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RL circuit

An RL circuit is an electrical circuit composed of a and an connected in series or parallel, driven by a voltage or source, and is fundamental for analyzing transient and steady-state behaviors in . The limits current flow and dissipates energy as heat, while the stores energy in a magnetic field and opposes changes in current through induced voltage. In direct current (DC) applications, the circuit exhibits transient responses where current grows or decays exponentially after a switch event, characterized by a time constant \tau = L/R, where L is inductance in henries and R is resistance in ohms; after approximately five time constants, the circuit reaches steady state with the behaving as a short . The current in such transients follows i(t) = I(\infty) + [i(0^+) - I(\infty)] e^{-t/\tau}, where i(0^+) is the initial current and I(\infty) is the final steady-state current. In alternating current (AC) scenarios, the introduces reactance X_L = 2\pi f L, where f is frequency, resulting in an impedance Z = \sqrt{R^2 + X_L^2} for series configurations and a phase difference where voltage leads current by \theta = \tan^{-1}(X_L / R). RL circuits are essential in applications such as filters, where series configurations act as low-pass filters attenuating high frequencies, and in power electronics for managing current surges and energy storage.

Fundamentals

Components and Configuration

An RL circuit is an electrical circuit composed of a and an , typically connected to a voltage or current source. The limits the flow of and dissipates electrical energy as heat according to , which states that the voltage drop across the is V_R = I R, where I is the current and R is the . The , often a coil of wire, stores energy in a magnetic field and opposes changes in current; its voltage is given by Faraday's law of induction as V_L = L \frac{dI}{dt}, where L is the inductance. Resistance is measured in ohms (\Omega), while is measured in henries (H). A key property of inductors is that through them cannot change instantaneously, so the immediately before and after a switching remains the same: i(0^-) = i(0^+). Standard configurations include the series RL circuit, where the and share the same in a single path connected to a , such that the total voltage is the sum of the drops across each component (as in a loop diagram with source, R, and L in sequence). In the parallel RL circuit, the and share the same voltage across them from the source, with currents adding up through each branch (as in a diagram with source connected to both R and L in parallel branches). The ratio \tau = L/R serves as a for the circuit's response.

Basic Equations

In a series RL circuit, the fundamental equation is derived from Kirchhoff's voltage law (KVL), which states that the supplied voltage equals the sum of the voltage drops across the resistor and inductor. The voltage across the resistor is Ri(t), and across the inductor is L \frac{di(t)}{dt}, yielding V_s(t) = R i(t) + L \frac{di(t)}{dt}. This equation reflects the inductor's property that its voltage opposes changes in current, consistent with Lenz's law. Rearranging the KVL equation produces the standard first-order linear differential equation for the current: \frac{di(t)}{dt} + \frac{R}{L} i(t) = \frac{1}{L} V_s(t). The general solution to this differential equation consists of the homogeneous solution plus a particular solution. The homogeneous equation \frac{di_h(t)}{dt} + \frac{R}{L} i_h(t) = 0 has the solution i_h(t) = A e^{-(R/L)t}, where A is a constant determined by initial conditions and \tau = L/R is the time constant. The particular solution i_p(t) depends on the form of the source voltage V_s(t); for a constant input, i_p(t) = V_s / R, while for an exponential input V_s(t) = V_0 e^{st}, assume i_p(t) = K e^{st} and solve for K. For a parallel RL circuit driven by a , Kirchhoff's current law (KCL) at the common node equates the source to the sum of the and branch currents. The is V(t)/R, and the is (1/L) \int V(t) \, dt, resulting in the integral-differential equation I_s(t) = \frac{V(t)}{R} + \frac{1}{L} \int_{-\infty}^{t} V(\tau) \, d\tau. Differentiating this equation yields a first-order differential form for the voltage, analogous to the series case but with roles of voltage and current interchanged. The time constant remains \tau = L/R. Energy considerations in RL circuits involve power dissipation and storage. The instantaneous power dissipated as heat in the resistor is P_R(t) = i^2(t) R. The instantaneous power associated with the inductor, representing the rate of change of its stored magnetic energy, is P_L(t) = v_L(t) i(t) = L i(t) \frac{di(t)}{dt}. Over time, the total energy stored in the inductor's magnetic field is (1/2) L i^2(t), while all energy input is eventually dissipated in the resistor.

