An In-Depth Study of Current Flow in an LR Circuit: An Understanding

The understanding of technology in the modern world greatly relies on electrical circuits. Advancements in engineering and technology are made possible with a proper understanding of their behavior. There are many forms of circuits, one of the commonly uses is an LR Circuit made up of an inductor and a resistive load connected in series. It is called an LR circuit because of its inductor-capacity nature. In this paper we will examine current flow with respect to the two major components of an LR circuit. The focus will be on the relationship between windings and time. Thus, whatever the nature of the reader, learner, or professional they will find this guide useful to expand their foundational understanding of the topic of electrical circuits.

What are LR Circuits and How Do They Function?

An LR circuit comprises of an inductor, L, and a resistor, R, linked either in series or parallel within an electric circuit. The inductor’s property of resisting changes in current with current induction, known as inductance, and the resistor opposing current flow and supplying energy in the form of heat acts as a dissipative element. Applying voltage does not let the current in the circuit rise to the maximum value theoretically; rather, it reaches the maximum value gradually over time with the time constant, τ = L/R. This time constant indicates the speed of a circuit’s reaction to a change in voltage. These circuits are commonly used in electrical systems for filtering, signal processing, and controlling transients.

Identifying the Parts: A Series Resistor and Inductor

The time constant denoted as τ, is particularly important in a series inductor-resistor (LR) circuit as it determines the rate at which the circuit may react to a change in voltage. It is defined as τ = L/R where L is the inductance in henries (H) and R is the resistance in ohms (Ω). The response is fastest for low time constants and achieves the steady state current level in less time, whereas a larger time constant slow system response. This is important where control of transient response is critical, such as in signal and electronic filters.

Considerations of Voltage in LR Series

To appreciate the role of voltage in an LR series circuit, one needs to consider the transient and steady state behavior of the circuit. While a voltage is being applied to the circuit, the inductor resist all sudden changes in current due to its inductance. This causes a time varying voltage drop across the inductor and the resistor.

• Voltage Across the Inductor ($V_\mathsf{L}$):
• The voltage across the inductor can be determined using the below equation.
• VL = L * (dI/dt)
• L is the voltage in henries (H),
• dI/dt is current’s rate of change with respect to time.
• $V_\mathsf{L}$ is at its peak when t =0, this is the moment the circuit has been energized, because current’s rate of increase is at its highest which results in maximum voltage.
• Voltage Across the Resistor (VR):
• The voltage across the resistor follows Ohm’s Law:
• VR = I * R
• Voltage across the resistor is determined by instantaneous current in amperes (A) (I) and resistance in ohms (Ω) (R).
• There is an initial transient period during which current increases exponentially. This leads to gradual and reaches a voltage steady state.
• Numerical Example:
• L = 5 H
• R = 10 Ω
• Applied Voltage (V) = 50 V
• As for the time constant, we have:
• τ = L / R = 5 / 10 = 0.5 seconds
• Also, according to the law of growth of current, at a certain time t, the current is equal to:
• I(t) = (V / R) * (1 – e^(-t/τ))
• At t=0, I(0) = 0 A, V_{L} = 50V
• At t= τ = 0.5 s I(τ) = 3.16A, V_{R} = 31.6V
• At stady state, t → ∞ I(τ) = 5A, V_{R} = 50V

This example shows how the voltage across the inductor and the resistor changes over time, illustrating the impact that both inductance and resistance have on the circuit’s operation.

LR Circuit vs. Other Circuit Types

At the bottom of this page, we provide the most important parameters of the LR circuit and their values in the order of occurrence.

• At t=0:
• Current, I(0) : 0A
• Voltage across Inductor, VL(0) : 50V
• Voltage across Resistor, VR(0): 0V
• At t = τ (0.5s)
• Required parameters:
• Current, I(τ): 3.16A
• Voltage across Inductor, VL(τ): 18.4V
• Voltage across Resistor, VR(τ): 31.6 V
• At Steady state(t: ∞)
• Required Parameters:
• Current, I(∞): 5 A
• Voltage across Inductor, VL(∞): 0
• Voltage across Resistor, VR(∞): 50 V
• Current is shown to increase exponentially over time, with a constant value, 5A, with time.
• Voltage across the Inductor in this case is shown to decrease over time starting at its maximum value (50V) and reaching 0V at steady state.
• In this case at steady state, Voltage across the Resistor shows an incrementing trend with a maximum value of 50V at steady state.

The information available here shows the energy exchange taking place in the various components i.e. inductor and resistor (along with time) which with time stabilizes.

How Does Current Change in an LR Circuit Over Time?

