect of the inductor diminishes.
123456789101112131415161718192021222324import matplotlib.pyplot as plt import numpy as np # Circuit parameters V = 10.0 # Step input voltage in volts R = 5.0 # Resistance in ohms L = 1.0 # Inductance in henries # Time array t = np.linspace(0, 2, 200) # 0 to 2 seconds # Get current response I = rl_step_response(V, R, L, t) # Plotting plt.figure(figsize=(8, 4)) plt.plot(t, I, label='Current through Inductor') plt.axhline(V/R, color='r', linestyle='--', label='Steady-State Current') plt.title('RL Circuit Step Response') plt.xlabel('Time (s)') plt.ylabel('Current (A)') plt.legend() plt.grid(True) plt.show()
1. What happens to current in an RL circuit immediately after a step voltage is applied?
2. How does inductance affect the speed of current change?
3. In what applications are RL circuits commonly used?
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RL circuits—circuits containing a resistor (R) and an inductor (L) in series—are fundamental building blocks in electrical engineering. When a step voltage is suddenly applied to such a circuit, the current does not instantly reach its maximum value. Instead, the inductor initially resists changes in current, causing a gradual increase until a steady state is reached. This chapter explores how to model and simulate the current response of RL circuits to a step voltage input, focusing on the transient (time-dependent) behavior that occurs immediately after the voltage is applied.
1234567891011121314151617181920212223import numpy as np def rl_step_response(V, R, L, t): """ Calculate current through an inductor in an RL circuit after a step input. Parameters: V : float Step input voltage (volts) R : float Resistance (ohms) L : float Inductance (henries) t : array_like Time points (seconds) Returns: I : ndarray Current at each time point (amperes) """ tau = L / R # Time constant return (V / R) * (1 - np.exp(-t / tau))
The time-dependent behavior of an RL circuit during a step response is governed by the inductor's opposition to changes in current. When the voltage is first applied, the inductor generates a back electromotive force (EMF) that resists the increase in current. The rate at which current rises is determined by the time constant, given by tau = L / R. A larger inductance or a smaller resistance leads to a slower current increase, while a smaller inductance or larger resistance allows the current to rise more quickly. Over time, the current approaches its steady-state value of V / R as the effect of the inductor diminishes.
123456789101112131415161718192021222324import matplotlib.pyplot as plt import numpy as np # Circuit parameters V = 10.0 # Step input voltage in volts R = 5.0 # Resistance in ohms L = 1.0 # Inductance in henries # Time array t = np.linspace(0, 2, 200) # 0 to 2 seconds # Get current response I = rl_step_response(V, R, L, t) # Plotting plt.figure(figsize=(8, 4)) plt.plot(t, I, label='Current through Inductor') plt.axhline(V/R, color='r', linestyle='--', label='Steady-State Current') plt.title('RL Circuit Step Response') plt.xlabel('Time (s)') plt.ylabel('Current (A)') plt.legend() plt.grid(True) plt.show()
1. What happens to current in an RL circuit immediately after a step voltage is applied?
2. How does inductance affect the speed of current change?
3. In what applications are RL circuits commonly used?
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