Time-Domain Response

DC Steady-State Behavior

In the DC steady-state regime of an RL circuit, the response to a constant voltage or stabilizes such that the rate of change of current through the is zero (di/dt = 0). This condition arises after sufficient time has passed for transients to decay, rendering the 's inductive ineffective. The then behaves as a with zero across it (V_L = L \, di/dt = 0), allowing the circuit to function as if the were replaced by a wire of negligible . This equivalence simplifies analysis, as the steady-state currents and voltages are determined solely by the resistive elements and sources. For a series RL circuit connected to a voltage source V_s, the steady-state current is i(\infty) = V_s / R, where R is the total . The source voltage appears entirely across the (V_R = V_s), while the carries this constant current with no opposing voltage. In a parallel RL configuration driven by a I_s, the similarly acts as a at , shunting all current such that i_L(\infty) = I_s (constant) and the current through the drops to zero (i_R(\infty) = 0), resulting in zero voltage across the parallel branches. This current distribution highlights the inductor's role in maintaining a constant flow without diverting to the resistive path once equilibrium is achieved. At steady state, the inductor stores magnetic energy given by \frac{1}{2} L I^2, where I is the constant current through it; this energy remains fixed as long as the current persists, representing the peak accumulation from the initial energization. For example, in charging a series RL circuit from zero initial current with V_s, the steady-state energy is \frac{1}{2} L (V_s / R)^2, fully stored in the inductor's magnetic field. In discharging, such as when the source is removed and the inductor current flows through a resistor, the steady-state condition is zero current (i(\infty) = 0) and zero stored energy, with all magnetic energy dissipated as heat in the resistor. The time to reach these steady states is governed by the circuit time constant \tau = L/R, typically requiring about five time constants for near-complete settling. Practically, this steady-state behavior enables inductors to permit unimpeded current flow in circuits once stabilized, limited only by , which is essential in power supplies and filters where components must pass while transients are managed. In applications like converters, the inductor's short-circuit equivalence at ensures efficient steady current delivery without additional voltage drops.

