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RL circuits combine resistors and inductors, creating unique electrical behavior. When voltage is applied, current doesn't instantly reach its maximum. Instead, it grows exponentially, governed by the . This gradual change is due to the 's opposition to current changes.

The circuit's response depends on the balance between and . A larger or smaller resistor slows current changes, while the opposite speeds them up. Energy is stored in the inductor's , affecting how the circuit charges and discharges over time.

RL Circuits

Behavior in RL circuits

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  • an
    • Connecting a voltage source to an RL circuit causes current to begin flowing
    • Current increases exponentially over time, approaching a maximum value of Imax=VRI_{max} = \frac{V}{R}
    • The rate of current increase is determined by the inductive τ=LR\tau = \frac{L}{R}
    • The current equation during charging is I(t)=VR(1−e−tτ)I(t) = \frac{V}{R}(1 - e^{-\frac{t}{\tau}})
  • an RL circuit
    • Removing the voltage source and shorting the circuit causes current to begin decreasing
    • Current decreases exponentially over time, approaching zero (exhibiting )
    • The rate of current decrease is determined by the inductive time constant τ=LR\tau = \frac{L}{R}
    • The current equation during discharging is I(t)=I0e−tτI(t) = I_0e^{-\frac{t}{\tau}}, where I0I_0 is the initial current
  • Voltage across the resistor
    • During charging, the voltage across the resistor is [VR(t)](https://www.fiveableKeyTerm:VR(t))=RI(t)=V(1−e−tτ)[V_R(t)](https://www.fiveableKeyTerm:V_R(t)) = RI(t) = V(1 - e^{-\frac{t}{\tau}})
    • During discharging, the voltage across the resistor is VR(t)=RI(t)=RI0e−tτV_R(t) = RI(t) = RI_0e^{-\frac{t}{\tau}}
  • Voltage across the inductor
    • During charging, the voltage across the inductor is [VL(t)](https://www.fiveableKeyTerm:VL(t))=LdIdt=Ve−tτ[V_L(t)](https://www.fiveableKeyTerm:V_L(t)) = L\frac{dI}{dt} = Ve^{-\frac{t}{\tau}}
    • During discharging, the voltage across the inductor is VL(t)=LdIdt=−RI0e−tτV_L(t) = L\frac{dI}{dt} = -RI_0e^{-\frac{t}{\tau}}
  • Kirchhoff's voltage law applies to RL circuits, stating that the sum of voltages around any closed loop in the circuit must equal zero

Inductive time constant effects

  • The inductive time constant τ=LR\tau = \frac{L}{R} determines the rate of current change in an RL circuit
    • A larger time constant results in a slower rate of current change (larger inductor or smaller resistor)
    • A smaller time constant results in a faster rate of current change (smaller inductor or larger resistor)
  • The time constant represents the time required for the current to reach approximately 63.2% of its final value during charging or to decrease to approximately 36.8% of its initial value during discharging
  • After one time constant, the current in an RL circuit reaches I(t)=Imax(1−e−1)≈0.632ImaxI(t) = I_{max}(1 - e^{-1}) \approx 0.632I_{max} during charging
  • After one time constant, the current in an RL circuit decreases to I(t)=I0e−1≈0.368I0I(t) = I_0e^{-1} \approx 0.368I_0 during discharging
  • The inductor opposes changes in current, causing the gradual increase or decrease in current over time due to its stored magnetic energy

Energy in inductor's magnetic field

  • The energy stored in the magnetic field of an inductor is given by E=12LI2E = \frac{1}{2}LI^2
    • LL is the inductance in henries (H)
    • II is the current flowing through the inductor in amperes (A)
  • The energy is stored in the magnetic field surrounding the inductor when current flows through it
  • As the current increases during charging, the energy stored in the magnetic field increases
  • As the current decreases during discharging, the energy stored in the magnetic field decreases
  • The maximum energy stored in the inductor occurs when the current reaches its maximum value Imax=VRI_{max} = \frac{V}{R}, resulting in Emax=12LImax2=12L(VR)2E_{max} = \frac{1}{2}LI_{max}^2 = \frac{1}{2}L(\frac{V}{R})^2
  • The stored energy in the inductor's magnetic field can be released back into the circuit during discharging, causing the current to decrease gradually instead of instantaneously

