FACTOID # 53: If you thought Antarctica was inhospitable, think again - its land area is only ninety-eight percent ice. Reassuringly, the other 2% is categorised as "barren rock".
 
 Home   Encyclopedia   Statistics   Countries A-Z   Flags   Maps   Education   Forum   FAQ   About 
 
WHAT'S NEW
RECENT ARTICLES
More Recent Articles »
 

FACTS & STATISTICS    Simple view

  1. Select countries to view: (hold down Control key and click to select several)

     

     

    Compare:

     

     
  1. Select fact or statistic: (* = graphable)

     

     

     

  2. (OPTIONAL) Compare to statistic: (both need to be graphable)

     

     

     

  3. View result as:

     

       
(OR) SEARCH ALL encyclopedia, stats & forums:   

Encyclopedia > RL circuit
This article does not cite any references or sources.
Please help improve this article by adding citations to reliable sources. (help, get involved!)
Unverifiable material may be challenged and removed.
This article has been tagged since December 2006.

A resistor-inductor circuit (RL circuit), or RL filter or RL network, is one of the simplest analogue infinite impulse response electronic filters. It consists of a resistor and an inductor, either in series or in parallel, driven by a voltage source. An analog filter handles analog stimuli (e. ... IIR (infinite impulse response) is a property of signal processing systems. ... Television signal splitter consisting of a hi-pass and a low-pass filter. ... Resistor symbols (non-European) Resistor symbols (Europe, IEC) Axial-lead resistors on tape. ... An inductor is a passive electrical device employed in electrical circuits for its property of inductance. ... This article or section does not adequately cite its references or sources. ... This article or section does not adequately cite its references or sources. ... It has been suggested that this article or section be merged with Current source. ...

Contents

Introduction

The fundamental passive linear circuit elements are the resistor (R), capacitor (C) and inductor (L). These circuit elements can be combined to form an electrical circuit in four distinct ways: the RC circuit, the RL circuit, the LC circuit and the RLC circuit with the abbreviations indicating which components are used. These circuits exhibit important types of behaviour that are fundamental to analogue electronics. In particular, they are able to act as passive filters. This article considers the RL circuit in both series and parallel as shown in the diagrams. A passive component is an electronic component that does not require a source of energy to perform its intended function. ... The word linear comes from the Latin word linearis, which means created by lines. ... Resistor symbols (non-European) Resistor symbols (Europe, IEC) Axial-lead resistors on tape. ... Capacitors: SMD ceramic at top left; SMD tantalum at bottom left; through-hole tantalum at top right; through-hole electrolytic at bottom right. ... An inductor is a passive electrical device employed in electrical circuits for its property of inductance. ... An electrical network or electrical circuit is an interconnection of analog electrical elements such as resistors, inductors, capacitors, diodes, switches and transistors. ... resistor-capacitor circuit (RC circuit), or RC filter or RC network, is one of the simplest analogue electronic filters. ... An LC circuit consists of an inductor, represented by the letter L, and a capacitor, represented by the letter C. When connected together, an electrical current can alternate between them at an angular frequency of where L is the inductance in henries, and C is the capacitance in farads. ... An RLC circuit (also known as a resonant circuit or a tuned circuit) is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. ... Analog electronics are those electronic systems with a continuously variable signal. ... Television signal splitter consisting of a hi-pass and a low-pass filter. ... This article or section does not adequately cite its references or sources. ... This article or section does not adequately cite its references or sources. ...


In practice, however, capacitors (and RC circuits) are usually preferred to inductors since they can be more easily manufactured and are generally physically smaller, particularly for higher values of components.

This article relies on knowledge of the complex impedance representation of inductors and on knowledge of the frequency domain representation of signals.

Electrical impedance, or simply impedance, is a measure of opposition to a sinusoidal alternating electric current. ... An inductor is a passive electrical device employed in electrical circuits for its property of inductance. ... Frequency domain is a term used to describe the analysis of mathematical functions with respect to frequency. ...