Transient Response to Step Input

The of an RL circuit to a step input describes the dynamic evolution of currents and voltages as the circuit transitions from an initial state to a new following the abrupt application of a constant voltage or . This behavior is governed by the inductor's opposition to sudden changes in current, leading to transients characterized by the circuit's . In a series RL circuit subjected to a voltage step, the current builds up gradually, while in a parallel RL circuit driven by a current step, the voltage across the components decays exponentially. Consider a series RL circuit where a voltage step V_s(t) = V_0 u(t) is applied at t = 0, with the resistor R and inductor L connected in series; the differential equation arises from Kirchhoff's voltage law as L \frac{di(t)}{dt} + R i(t) = V_0 for t \geq 0. The general solution for the inductor current is i(t) = \frac{V_0}{R} \left(1 - e^{-(R/L)t}\right) + i(0) e^{-(R/L)t}, where i(0) is the initial current at t = 0^-. This combines the steady-state particular solution \frac{V_0}{R} with the homogeneous solution A e^{-(R/L)t}, where the constant A is determined by the initial condition to ensure current continuity: i(0^+) = i(0). At t = 0^+, the inductor voltage is v_L(0^+) = V_0 - R i(0), meaning the full step voltage initially drops across the inductor if i(0) = 0. The time constant \tau = \frac{L}{R} defines the rate of this exponential response, representing the time for the current to reach approximately 63% of its final change from the initial value. For instance, with i(0) = 0, the current reaches 63% of \frac{V_0}{R} at t = \tau, about 95% at t \approx 3\tau, and nearly 99% at t \approx 5\tau. The voltage across the inductor follows v_L(t) = V_0 e^{-(R/L)t} when i(0) = 0, decaying exponentially from V_0 to 0. Typical time-domain plots show the current starting at i(0) and asymptotically approaching \frac{V_0}{R}, while the inductor voltage starts at V_0 and decays to 0, illustrating the energy storage and release in the inductor. For a parallel RL circuit, consider a current step I_s(t) = I_0 u(t) applied across the parallel combination of R and L at t = 0, assuming initial inductor current i_L(0) = 0; the voltage response across the parallel branches is v(t) = I_0 R e^{-(R/L)t} for t \geq 0. This arises from the relation I_0 = \frac{v(t)}{R} + i_L(t) and v(t) = L \frac{di_L(t)}{dt}, yielding the exponential decay with the inductor current building to I_0 as i_L(t) = I_0 \left(1 - e^{-(R/L)t}\right). At t = 0^+, the voltage jumps to v(0^+) = I_0 R (fully across the resistor, as inductor current cannot change instantly), and the inductor current remains 0. The same time constant \tau = \frac{L}{R} applies, with the voltage decaying to 63% of I_0 R at t = \tau and approaching 5% at t \approx 3\tau. Plots for this configuration depict the voltage starting at I_0 R and exponentially falling to 0, while the inductor current rises from 0 to I_0, highlighting the current division and transient energy transfer.

Frequency-Domain Analysis

Complex Impedance

In the frequency domain, the behavior of an RL circuit under sinusoidal excitation is analyzed using complex impedance, which extends Ohm's law to account for both resistive and reactive effects. The impedance of a resistor, denoted Z_R, is purely real and frequency-independent, given by Z_R = R, where R is the resistance in ohms. This reflects that the voltage across a resistor is in phase with the current, with no energy storage involved. The impedance of an , Z_L, is purely imaginary and proportional to , expressed as Z_L = j \omega L, where j = \sqrt{-1} is the , \omega = 2\pi f is the in radians per second, f is the in hertz, and L is the in henries. This form arises because the opposes changes in , causing the voltage to lead the by 90 degrees, with the |\omega L| increasing linearly with . For a series RL circuit, the total impedance Z is the vector sum of the individual impedances: Z = R + j \omega L. The magnitude is |Z| = \sqrt{R^2 + (\omega L)^2}, representing the effective opposition to current flow, while the phase angle \phi = \tan^{-1}(\omega L / R) indicates the inductive lag, where current trails voltage. This complex representation simplifies the analysis of sinusoidal steady-state responses, as sinusoidal signals serve as eigenfunctions of linear time-invariant systems, satisfying the circuit's differential equations in the phasor domain without distortion. The admittance Y for a parallel RL configuration is the sum of the individual admittances: Y = \frac{1}{R} + \frac{1}{j \omega L} = \frac{1}{R} - j \frac{1}{\omega L}. This expression highlights the conductive (real) and susceptive (imaginary) components, facilitating current division analysis. The complex impedance framework is derived from time-domain differential equations, such as L \frac{di}{dt} + R i = v(t), by substituting the phasor form e^{j \omega t} (via Fourier transform) or setting s = j \omega in the Laplace domain, yielding algebraic equations for steady-state sinusoidal inputs.

