23.10 RL Circuits
3 min read•june 18, 2024
showcase the fascinating interplay between resistance and . When voltage is applied or removed, doesn't change instantly. Instead, it follows a smooth, due to the 's opposition to current changes.
The , τ = L/R, is key in RL circuits. It tells us how quickly the circuit responds to voltage changes. A larger τ means a slower response, while a smaller τ indicates a faster one. This behavior is crucial in many electrical systems.
RL Circuits
Current behavior in RL circuits
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- In an RL circuit, the current does not change instantaneously when the circuit is connected or disconnected from a because the inductor opposes the change in current, causing a gradual increase or decrease in current over time (e.g., turning on or off a motor)
- The current in an RL circuit follows an exponential curve
- When the circuit is connected to a voltage source, the current starts at zero and increases exponentially towards its maximum value, (e.g., charging an inductor)
- When the circuit is disconnected from the voltage source, the current decreases exponentially from its maximum value to zero (e.g., discharging an inductor)
- This behavior is related to the inductor's ability to store energy in its magnetic field
Time constant of RL circuits
- The time constant, denoted by (tau), characterizes the response of an RL circuit to changes in the applied voltage, representing the time required for the current to reach approximately 63.2% of its final value when the circuit is connected to a voltage source, or to decrease to approximately 36.8% of its initial value when disconnected (e.g., time for a relay to switch on or off)
- The time constant is calculated using the formula:
- is the of the inductor in (H)
- is the total resistance of the circuit in ()
- A larger time constant indicates a slower response to changes in the applied voltage, while a smaller time constant indicates a faster response (e.g., a large inductor with a small resistance will have a slower response compared to a small inductor with a large resistance)
Current calculations at specific intervals
- The current in an RL circuit at any given time can be calculated using the following equations:
- When the circuit is connected to a voltage source:
- When the circuit is disconnected from the voltage source:
- In these equations:
- is the current at time
- is the maximum current, equal to
- is the mathematical constant (approximately 2.718)
- is the time elapsed since the circuit was connected or disconnected
- is the time constant of the circuit
- To solve for current values at specific time intervals, substitute the given values into the appropriate equation and calculate the result (e.g., find the current 5ms after connecting a 12V battery to an RL circuit with a 100mH inductor and 50 )
Electromagnetic Induction in RL Circuits
- describes the relationship between changing and induced electromotive force (emf) in RL circuits
- The changing current in an RL circuit creates a time-varying , which induces a in the inductor
- This back emf opposes the change in current, contributing to the circuit's
- The transient response describes the circuit's behavior during the transition between its initial state and
- After sufficient time has passed, the circuit reaches steady state, where the current remains constant in DC circuits or follows the driving voltage in AC circuits
Key Terms to Review (30)
$ ext{ω}$: $ ext{ω}$ is a Greek letter that represents angular velocity, a measure of how quickly an object rotates around a fixed axis. It is a fundamental concept in the study of rotational motion and is closely related to other important physical quantities such as angular acceleration and rotational kinetic energy.
$\tau$: $\tau$ is a symbol used to represent torque, which is the rotational equivalent of force. Torque is a measure of the rotational force that causes an object to rotate about an axis, fulcrum, or pivot. This term is crucial in understanding the concepts of angular acceleration, rotational kinetic energy, and RL circuits.
$e$: $e$ is a mathematical constant that is the base of the natural logarithm. It is an irrational number, approximately equal to 2.71828, and is fundamental to many areas of mathematics and physics, including the study of RL circuits.
$I_{max}(1 - e^{-t/\tau})$: $I_{max}(1 - e^{-t/\tau})$ is a mathematical expression that describes the current in an RL (Resistor-Inductor) circuit as a function of time. It represents the maximum current ($I_{max}$) that the circuit will reach, and how the current approaches this maximum value over time, as determined by the time constant ($\tau$) of the circuit.
$I_{max}$: $I_{max}$ is the maximum current that can flow through an RL (resistor-inductor) circuit. It represents the maximum value of the current in the circuit, which is reached when the circuit has reached a steady state condition and the current is no longer changing with time.
$I_{max}e^{-t/\tau}$: $I_{max}e^{-t/\tau}$ is a mathematical expression that describes the current in an RL (resistor-inductor) circuit as a function of time. It represents the exponential decay of the current from its maximum value ($I_{max}$) to zero, with the time constant ($\tau$) determining the rate of decay.
$I(t)$: $I(t)$ represents the current as a function of time in an electrical circuit. It is a fundamental quantity that describes the flow of electric charge over time, and is a crucial concept in the study of RL (Resistor-Inductor) circuits.
$L/R$: $L/R$ is the ratio of the inductance (L) to the resistance (R) in an RL circuit. This ratio is a crucial parameter that determines the behavior and time-dependent characteristics of the circuit, such as the time constant and the rate of change of current.