Complex Impedance

The complex impedance ZL (in ohms) of an inductor with inductance L (in henries) is Electrical impedance or simply impedance is a measure of opposition to a sinusoidal electric current. ... A multimeter can be used to measure resistance in ohms. ... The henry (symbol H) is the SI unit of inductance. ...

Z_L = Ls

The complex frequency s is a complex number, In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...

s = sigma + j omega

where

j2 = − 1
  • omega is the angular frequency (in radians per second).

In mathematics, the imaginary unit (or sometimes the Latin or the Greek iota, see below) allows the real number system to be extended to the complex number system . ... A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value. ... Angular frequency is a measure of how fast an object is rotating In physics (specifically mechanics and electrical engineering), angular frequency ω (also called angular speed) is a scalar measure of rotation rate. ...

Eigenfunctions

The complex-valued eigenfunctions of ANY linear time-invariant (LTI) system are of the following forms: In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... The word linear comes from the Latin word linearis, which means created by lines. ... A time-invariant system is one whose output does not depend explicitly on time. ...

V(t) = mathbf{A}e^{st} = mathbf{A}e^{(sigma + j omega) t} , or letting mathbf{A} = A e^{j phi} and rewriting; = A e^{j phi}e^{(sigma + j omega) t} , and collecting terms is = A e^{sigma t}e^{j ( omega t + phi )}

From Euler's formula, the real-part of these eigenfunctions are exponentially-decaying sinusoids: Eulers formula, named after Leonhard Euler, is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. ...

v(t) = Re left { V(t) right } = A e^{sigma t} cos(omega t + phi )

Sinusoidal Steady State

Sinusoidal steady state is a special case in which the input voltage consists of a pure sinusoid (with no exponential decay). As a result,

sigma = 0

and the evaluation of s becomes

s = j omega

Series circuit

Series RL circuit
Series RL circuit

By viewing the circuit as a voltage divider, we see that the voltage across the inductor is: Image File history File links Series-RL.png Series RL circuit. ... Image File history File links Series-RL.png Series RL circuit. ... This article or section does not adequately cite its references or sources. ... In electronics, a voltage divider is a simple device designed to create a voltage (Vout) which is proportional to another voltage (Vin). ... International safety symbol Caution, risk of electric shock (ISO 3864), colloquially known as high voltage symbol. ...

V_L(s) = frac{Ls}{R+Ls}V_{in}(s)

and the voltage across the resistor is:

V_R(s) = frac{R}{R+Ls}V_{in}(s).

Transfer functions

The transfer function for the inductor is A transfer function is a mathematical representation of the relation between the input and output of a linear time-invariant system. ...

H_L(s) = { V_L(s) over V_{in}(s) } = { Ls over R + Ls } = G_L e^{j phi_L}

Similarly, the transfer function for the resistor is

H_R(s) = { V_R(s) over V_{in}(s) } = { R over R + Ls } = G_R e^{j phi_R}

Poles and zeros

Both transfer functions have a single pole located at

s = - {R over L }

In addition, the transfer function for the inductor has a zero located at the origin. In complex analysis, a zero of a holomorphic function f is a complex number a such that f(a) = 0. ... In mathematics, the origin of a coordinate system is the point where the axes of the system intersect. ...


Gain and phase angle

The gains across the two components are found by taking the magnitudes of the above expressions:

G_L = | H_L(s) | = left|frac{V_L(s)}{V_{in}(s)}right| = frac{omega L}{sqrt{R^2 + left(omega Lright)^2}}

and

G_R = | H_R(s) | = left|frac{V_R(s)}{V_{in}(s)}right| = frac{R}{sqrt{R^2 + left(omega Lright)^2}},

and the phase angles are: This article is about a portion of a periodic process. ...

phi_L = angle H_L(s) = tan^{-1}left(frac{R}{omega L}right)

and

phi_R = angle H_R(s) = tan^{-1}left(-frac{omega L}{R}right).