Phasor Representation

In the phasor representation, sinusoidal steady-state signals in RL circuits are analyzed by modeling voltages and currents as complex numbers, or , which capture both and information. A sinusoid such as v(t) = V_m \cos(\omega t + \theta) is represented by the phasor \mathbf{V} = V_m e^{j\theta}, where j is the and \omega is the . This approach assumes sinusoidal steady-state conditions, where the time derivative operator d/dt in the circuit's differential equations is replaced by j\omega, transforming the time-domain analysis into algebraic operations in the . For a series RL circuit driven by a sinusoidal voltage source \mathbf{V}_s = V_m e^{j\theta}, the current phasor is \mathbf{I} = \mathbf{V}_s / (R + j\omega L), where R is resistance and L is inductance. The phase angle \phi between voltage and current is given by \phi = \tan^{-1}(\omega L / R), with the voltage leading the current by \phi (or equivalently, the current lagging the voltage by \phi). The voltage across the resistor is in phase with the current, while the voltage across the inductor leads the current by 90°. In a parallel RL circuit, the voltage phasor \mathbf{V} is the same across both the and . The through the is \mathbf{I}_R = \mathbf{V} / R, in with \mathbf{V}, while the through the is \mathbf{I}_L = \mathbf{V} / (j\omega L), lagging \mathbf{V} by 90°. The total is the vector sum \mathbf{I} = \mathbf{I}_R + \mathbf{I}_L. Phasor analysis facilitates the construction of Bode plots, which display the magnitude (in decibels) and phase (in degrees) of the frequency response as functions of logarithmic angular frequency \log \omega. For the voltage across the resistor in a series RL circuit, the transfer function exhibits low-pass filter behavior: at low frequencies (\omega \ll R/L), the magnitude is approximately 0 dB with 0° phase shift, while at high frequencies (\omega \gg R/L), the magnitude rolls off at -20 dB per decade with a -90° phase shift. The corner frequency occurs at \omega_c = R/L, where the magnitude is -3 dB. Consider an example with input v_s(t) = 5 \cos(10 t) V, so \mathbf{V}_s = 5 \angle 0^\circ. In a series RL circuit with R = 4 \, \Omega and L = 0.1 \, \text{H} at \omega = 10 \, \text{rad/s}, the current is i(t) \approx 1.213 \cos(10t - 14.04^\circ) \, \text{A}, and the voltage is v_R(t) \approx 4.852 \cos(10t - 14.04^\circ) \, \text{V}.

Series RL Circuit

Voltage and Current Relationships

In a series RL circuit, the source voltage V_s equals the sum of the voltages across the resistor and inductor, V_s = V_R + V_L, where V_R = i R and V_L = L \frac{di}{dt}. This relationship follows from Kirchhoff's voltage law applied to the loop. In the time domain, the current i(t) satisfies the first-order linear differential equation L \frac{di}{dt} + R i(t) = V_s(t), derived by substituting the component voltage expressions into the loop equation. The solution for i(t) depends on the form of V_s(t) and initial conditions; for instance, with a constant DC source V_s(t) = V for t \geq 0 and zero initial current, the general solution is i(t) = \frac{V}{R} (1 - e^{-(R/L)t}). In the for sinusoidal steady-state analysis, the voltages relate as \mathbf{V}_R(\omega) = \mathbf{I}(\omega) R and \mathbf{V}_L(\omega) = \mathbf{I}(\omega) j \omega L, with the source given by \mathbf{I}(\omega) = \frac{\mathbf{V}_s(\omega)}{R + j \omega L}. The denominator represents the complex impedance Z = R + j \omega L. The power factor of the series RL circuit is \cos \phi = \frac{R}{|Z|}, where \phi is the phase angle by which the lags the source voltage, and |Z| = \sqrt{R^2 + (\omega L)^2}. This quantifies the fraction of apparent power that is real power. For a numerical example, consider R = 100 \, \Omega, L = 0.1 \, \text{H}, and V_s(t) = 10 \sin(100 t) \, \text{V} (\omega = 100 \, \text{rad/s}). The impedance magnitude is |Z| = \sqrt{100^2 + (100 \cdot 0.1)^2} = \sqrt{10100} \approx 100.5 \, \Omega, so the peak is I = \frac{10}{100.5} \approx 0.0995 \, \text{A}. The phase angle is \phi = \tan^{-1} \left( \frac{\omega L}{R} \right) = \tan^{-1}(0.1) \approx 5.71^\circ, with lagging voltage; the power factor is \cos \phi \approx 0.995. The peak voltage across the is V_R = I R \approx 9.95 \, \text{V}, and across the is V_L = I \omega L \approx 0.995 \, \text{V}.