Concept of the Time Constant With Regards to an LR Circuit

The time constant (\( \tau \)) in an LR circuit quantifies the time taken for the system’s current to stabilize after a step increase in voltage. It can be computed from the quotient of inductance and resistance as \(\tau = \frac{L}{R}\), meaning that \({L}\) is the Inudctance in Henries and \((R}\) is the resistance in Ohm.

At no time does the current reach a maximum value immediately after a voltage is applied to the circuit because of the inductor does not allow change in current. But current increases exponentially and follows the below mentioned equation.

I(t) = I_{\text{max}} \left( 1 – e^{-\frac{t}{\tau}} \right)

\( t \) denotes time while \( e \) describes the base of the natural log. As a rule of thumb, every 5 time constant, the current becomes 99% of timer state value and achieves steady state value, thus completing its shall transient phase. Reaching a value of approximately 5 time constants does indicate effectively completing the transient phase, which, underscores the importance of time constand in studying the behavior of LR circuits.

The Exponential Growth and Decay of Current

Below are the critical parameters and details associated with the exponential growth and decay of current in an LR circuit:

\textbf{Maximum Steady-State Current ($I_{\text{max}}$)}

Indicates the maximum current the system can reach based on the voltage and resistance applied to the circuit.

I_{\text{max}} = \frac{V}{R}

where \( V \) is the voltage across the circuit and \( R \) is the total resistance.

\textbf{Time Constant ($\tau$)}

Refers to the time required to reach approximately 63.2% of the steady state value during growth phase, or 36.8% during decay phase.

\tau = \frac{L}{R}

where \( L \) is the inductance and \( R \) is the resistance.

I(t) = I_{\text{max}} \left(1 – e^{-\frac{t}{\tau}}\right)

As current begins to flow, it increases principally at first and then approaches \( I_{\text{max}} \) as it levels off.

Current governed by the equation:

I(t) = I_{\text{max}} e^{-\frac{t}{\tau}}

When the supply is unhooked, current reduces to nearly zero.

\textbf{Steady-State Value of Current}\\

No matter what the initial conditions are, approximately 5 time constants (\( 5\tau \)) are required to reach to the value of steady state.

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Natural Logarithm Base (\( e \))

\( e \approx 2.718 \) is the mathematical constant used in the exponential functions that describe growth and decay.

This allows for accurate evaluation and forecasting of the behavior of an LR circuit, which helps in the designing and troubleshooting of electrical systems.

Circuits With an Inductor and Resistor Attached in Series

The two factors affecting the flow of current in an LR circuit are:

Resistance (\( R \)): Resistance opposes the current, impacting how rapidly the current flows in the circuit. A higher resistance would delay the response of the current and reduce the steady state current.

Inductance (\( L \)): Inductance tends to oppose any increase in current by creating a counter-electromotive force that resists the current flow. Higher inductance results in a greater time constant (\( \tau = L / R \)), which increases the time taken for the current to attain its steady state value.

These factors interact to create the various changes in the circuit’s operations over time.

What is the Time Constant in an LR Circuit?

Finding the Time Constant when Inductance and Resistance are Present

The Time Constant (\( τ \)) in an LR circuit can be defined as the time required for a current to change to a value about 63.2 percent of its final value. The time is measured after a step change in voltage is applied. In such cases, \( τ \) may be calculated as:

τ = L/R

Where L is the inductance measured in henries (H) and R is the resistance in ohms (\(Ω\)).

This relation brings out the fact that an increase in inductance or a decrease in resistance results in an increased time constant, which means that the circuit is not responding well to changes. On the other hand, a decrease in inductance or an increase in resistance results in decreased time constant, which means the circuit is responsive. In both cases the time constant is a crucial parameter in analyzing the transient behavior of the LR circuits.

Effects of Time Constant on Growth and Decay of Current

In order to analyze the impact of time constant in detail for LR circuits, the following key features are needed together with their corresponding characteristics of Time constant.

Unit of Measurement: Henrys (H)

Role in Circuit Behavior:

Increased value of inductance leads to increased time constant that slows the current growth and decay rate.

Lower inductance decreases the time constant, enabling faster growth and decay of the current.

Unit of Measurement: Ohms (\( \Omega \))

Role in Circuit Behavior:

Greater resistance leads to reduced time constant value and quicker rate of response during the transient phase.

Lower resistance increases the time constant, leading to slower response during the transients.

Formula: \( \tau = \frac{L}{R} \)

Unit of Measurement: Seconds (s)

Significance:

Determines the rate at which current changes, both in the ‘growth phase’ (when voltage is applied) and in the ‘decay phase’ (when voltage is removed).

