A resistor-inductor ambit (RL circuit), or RL clarify or RL network, is one of the simplest alternation absolute actuation acknowledgment cyberbanking filters. It consists of a resistor and an inductor, either in alternation or in parallel, apprenticed by a voltage source.
Monday, February 27, 2012
Introduction
The axiological acquiescent beeline ambit elements are the resistor (R), capacitor (C) and inductor (L). These ambit elements can be accumulated to anatomy an electrical ambit in four audible ways: the RC circuit, the RL circuit, the LC ambit and the RLC ambit with the abbreviations advertence which apparatus are used. These circuits display important types of behaviour that are axiological to alternation electronics. In particular, they are able to act as acquiescent filters. This commodity considers the RL ambit in both alternation and alongside as apparent in the diagrams.
In practice, however, capacitors (and RC circuits) are usually adopted to inductors back they can be added calmly bogus and are about physically smaller, decidedly for college ethics of components.
This commodity relies on ability of the circuitous impedance representation of inductors and on ability of the abundance area representation of signals.
In practice, however, capacitors (and RC circuits) are usually adopted to inductors back they can be added calmly bogus and are about physically smaller, decidedly for college ethics of components.
This commodity relies on ability of the circuitous impedance representation of inductors and on ability of the abundance area representation of signals.
Complex Impedance
The circuitous impedance ZL (in ohms) of an inductor with inductance L (in henries) is
Z_L \ = \ Ls
The circuitous abundance s is a circuitous number,
s \ = \ \sigma + j \omega
where
j represents the abstract unit:
j2 = − 1
\sigma \ is the exponential adulteration connected (in radians per second), and
\omega \ is the angular abundance (in radians per second).
edit Eigenfunctions
The complex-valued eigenfunctions of ANY beeline time-invariant (LTI) arrangement are of the afterward forms:
V(t) \ = \ \mathbf{A}e^{st} \ = \ \mathbf{A}e^{(\sigma + j \omega) t} \ , or absolution \mathbf{A} \ = \ A e^{j \phi} and rewriting; \ = \ A e^{j \phi}e^{(\sigma + j \omega) t} , and accession agreement is \ = \ A e^{\sigma t}e^{j ( \omega t + \phi )}
From Euler's formula, the real-part of these eigenfunctions are exponentially-decaying sinusoids:
v(t) \ = \ \mathrm{Re} \left \{ V(t) \right \} \ = \ A e^{\sigma t} \cos(\omega t + \phi )
edit Sinusoidal Abiding State
Sinusoidal abiding accompaniment is a appropriate case in which the ascribe voltage consists of a authentic sinusoid (with no exponential decay). As a result,
\sigma \ = \ 0
and the appraisal of s becomes
s \ = \ j \omega
edit Series circuit
Z_L \ = \ Ls
The circuitous abundance s is a circuitous number,
s \ = \ \sigma + j \omega
where
j represents the abstract unit:
j2 = − 1
\sigma \ is the exponential adulteration connected (in radians per second), and
\omega \ is the angular abundance (in radians per second).
edit Eigenfunctions
The complex-valued eigenfunctions of ANY beeline time-invariant (LTI) arrangement are of the afterward forms:
V(t) \ = \ \mathbf{A}e^{st} \ = \ \mathbf{A}e^{(\sigma + j \omega) t} \ , or absolution \mathbf{A} \ = \ A e^{j \phi} and rewriting; \ = \ A e^{j \phi}e^{(\sigma + j \omega) t} , and accession agreement is \ = \ A e^{\sigma t}e^{j ( \omega t + \phi )}
From Euler's formula, the real-part of these eigenfunctions are exponentially-decaying sinusoids:
v(t) \ = \ \mathrm{Re} \left \{ V(t) \right \} \ = \ A e^{\sigma t} \cos(\omega t + \phi )
edit Sinusoidal Abiding State
Sinusoidal abiding accompaniment is a appropriate case in which the ascribe voltage consists of a authentic sinusoid (with no exponential decay). As a result,
\sigma \ = \ 0
and the appraisal of s becomes
s \ = \ j \omega
edit Series circuit
Series circuit
By examination the ambit as a voltage divider, we see that the voltage beyond the inductor is:
V_L(s) = \frac{Ls}{R+Ls}V_{in}(s)
and the voltage beyond the resistor is:
V_R(s) = \frac{R}{R+Ls}V_{in}(s).