$V/R$: $V/R$ is the ratio of the voltage ($V$) to the resistance ($R$) in an electrical circuit. This fundamental relationship is a crucial concept in understanding the behavior and analysis of various circuit types, particularly RL (Resistor-Inductor) circuits.
Back EMF: Back EMF, or back electromotive force, is an induced voltage that opposes the change in current flowing through an inductor. It is a fundamental concept in understanding the behavior of electrical circuits involving inductors, such as in the context of 23.6 Back EMF, 23.9 Inductance, and 23.10 RL Circuits.
Characteristic time constant: The characteristic time constant in an RL circuit, denoted as $\tau$, is the time it takes for the current to reach approximately 63% of its final value after a sudden change in voltage. It is calculated as the ratio of inductance $L$ to resistance $R$, i.e., $\tau = \frac{L}{R}$.
Current: Current is the flow of electric charge in a circuit, typically measured in amperes (A). It represents how much charge passes through a point in the circuit per unit of time, and it plays a crucial role in determining how electrical energy is distributed and consumed in various applications.
Energy Storage: Energy storage refers to the ability to store energy in various forms, such as chemical, electrical, or mechanical, for later use. It is a crucial concept in the context of both world energy use and electrical circuits, as it allows for the efficient management and utilization of energy resources.
Energy stored in an inductor: Energy stored in an inductor is the potential energy due to the magnetic field created by current flowing through it. This energy can be expressed mathematically as $E = \frac{1}{2}LI^2$, where $L$ is inductance and $I$ is current.
Exponential Curve: An exponential curve is a mathematical function that exhibits a characteristic shape where the rate of change increases or decreases at a constant rate over time. This type of curve is commonly observed in various natural and technological phenomena.
Faraday's Law: Faraday's law describes the relationship between a changing magnetic field and the electric field it induces. It states that the magnitude of the induced electromotive force (emf) in a circuit is proportional to the rate of change of the magnetic flux through the circuit.
Henries: Henries (H) is the unit of inductance, which is a measure of the amount of magnetic flux produced by an electric current. Inductance is a fundamental property of electrical circuits and is crucial in understanding the behavior of RL (Resistor-Inductor) circuits.
Inductance: Inductance is a property of an electrical conductor that opposes a change in current. It is measured in henrys (H) and results from the magnetic field generated by the current flowing through the conductor.
Inductance: Inductance is a property of an electrical circuit or component that opposes changes in the electric current flowing through it. It is a measure of the magnetic field generated by a current-carrying conductor, which in turn induces a voltage that opposes the change in current.Inductance is a fundamental concept in understanding the behavior of electrical circuits, particularly in the context of RL circuits and RLC series AC circuits.
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 electrical circuits, particularly in the context of RL circuits and reactance.
Magnetic flux: Magnetic flux is the measure of the quantity of magnetism, taking into account the strength and extent of a magnetic field. It is calculated as the product of the magnetic field and the area through which it passes, perpendicular to the field.
Magnetic Flux: Magnetic flux is a measure of the total amount of magnetic field passing through a given surface or area. It represents the strength and distribution of a magnetic field and is a fundamental concept in the study of electromagnetism and its applications.
Ohms: Ohms is the unit of electrical resistance in the International System of Units (SI). It quantifies how much a material opposes the flow of electric current, allowing for the understanding of how voltage and current are related in a circuit. The relationship is described by Ohm's Law, which states that the voltage across a conductor is directly proportional to the current flowing through it, provided the temperature remains constant.
Resistor: A resistor is an electrical component that limits or regulates the flow of electrical current in a circuit. It provides resistance, measured in ohms ($\Omega$), to control voltage and current levels.
Resistor: A resistor is a passive electronic component that is used to control or limit the flow of electric current in a circuit. It is a fundamental element in electrical and electronic systems, playing a crucial role in various applications such as voltage division, current regulation, and signal processing.
RL Circuits: An RL circuit is an electrical circuit that consists of a resistor (R) and an inductor (L) connected in series. RL circuits are used to study the behavior of electrical systems that involve both resistance and inductance, which are fundamental properties of electronic components and devices.
Steady State: Steady state refers to a condition in which a system or process has reached a stable, unchanging state, where the input and output values remain constant over time. This concept is particularly relevant in the context of electrical circuits, where it describes the point at which the circuit has reached a stable, predictable behavior.
Time Constant: The time constant is a fundamental concept in the study of electrical circuits, particularly those involving resistors and capacitors (RC circuits) or inductors and resistors (RL circuits). It represents the time required for a circuit to reach a specific percentage of its final value when subjected to a step change in input.
Transient Response: The transient response refers to the temporary, initial behavior of a system when it is subjected to a change in input or initial conditions. It describes the system's dynamic response before it settles into a steady-state or equilibrium condition.
Voltage Source: A voltage source is a device that maintains a constant potential difference, or voltage, between two points in an electrical circuit. It is the driving force that pushes electric charge through the circuit, providing the energy necessary for the circuit to function.