Phasor notation

These expressions together may be substituted into the usual expression for the phasor representing the output: See wikibooks book on Phasors A phasor is a constant complex number representing the complex amplitude (magnitude and phase) of a sinusoidal function of time. ...

V_L = G_{L}V_{in} e^{j phi_L}
V_R = G_{R}V_{in}e^{j phi_R}.

Current

The current in the circuit is the same everywhere since the circuit is series:

I(s) = frac{V_{in}(s)}{R + Ls}.

Impulse Response

The impulse response for each voltage is the inverse Laplace transform of the corresponding transfer function. It represents the response of the circuit to an input voltage consisting of an impulse or delta function. The Impulse response from a simple audio system. ... In mathematics, the Laplace transform is a technique for analyzing linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems. ... The Dirac delta function, introduced by Paul Dirac, can be informally thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere, and a total integral of one. ...


The impulse response for the inductor voltage is

h_L(t) = - { R over L} e^{-tR / L} u(t) = - { 1 over tau} e^{-t / tau} u(t)

where u(t) is the Heaviside step function and The Heaviside step function, using the half-maximum convention The Heaviside step function, sometimes called the unit step function and named in honor of Oliver Heaviside, is a discontinuous function whose value is zero for negative argument and one for positive argument: The function is used in the mathematics of...

tau = { L over R}

is the time constant. In physics and engineering, the time constant usually denoted by the Greek letter , (tau), characterizes the frequency response of a first-order, linear time-invariant (LTI) system. ...


Similarly, the impulse response for the resistor voltage is

h_R(t) = {R over L} e^{-tR / L} u(t) = { 1 over tau} e^{-t / tau} u(t)

Zero input response (ZIR)

The Zero input response, also called the natural response, of an RL circuit describes the behavior of the circuit after it has reached constant voltages and currents and is disconnected from any power source. It is called the zero-input response because it requires no input.


The ZIR of an RL circuit is:

i(t) = i(0)e^{-(R/L) t} = i(0)e^{-t/ tau} ! .

Frequency domain considerations

These are frequency domain expressions. Analysis of them will show which frequencies the circuits (or filters) pass and reject. This analysis rests on a consideration of what happens to these gains as the frequency becomes very large and very small. Frequency domain is a term used to describe the analysis of mathematical functions with respect to frequency. ...


As omega to infty:

G_L to 1
G_R to 0.

As omega to 0:

G_L to 0
G_R to 1.

This shows that, if the output is taken across the inductor, high frequencies are passed and low frequencies are attenuated (rejected). Thus, the circuit behaves as a high-pass filter. If, though, the output is taken across the resistor, high frequencies are rejected and low frequencies are passed. In this configuration, the circuit behaves as a low-pass filter. Compare this with the behaviour of the resistor output in an RC circuit, where the reverse is the case. A high-pass filter is a filter that passes high frequencies well, but attenuates (or reduces) frequencies lower than the cutoff frequency. ... A low-pass filter is a filter that passes low frequencies but attenuates (or reduces) frequencies higher than the cutoff frequency. ... resistor-capacitor circuit (RC circuit), or RC filter or RC network, is one of the simplest analogue electronic filters. ...


The range of frequencies that the filter passes is called its bandwidth. The point at which the filter attenuates the signal to half its unfiltered power is termed its cutoff frequency. This requires that the gain of the circuit be reduced to This article does not cite any references or sources. ... A bode plot of the Butterworth filters frequency response, with corner frequency labeled. ...

G_L = G_R = frac{1}{sqrt{2}}.

Solving the above equation yields

omega_{c} = frac{R}{L}rad/s

or Some common angles, measured in radians. ... Look up second in Wiktionary, the free dictionary. ...

f_c = frac{R}{2pi L}Hz

which is the frequency that the filter will attenuate to half its original power. MHZ redirects here. ...


Clearly, the phases also depend on frequency, although this effect is less interesting generally than the gain variations.