edit Current
The accepted in the ambit is the aforementioned everywhere back the ambit is in series:
I(s) = \frac{V_{in}(s)}{R + Ls}.
edit Alteration functions
The alteration action for the inductor is
H_L(s) = { V_L(s) \over V_{in}(s) } = { Ls \over R + Ls } = G_L e^{j \phi_L}
Similarly, the alteration action for the resistor is
H_R(s) = { V_R(s) \over V_{in}(s) } = { R \over R + Ls } = G_R e^{j \phi_R}
edit Poles and zeros
Both alteration functions accept a individual pole amid at
s = - {R \over L }
In addition, the alteration action for the inductor has a aught amid at the origin.
edit Accretion and appearance angle
The assets beyond the two apparatus are begin by demography the magnitudes of the aloft expressions:
G_L = | H_L(s) | = \left|\frac{V_L(s)}{V_{in}(s)}\right| = \frac{\omega L}{\sqrt{R^2 + \left(\omega L\right)^2}}
and
G_R = | H_R(s) | = \left|\frac{V_R(s)}{V_{in}(s)}\right| = \frac{R}{\sqrt{R^2 + \left(\omega L\right)^2}},
and the appearance angles are:
\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).
edit Phasor notation
These expressions calm may be commissioned into the accepted announcement for the phasor apery the output:
V_L = G_{L}V_{in} e^{j \phi_L}
V_R = G_{R}V_{in}e^{j \phi_R}.
edit Actuation Response
The actuation acknowledgment for anniversary voltage is the changed Laplace transform of the agnate alteration function. It represents the acknowledgment of the ambit to an ascribe voltage consisting of an actuation or Dirac basin function.
The actuation acknowledgment for the inductor voltage is
h_L(t) = \delta(t)- { R \over L} e^{-tR / L} u(t) = \delta(t) - { 1 \over \tau} e^{-t / \tau} u(t)
where u(t) is the Heaviside footfall action and
\tau = { L \over R}
is the time constant.
Similarly, the actuation acknowledgment for the resistor voltage is
h_R(t) = {R \over L} e^{-tR / L} u(t) = { 1 \over \tau} e^{-t / \tau} u(t)
edit Aught ascribe acknowledgment (ZIR)
The Aught ascribe response, aswell alleged the accustomed response, of an RL ambit describes the behavior of the ambit afterwards it has accomplished connected voltages and currents and is broken from any ability source. It is alleged the zero-input acknowledgment because it requires no input.
The ZIR of an RL ambit is:
i(t) = i(0)e^{-(R/L) t} = i(0)e^{-t/ \tau} \!\ .
edit Abundance area considerations
These are abundance area expressions. Assay of them will appearance which frequencies the circuits (or filters) canyon and reject. This assay rests on a application of what happens to these assets as the abundance becomes actual ample and actual small.
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 achievement is taken beyond the inductor, top frequencies are anesthetized and low frequencies are attenuated (rejected). Thus, the ambit behaves as a high-pass filter. If, though, the achievement is taken beyond the resistor, top frequencies are alone and low frequencies are passed. In this configuration, the ambit behaves as a low-pass filter. Compare this with the behaviour of the resistor achievement in an RC circuit, area the about-face is the case.
The ambit of frequencies that the clarify passes is alleged its bandwidth. The point at which the clarify attenuates the arresting to bisected its unfiltered ability is termed its blow frequency. This requires that the accretion of the ambit be bargain to
G_L = G_R = \frac{1}{\sqrt{2}}.
Solving the aloft blueprint yields
\omega_{c} = \frac{R}{L}rad/s
or
f_c = \frac{R}{2\pi L}Hz
which is the abundance that the clarify will abate to bisected its aboriginal power.
Clearly, the phases aswell depend on frequency, although this aftereffect is beneath absorbing about than the accretion 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 appearance with the arresting voltage while the inductor voltage leads it by 90°. As abundance increases, the resistor voltage comes to accept a 90° lag about to the arresting and the inductor voltage comes to be in-phase with the signal.
edit Time area considerations
This area relies on ability of e, the accustomed logarithmic constant.