Transfer Function and Poles

The transfer function of a series RL circuit in the Laplace domain is derived by applying the Laplace transform to the circuit equations, treating the input as the source voltage V_{in}(s) and considering the output as the voltage across either the resistor or the inductor. For the voltage across the inductor V_L(s), the transfer function is H(s) = \frac{V_L(s)}{V_{in}(s)} = \frac{s}{s + \frac{R}{L}}. For the voltage across the resistor V_R(s), it is H(s) = \frac{V_R(s)}{V_{in}(s)} = \frac{R}{sL + R}. These expressions arise from the s-domain equivalent circuit, where the inductor impedance is sL and the resistor is R, leading to the total impedance Z(s) = R + sL. The series RL circuit exhibits a single at s = -\frac{R}{L}, which is a negative real value assuming positive R and L, indicating a first-order . There are no zeros in the when the output is taken across the , while the inductor output introduces a zero at s = 0. The pole location determines the system's of decay, with the magnitude \frac{R}{L} governing the exponential rate in the . The zero-input response, or natural response, of the circuit is given by i_{zi}(t) = i(0) e^{-\frac{R}{L} t} for t \geq 0, obtained from solving the derived via Laplace methods with zero input. This represents the transient decay due to initial conditions without external forcing. The , corresponding to a unit voltage input, for the output is h(t) = \frac{1}{L} e^{-\frac{R}{L} t} u(t), where u(t) is the step function; this is the of the \frac{1}{sL + R}. Stability is ensured by the pole's position in the left half of the s-plane, where the negative real part guarantees that bounded inputs produce bounded outputs, with the time constant \tau = \frac{L}{R} derived from the pole magnitude. The frequency response, evaluated at s = j\omega, for the resistor output is H(j\omega) = \frac{1}{1 + j \omega \tau}, characterizing the circuit as a low-pass filter with cutoff angular frequency \omega_c = \frac{R}{L}.

Parallel RL Circuit

Admittance and Current Division

In a parallel RL circuit, the resistor R and inductor L share the same voltage V across their terminals. The total input current I_s divides between the branches according to Kirchhoff's current law, such that I_s = I_R + I_L, where I_R = V / R is the resistive branch current and I_L = \frac{1}{L} \int V \, dt is the inductive branch current in the time domain. This topology contrasts with the series RL configuration, where currents are identical but voltages add. In the frequency domain, analysis employs phasors and admittances for parallel elements. The total admittance Y is the sum of the branch admittances: Y = \frac{1}{Z} = G + \frac{1}{j \omega L}, where G = 1/R is the conductance and $1/(j \omega L) is the inductive susceptance (with negative imaginary part). The equivalent impedance follows as the reciprocal: Z_\text{eq} = R \parallel j \omega L = \frac{R \cdot j \omega L}{R + j \omega L}. This yields a magnitude |Z_\text{eq}| = \left( \frac{1}{R^2} + \frac{1}{(\omega L)^2} \right)^{-1/2}. Current division in the phasor domain allocates I_s based on branch impedances. The resistive current is I_R = I_s \cdot \frac{j \omega L}{R + j \omega L}, while the inductive current is I_L = I_s \cdot \frac{R}{R + j \omega L}. Equivalently, using admittances, I_R = I_s \cdot G / Y and I_L = I_s \cdot (1/(j \omega L)) / Y. The phase of I_R aligns with the voltage (in phase), whereas I_L lags the voltage—and thus I_R—by 90°, with the lag approaching exactly 90° at higher frequencies where resistive effects diminish. In the time domain, consider a unit step current input I_s u(t) applied at t=0 to an initially unenergized parallel RL circuit. The voltage v(t) across the branches satisfies the differential equation derived from KCL and the inductor relation: I_s = v/R + (1/L) \int v \, dt. Solving yields v(t) = I_s R e^{-t/\tau}, \quad t \geq 0, where \tau = L/R is the time constant. At t=0^+, v(0^+)=I_s R due to the inductor acting as an open circuit; it decays exponentially to the steady-state v(\infty) = 0. Frequency-dependent behavior of Z_\text{eq} highlights the inductor's role. At low frequencies (\omega \to 0), |j \omega L| \to 0, so the inductor shorts the circuit and Z_\text{eq} \to 0. At high frequencies (\omega \to \infty), |j \omega L| \to \infty, so the inductor opens and Z_\text{eq} \to R. This duality underscores the parallel RL circuit's utility in filters and timing applications.