As omega to 0:

phi_L to 90^{circ} = pi/2^{c}
phi_R to 0.

As omega to infty:

phi_L to 0
phi_R to -90^{circ} = -pi/2^{c}

So at DC (0 Hz), the resistor voltage is in phase with the signal voltage while the inductor voltage leads it by 90°. As frequency increases, the resistor voltage comes to have a 90° lag relative to the signal and the inductor voltage comes to be in-phase with the signal. Direct current (DC or continuous current) is the continuous flow of electricity through a conductor such as a wire from high to low potential. ... MHZ redirects here. ...


Time domain considerations

This section relies on knowledge of e, the natural logarithmic constant.

The most straightforward way to derive the time domain behaviour is to use the Laplace transforms of the expressions for VL and VR given above. This effectively transforms jomega to s. Assuming a step input (i.e. Vin = 0 before t = 0 and then Vin = V afterwards): The title given to this article is incorrect due to technical limitations. ... In mathematics, the Laplace transform is a technique for analyzing linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems. ... The Heaviside step function, using the half-maximum convention The Heaviside step function, sometimes called the unit step function and named in honor of Oliver Heaviside, is a discontinuous function whose value is zero for negative argument and one for positive argument: The function is used in the mathematics of...

V_{in}(s) = Vfrac{1}{s}
V_L(s) = Vfrac{sL}{R + sL}frac{1}{s}

and

V_R(s) = Vfrac{R}{R + sL}frac{1}{s}.
Inductor voltage step-response.
Inductor voltage step-response.
Resistor voltage step-response.
Resistor voltage step-response.

Partial fractions expansions and the inverse Laplace transform yield: Image File history File links Series_RC_resistor_voltage. ... Image File history File links Series_RC_resistor_voltage. ... Image File history File links Series_RC_capacitor_voltage. ... Image File history File links Series_RC_capacitor_voltage. ... In algebra, the partial fraction decomposition of a rational function expresses the function as a sum of fractions, in each term of which, the denominator is an irreducible (i. ... In mathematics, the Laplace transform is a technique for analyzing linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems. ...

,!V_L(t) = Ve^{-tR/L}
,!V_R(t) = Vleft(1 - e^{-tR/L}right).

Thus, the voltage across the inductor tends towards 0 as time passes, while the voltage across the resistor tends towards V, as shown in the figures. This is in keeping with the intuitive point that the inductor will only have a voltage across as long as the current in the circuit is changing — as the circuit reaches its steady-state, there is no further current change and ultimately no inductor voltage.


These equations show that a series RL circuit has a time constant, usually denoted τ = L / R being the time it takes the voltage across the component to either fall (across L) or rise (across R) to within 1 / e of its final value. That is, τ is the time it takes VL to reach V(1 / e) and VR to reach V(1 − 1 / e).


The rate of change is a fractional left(1 - frac{1}{e}right) per τ. Thus, in going from t = Nτ to t = (N + 1)τ, the votage will have moved about 63% of the way from its level at t = Nτ toward its final value. So the voltage across L will have dropped to about 37% after τ, and essentially to zero (0.7%) after about 5τ. Kirchhoff's voltage law implies that the voltage across the resistor will rise at the same rate. When the voltage source is then replaced with a short-circuit, the voltage across R drops exponentially with t from V towards 0. R will be discharged to about 37% after τ, and essentially fully discharged (0.7%) after about 5τ. Note that the current, I, in the circuit behaves as the voltage across R does, via Ohm's Law. Kirchhoffs circuit laws are a pair of laws that deal with the conservation of charge and energy in electrical circuits, and were first described in 1845 by Gustav Kirchhoff. ... A voltage source, V, drives an electric current, I , through resistor, R, the three quantities obeying Ohms law: V = IR Ohms law states that, in an electrical circuit, the current passing through a conductor from one terminal point on the conductor to another terminal point on the conductor...