A larger \( \tau \) value denotes slower transitions while a smaller \( \tau \) value indicates a faster change.

Growth Phase:

The current, \(I\) during the time \(t\), can be approximated with the relation \( I(t) = I_{max}(1 – e^{-\frac{t}{\tau}}) \) to where \( I_{max} \) is the maximum (asymptotic) value of constant current in steady state.

The time required to achieve steady state is prolonged when \( \tau \) increases.

Decay Phase:

The decrease in current can be stated as \( I(t) = I_{max} e^{-\frac{t}{\tau}} \).

A rapid decay is noted when there is a smaller \( \tau \) value.

These parameters in conjunction with each other defining precise mathematics are essential in analyzing the patterns of transient behavior within LR circuits. Such information helps greatly in the development of circuits having particular design requirements.

How Does the Voltage Across Components Affect Current?

Examining Voltage Drop Over an Inductor

The voltage over an inductor in an LR circuit is given by the following relation:

V_L = L \frac{dI}{dt}

V_L: Voltage over the inductor (in volts).

L: Inductance of the inductor (in henries, H).

\frac{dI}{dt}: Change of current with respect to time (in Amperes per second).

Inductance (L):

With higher inductance, the rate of change in the current is slower and hence the voltage drop more significantly.

Measured in Henries (H).

Initial Current (I_{max}):

The current maximum at t = 0 affects the rest of the response during steady-state.

Time Constant (\tau):

Defined as \tau = \frac{L}{R} , where R is resistance.

Larger \tau is associated with slower changes in current and voltage

Time(t):

Denotes the period during which the system undergoing the transient response.

These parameters help to explain the behavior of voltage drop over the inductor during the circuit’s transient phase. Precise control and manipulation of these parameters aids the design and performance of the circuit for the desired purpose.

Impact of Voltage on Resistor on Circuit Operation

Both in steady state and in transients, the voltage across the resistor \( V_R \) affects the operation and performance of the circuit. Ohm’s law states that \( V_R = I \cdot R \), where \( I \) stands for current and \( R \) is resistance. Changing any value of resistance or current in the system will affect \( V_R \) leading to change in power dissipation, which can be expressed as \( P_R = \frac{V_R^2}{R} \). Also, using too high of a value for voltage drops across the resistor may lower the operational voltage for other parts of the circuit, leading to performance problems. It is a must that engineers have balanced values of resistors and have set values of voltage so as to mitigate losses in the performance of the circuit.

What Happens to Current During the Steady State of an LR Circuit?

Characteristics of the Steady State in RL Circuits

Current Stabilization:

In an RL circuit, during the steady state, the inductor current reaches its maximum value which is constant and is given by ohm’s law as \(I = V/R\) where \(V\) is the applied voltage and \(R\) is the total resistance of the circuit.

Inductive Reactance:

The term \(X_L\) becomes insignificant and behaves like a short circuit.

Since \(X_L = 2\pi f L\) and \(f = 0\) for D.C. signals in the steady state, \(X_L = 0\).

Voltage across the Inductor:

The voltage drop across the inductor becomes zero when the current is constant.

This follows the equation \(V_L = L \frac{dI}{dt}\), where during the steady state, \(\frac{dI}{dt}\) is zero.

Power Distribution:

As current flows through the inductor, no energy is stored, hence power is dissipated only through the resistor as \(P=I^2R\).

Energy Stored in the Magnetic Field:

The energy stored in the magnetic field of the inductor gets released when the circuit approaches steady state conditions.

The energy is provided in the form of \( E = \frac{1}{2} L I^2 \), which becomes inconsequential when the rate of current alteration ceases.

These parameters mark the functioning of RL electrical circuits when in a steady state and are pivotal in any system which requires the automated or manual control of electrical components and devices.

What Causes The Increase in Current

In an RL circuit, the current level is constant and at its uppermost peak because the inductor will at some point let the current persist at a certain level. At first, the inductor constantly opposes changes in the current flow due to self-induction, though eventually with time, the magnetic field becomes stable, thus reducing resistance (inductive reactance). In a steady-state condition, the inductor acts as a simple wire, which allows current to freely pass through. The level of current is dependent upon resistance \( R \) and voltage \( V \), according to Ohm’s Law: \( I = \frac{V}{R} \).

Frequently Asked Questions (FAQs)

Q: What is a series RL circuit?

A: A series RL circuit is a type of electric circuit that includes a resistor (R), an inductor (L), and a power source (battery) connected in series. This circuit helps one in analyzing the behavior of electric current under varying voltage conditions.

Q: In what way is the current in a series RL circuit time dependent?