Power and Phase in RL Circuits

  • in an RL circuit occurs primarily in the resistor, converting electrical energy to heat
  • The in an RL circuit represents the time delay between the voltage and current waveforms
  • In AC circuits, the inductor causes the current to lag behind the voltage by a determined by the circuit's characteristics

Key Terms to Review (31)

Charging: Charging is the process of adding or removing electric charge from an object, resulting in an imbalance of positive and negative charges. This concept is fundamental to understanding the behavior of electrical circuits, particularly in the context of capacitors and inductors.
Charging by induction: Charging by induction involves transferring electric charge to an object without direct contact. It relies on the influence of a nearby charged object to redistribute electrons within a conductor.
DI/dt: The term dI/dt represents the rate of change of current over time in an electrical circuit. It is a crucial concept that describes how quickly the electric current is increasing or decreasing, which directly relates to the behavior of inductors and self-inductance. Understanding dI/dt helps explain how inductors resist changes in current and how RL circuits respond when connected to a voltage source or switched on and off.
Discharging: Discharging refers to the process by which an electric charge is released from a capacitor or an inductor, leading to a decrease in stored energy and current flow in the circuit. This action can cause the voltage across the component to drop and the energy to be dissipated as heat or light. Understanding discharging is crucial as it highlights the behavior of electric circuits during the energy release phase, showcasing how stored electrical energy transforms into other forms.
E = 1/2LI²: E = 1/2LI² is an equation that represents the energy stored in an inductor in an RL (Resistor-Inductor) circuit. It describes the relationship between the inductance (L) of the circuit, the current (I) flowing through the inductor, and the energy (E) stored in the magnetic field of the inductor.
E_max: E_max refers to the maximum energy stored in an inductor within an RL circuit, which occurs when the current flowing through the inductor reaches its peak value. This term is crucial for understanding energy dynamics in circuits that include both resistors and inductors, as it highlights how energy is converted and stored in the magnetic field of the inductor during the charging phase of the circuit.
Equivalent resistance: Equivalent resistance is the total resistance of a combination of resistors connected either in series or parallel. It simplifies complex circuits into a single resistor value that has the same effect on the circuit.
Exponential Decay: Exponential decay is a process where a quantity decreases at a rate proportional to its current value, leading to a rapid decline over time. This behavior is commonly represented mathematically by the equation $$N(t) = N_0 e^{-kt}$$, where $$N(t)$$ is the quantity at time $$t$$, $$N_0$$ is the initial quantity, and $$k$$ is the decay constant. In various physical systems, such as circuits, exponential decay describes how voltages or currents diminish over time when energy is released or dissipated.
Faraday's law: Faraday's law states that a change in magnetic flux through a circuit induces an electromotive force (emf) in that circuit. This principle is crucial for understanding how magnetic fields interact with electric circuits and lays the foundation for many applications in electromagnetism.
Heinrich Lenz: Heinrich Lenz was a Russian physicist who formulated the fundamental principle that describes the direction of the induced current in an electromagnetic induction system. This principle, known as Lenz's Law, is a crucial concept in understanding the behavior of electromagnetic phenomena and its applications in various areas of physics.
Henry: The henry (H) is the unit of measurement for the physical quantity of inductance, which is a measure of the magnetic field created by an electric current passing through a coil or other inductor. It is a fundamental unit in the study of electromagnetism and is essential in understanding the behavior of circuits involving inductors.
I_max: I_max, or the maximum current, is a key concept in the study of RL (Resistor-Inductor) circuits. It represents the maximum value of the current that can flow through the circuit under specific conditions, and it is an important parameter in understanding the behavior and applications of RL circuits.
I(t) = I₀e^(-t/τ): The equation I(t) = I₀e^(-t/τ) describes the behavior of current in an RL circuit over time as it decays exponentially. Here, I(t) is the current at time t, I₀ is the initial current, and τ (tau) is the time constant, representing how quickly the current decreases. This equation highlights the transient response of the circuit when the current is interrupted or altered, showcasing how inductance and resistance affect the flow of electric current.
I(t) = V/R(1 - e^(-t/τ)): The equation I(t) = V/R(1 - e^(-t/τ)) describes the current through an inductor in an RL circuit over time as it responds to a constant voltage source. Here, I(t) represents the instantaneous current, V is the applied voltage, R is the resistance, and τ (tau) is the time constant defined as L/R, where L is the inductance. This relationship shows how the current increases from zero to a maximum value asymptotically, illustrating the transient behavior of RL circuits during the charging phase when connected to a voltage source.