The delay in the rise/fall time of the circuit is in this case caused by the back-EMF from the inductor which, as the current flowing through it tries to change, prevents the current (and hence the voltage across the resistor) from rising or falling much faster than the time-constant of the circuit. Since all wires have some self-inductance and resistance, all circuits have a time constant. As a result, when the power supply is switched on, the current does not instantaneously reach its steady-state value, V / R. The rise instead takes several time-constants to complete. If this were not the case, and the current were to reach steady-state immediately, extremely strong inductive electric fields would be generated by the sharp change in the magnetic field — this would lead to breakdown of the air in the circuit and electric arcing, probably damaging components (and users). Back Emf, or back torque, is an electromotive force that occurs in Electric motors and some generators where there is relative motion between the armature of the motor and the external magnetic field. ... Inductance (or electric inductance) is a measure of the amount of magnetic flux produced for a given electric current. ... Electricity arcs between the power rail and electrical pickup shoe on a London Underground train An electric arc is an electrical breakdown of a gas which produces an ongoing plasma discharge, similar to the instant spark, resulting from a current flowing through normally nonconductive media such as air. ...


These results may also be derived by solving the differential equation describing the circuit: A simulation of airflow into a duct using the Navier-Stokes equations A differential equation is a mathematical equation for an unknown function of one or several variables which relates the values of the function itself and of its derivatives of various orders. ...

V_{in} = IR + Lfrac{dI}{dt},

and

,!V_R = V_{in} - V_L.

The first equation is solved by using an integrating factor and yields the current which must be differentiated to give VL; the second equation is straightforward. The solutions are exactly the same as those obtained via Laplace transforms. In mathematics, one solves certain ordinary differential equations by using an integrating factor. ...


Parallel circuit

Please help improve this article by expanding this section.
Further information might be found on the talk page or at requests for expansion.
Please remove this message once the section has been expanded.
This section has been tagged since January 2007.
Parallel RL circuit
Parallel RL circuit

The parallel RL circuit is generally of less interest than the series circuit unless fed by a current source. This is largely because the output voltage Vout is equal to the input voltage Vin — as a result, this circuit does not act as a filter for a voltage input signal. Image File history File links Parallel-RL.png Parallel RL circuit. ... Image File history File links Parallel-RL.png Parallel RL circuit. ... This article or section does not adequately cite its references or sources. ...


With complex impedances:

I_R = frac{V_{in}}{R}

and

,!I_L = frac{V_{in}}{jomega L} = -frac{jV_{in}}{omega L}.

This shows that the inductor lags the resistor (and source) current by 90°.


The parallel circuit is seen on the output of many amplifier circuits, and is used to isolate the amplifier from capacitive loading effects at high frequencies. Because of the phase shift introduced by capacitance, some amplifiers become unstable at very high frequencies, and tend to oscillate. This affects sound quality and component life (especially the transistors), and is to be avoided.


See also


  Results from FactBites:
 
RL circuit - Wikipedia, the free encyclopedia (1303 words)
These may be combined in four important combinations: the RC circuit, the RL circuit, the LC circuit and the RLC circuit with the abbreviations indicating which components are used.
It represents the response of the circuit to an input voltage consisting of an impulse or delta function.
The delay in the rise/fall time of the circuit is in this case caused by the back-EMF from the inductor which, as the current flowing through it tries to change, prevents the current (and hence the voltage across the resistor) from rising or falling much faster than the time-constant of the circuit.
  More results at FactBites »


 

COMMENTARY     


Share your thoughts, questions and commentary here
Your name
Your comments
Please enter the 5-letter protection code

Want to know more?
Search encyclopedia, statistics and forums:
 


Lesson Plans | Student Area | Student FAQ | Reviews | Press Releases |  Feeds | Contact
The Wikipedia article included on this page is licensed under the GFDL.
Images may be subject to relevant owners' copyright.
All other elements are (c) copyright NationMaster.com 2003-5. All Rights Reserved.
Usage implies agreement with terms.