Transfer Function Characteristics

The transfer function for a parallel RL circuit is defined as the ratio of the Laplace transform of the output voltage V(s) across the parallel combination to the Laplace transform of the input current I_s(s) supplied to the parallel branches, assuming zero initial conditions for the zero-state response. Using the s-domain impedances of the resistor (R) and inductor (sL), the equivalent impedance is the parallel combination Z(s) = \frac{R \cdot (sL)}{R + sL}, yielding the transfer function H(s) = \frac{V(s)}{I_s(s)} = \frac{s L R}{s L + R}. This can be rewritten in normalized form as H(s) = R \cdot \frac{s \tau}{s \tau + 1}, where the time constant \tau = \frac{L}{R} represents the ratio of stored magnetic energy to dissipated power. The transfer function exhibits a single zero at s = 0 and a single pole at s = -\frac{1}{\tau} = -\frac{R}{L}, confirming its first-order nature and high-pass filtering behavior, as the zero at the origin blocks DC components while the left-half-plane pole ensures stability. The pole location indicates exponential decay in the time domain with rate \frac{R}{L}, guaranteeing bounded responses for bounded inputs in stable operation. The impulse response, corresponding to a unit current impulse input, is the inverse Laplace transform of H(s), resulting in h(t) = R \delta(t) - \frac{R^2}{L} e^{-(R/L) t} u(t) for t \geq 0, where \delta(t) is the Dirac delta function and u(t) is the unit step function; this describes the voltage across the parallel branches following the impulse, with the delta term representing the instantaneous response.[](https://math.stackexchange.com/questions/ 357391/inverse-laplace-transform-of-s-s1) For the zero-state response without initial inductor current, the output voltage is obtained by convolving the input current with this impulse response, providing the complete transient solution via Duhamel's integral. In the frequency domain, substituting s = j\omega yields the magnitude |H(j\omega)| = R \cdot \frac{\omega \tau}{\sqrt{1 + (\omega \tau)^2}}, which approaches 0 at low frequencies (\omega \to 0) and R at high frequencies (\omega \to \infty), characteristic of a high-pass filter with corner frequency \omega_c = 1/\tau. The phase response is \phi(\omega) = \arctan(1/(\omega \tau)), providing a phase lead that decreases from 90° at low frequencies to 0° at high frequencies, reflecting the transition from inductive dominance (current lags voltage) at low speeds to resistive behavior at higher speeds. In contrast to the series RL circuit, which functions as a for the voltage across the , the parallel configuration yields a high-pass response for the voltage across the branches to a input, emphasizing its utility in applications requiring of low-frequency signals.

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    this gives a explicit solution of the circuit. • these equations are identical to those for a linear static circuit. (except instead of real numbers we have ...
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    RL Circuit Transfer Function Time Constant RL Circuit as Filter
    May 27, 2024 · A transfer function is used to analyze an RL circuit. It is defined as the ratio of the output to the input of the system in the Laplace domain.Transfer Function of Series RL... · Time Constant in RL Circuit · RL Circuit as Filter