Inductance: Inductance is a fundamental property of electrical circuits that describes the ability of a component or circuit to store energy in the form of a magnetic field. It is a measure of the amount of magnetic flux produced by a current flowing through a circuit or component, and it plays a crucial role in the behavior of circuits, particularly in the context of solenoids, toroids, and RL (Resistor-Inductor) circuits.
Inductive time constant: The inductive time constant, denoted as $\tau_L$, is the time required for the current in an RL circuit to change significantly (about 63.2%) towards its final value after a change in voltage. It is calculated as $\tau_L = \frac{L}{R}$, where $L$ is the inductance and $R$ is the resistance.
Inductor: An inductor is a passive electrical component that stores energy in its magnetic field when electric current flows through it. It typically consists of a coil of wire and exhibits property known as inductance.
Inductor: An inductor is a passive electronic component that is used to store energy in the form of a magnetic field. It is a fundamental element in various electrical circuits and plays a crucial role in the behavior and functioning of these circuits.
Lenz's Law: Lenz's law is a fundamental principle in electromagnetic induction that describes the direction of the induced current in a conductor. It states that the direction of the induced current will be such that it opposes the change in the magnetic field that caused it, in accordance with Faraday's law of electromagnetic induction.
Magnetic Field: A magnetic field is a region of space where magnetic forces can be detected. It is a fundamental concept in electromagnetism, describing the invisible lines of force that surround and permeate magnetic materials, electric currents, and changing electric fields. The magnetic field plays a crucial role in various topics within the study of college physics.
Phase angle: Phase angle is the measure of the phase difference between the voltage and current in an AC circuit, usually expressed in degrees. It indicates whether the current leads or lags behind the voltage.
Phase Angle: The phase angle is the difference in the timing or displacement between two periodic signals, such as voltage and current, in an alternating current (AC) circuit. It represents the angular difference between the peak values of these signals and is a crucial parameter in understanding the behavior of AC circuits.
Power Dissipation: Power dissipation refers to the conversion of electrical energy into heat energy within a circuit or component. It is a fundamental concept in understanding the behavior and performance of electrical systems, particularly in the context of resistors and RL circuits.
Resistance: Resistance is a measure of the opposition to the flow of electric current in an electrical circuit. It is a fundamental concept in understanding the behavior of electric circuits and the relationship between voltage, current, and power.
RL Circuit: An RL circuit is an electrical circuit that consists of a resistor (R) and an inductor (L) connected in series. This type of circuit is used to study the behavior of current and voltage in the presence of both resistance and inductance, which are fundamental concepts in understanding the dynamics of electrical systems.
Steady State: Steady state refers to a condition in which the variables of a system, such as current or voltage, remain constant over time. This concept is particularly important in the analysis of electrical circuits, where it describes the long-term behavior of the circuit after any initial transient effects have subsided.
Time Constant: The time constant is a fundamental concept that describes the rate of change in various electrical and physical systems. It represents the time required for a system to reach approximately 63% of its final value when undergoing a step change in input.
Transient Response: The transient response refers to the temporary or short-lived behavior of a system as it transitions from one steady-state condition to another. It describes the initial, dynamic response of a system before it reaches a stable or equilibrium state.
V_L(t): V_L(t) is the time-dependent voltage across the inductor in an RL circuit. It represents the voltage drop across the inductor as a function of time, which is a crucial parameter in understanding the behavior of RL circuits.
V_R(t): V_R(t) represents the time-dependent voltage drop across the resistor in an RL circuit. It is a crucial parameter that describes the behavior of the circuit and the energy dissipation within the resistor as a function of time.
τ = L/R: The term τ = L/R represents the time constant of a resistor-inductor (RL) circuit, which describes the rate at which the current in the circuit changes over time. This time constant is a fundamental parameter that governs the behavior of RL circuits and their response to changes in the applied voltage or current.
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14.5 Oscillations in an LC Circuit