A: In a series RL circuit, when the switch is closed, the current increases proportionally with time until it reaches a steady value and the growth is often be described with the help of a differential equation. In the beginning, current is nonexistent, but due to the work put in by the inductor, current builds till a certain point.

Q: What does the series RL circuit inductor voltage oppose?

A: In an RL circuit, the varying inductor voltage obstruct flow changes of current. While powering the circuit, the inductor produces a magnetic flux to counter the electrical current flow to current with dynamics that prevent constant change. Inductors build an electromagnetic flux when rate changes occur and counter parts are needed.

Q: In what ways does circuit analysis help understand the behavior of the current in RL circuit?

A: Analyzing the circuit gives us the capability to predict the behavior of current and voltage in a series RL circuit. With voltage law and other electrical laws applied, we can obtain restricted values of current as well as the final values and study the transient and steady state responses.

Q: What does it mean to reach the maximum current in a series RL circuit?

A: The moment the maximum current is attained in a series RL circuit, current is said to be constant which means that the energy stored in the inductor now is equal to the energy being lost in the resistor. In this moment, the increase of current value stops and the circuit is in a steady state.

Q: How can the current in an RL series circuit be calculated at a certain point in time?

A: The current in the RL series circuit can be calculated at any point in time by solving the differential equation which describes the circuit. This requires applying the initial conditions and factoring in the time constant which depends on the resistor and inductor’s values.

Q: What is the reasoning behind the importance of the time constant in RL series circuit?

A: The time constant in series RL circuit, given by the formula L/R where L is the inductance and R is the resistance, defines how fast the current grow and decline. It signifies a duration in which the current will surge till approximately 63% of the maximum or will fall to 37% of the initial value during transient state.

Q: Is it possible to use the series RL circuit as a filter?

A: Yes, a series RL circuit can operate as an RL filter. It permits RL circuits to function as low pass filters which allows signals of lower frequency to pass through while reducing signals of higher frequency. The value of RL also defines the degree of filtering capabilites.

Q: How does the current behave in a series RL circuit when the switch is opened?

A: Once the switch is opened in a series RL circuit, the energy retained in the inductor is released through the resistor, which causes the current to decrease. The decay of current follows an exponential function, which approaches zero based on the circuit’s time constant.

Reference Sources

1. Title: Short-Circuit Current Contribution of Doubly-Fed Wind Turbines According to IEC and IEEE Standards
   Authors: Francisco Jimenez-Buendía et al.
   Journal: IEEE Transactions on Power Delivery
   Publication Date: October 1, 2021
   Citation Token: (Jimenez-Buendia et al., 2021, pp. 2904–2912)
   Summary:
   This paper discusses the calculation of short-circuit currents for doubly-fed wind turbines, which are increasingly integrated into power systems. The authors develop a methodology to calculate the short-circuit contribution of these turbines based on IEC and IEEE standards. The study emphasizes the importance of understanding the short-circuit behavior of wind turbines to ensure proper protection system design. The methodology is validated through field measurements and simulations, highlighting the need for accurate short-circuit current calculations in modern power systems.
2. Title: Short-Circuit Current Analysis of AC Distribution Systems Dominated by Voltage Source Converters Considering Converter Limitations
   Authors: Jie Song et al.
   Journal: IEEE Transactions on Smart Grid
   Publication Date: September 1, 2022
   Citation Token: (Song et al., 2022, pp. 3867–3878)
   Summary:
   This paper presents a novel methodology for analyzing short-circuit currents in distribution systems dominated by voltage source converters (VSCs). The authors consider the control modes and potential current-saturated operation of power converters. The study models the distribution system using an element-based steady-state formulation, identifying equilibrium points under different saturation states of the VSCs. The findings indicate that VSC operation significantly impacts short-circuit equilibrium points, providing insights for the design of protection systems in VSC-dominated networks.
3. Title: Short-Circuit Current Calculation and Performance Requirement of HVDC Breakers for MMC-MTDC Systems
   Authors: Zheren Zhang, Zheng Xu
   Journal: IEEJ Transactions on Electrical and Electronic Engineering
   Publication Date: March 1, 2016
   Citation Token: (Zhang & Xu, 2016)
   Summary:
   This paper focuses on calculating the maximum short-circuit current for high-voltage direct current (HVDC) breakers in modular multilevel converter (MMC)-based multi-terminal direct current (MTDC) systems. The authors present an equivalent model for calculating short-circuit currents, decomposing them into steady-state and fault components. The methodology is validated through simulations, providing insights into the performance requirements of HVDC breakers and their impact on system design.

RL circuit

Electrical network