The a lot of aboveboard way to acquire the time area behaviour is to use the Laplace transforms of the expressions for VL and VR accustomed above. This finer transforms j\omega \to s. Assuming a footfall ascribe (i.e. Vin = 0 afore t = 0 and again Vin = V afterwards):
V_{in}(s) = V\frac{1}{s}
V_L(s) = V\frac{sL}{R + sL}\frac{1}{s}
and
V_R(s) = V\frac{R}{R + sL}\frac{1}{s}.
Inductor voltage step-response.
Resistor voltage step-response.
Partial fractions expansions and the changed Laplace transform yield:
\,\!V_L(t) = Ve^{-tR/L}
\,\!V_R(t) = V\left(1 - e^{-tR/L}\right).
Thus, the voltage beyond the inductor tends appear 0 as time passes, while the voltage beyond the resistor tends appear V, as apparent in the figures. This is in befitting with the automatic point that the inductor will alone accept a voltage beyond as continued as the accepted in the ambit is alteration — as the ambit alcove its steady-state, there is no added accepted change and ultimately no inductor voltage.
These equations appearance that a alternation RL ambit has a time constant, usually denoted τ = L / R getting the time it takes the voltage beyond the basic to either abatement (across L) or acceleration (across R) to aural 1 / e of its final value. That is, τ is the time it takes VL to ability V(1 / e) and VR to ability V(1 − 1 / e).
The amount of change is a apportioned \left(1 - \frac{1}{e}\right) per τ. Thus, in traveling from t = Nτ to t = (N + 1)τ, the votage will accept confused about 63% of the way from its akin at t = Nτ against its final value. So the voltage beyond L will accept alone to about 37% afterwards τ, and about to aught (0.7%) afterwards about 5τ. Kirchhoff's voltage law implies that the voltage beyond the resistor will acceleration at the aforementioned rate. If the voltage antecedent is again replaced with a short-circuit, the voltage beyond R drops exponentially with t from V appear 0. R will be absolved to about 37% afterwards τ, and about absolutely absolved (0.7%) afterwards about 5τ. Note that the current, I, in the ambit behaves as the voltage beyond R does, via Ohm's Law.
The adjournment in the rise/fall time of the ambit is in this case acquired by the back-EMF from the inductor which, as the accepted abounding through it tries to change, prevents the accepted (and appropriately the voltage beyond the resistor) from ascent or falling abundant faster than the time-constant of the circuit. Back all affairs accept some self-inductance and resistance, all circuits accept a time constant. As a result, if the ability accumulation is switched on, the accepted does not anon ability its steady-state value, V / R. The acceleration instead takes several time-constants to complete. If this were not the case, and the accepted were to ability steady-state immediately, acutely able anterior electric fields would be generated by the aciculate change in the alluring acreage — this would advance to breakdown of the air in the ambit and electric arcing, apparently damaging apparatus (and users).
These after-effects may aswell be acquired by analytic the cogwheel blueprint anecdotic the circuit:
V_{in} = IR + L\frac{dI}{dt},
and
\,\!V_R = V_{in} - V_L.
The aboriginal blueprint is apparent by application an amalgam agency and yields the accepted which have to be differentiated to accord VL; the additional blueprint is straightforward. The solutions are absolutely the aforementioned as those acquired via Laplace transforms.
V_L(s) = \frac{Ls}{R+Ls}V_{in}(s)
and the voltage beyond the resistor is:
V_R(s) = \frac{R}{R+Ls}V_{in}(s).
edit Current
The accepted in the ambit is the aforementioned everywhere back the ambit is in series:
I(s) = \frac{V_{in}(s)}{R + Ls}.
edit Alteration functions
The alteration action for the inductor is
H_L(s) = { V_L(s) \over V_{in}(s) } = { Ls \over R + Ls } = G_L e^{j \phi_L}
Similarly, the alteration action for the resistor is
H_R(s) = { V_R(s) \over V_{in}(s) } = { R \over R + Ls } = G_R e^{j \phi_R}
edit Poles and zeros
Both alteration functions accept a individual pole amid at
s = - {R \over L }
In addition, the alteration action for the inductor has a aught amid at the origin.
edit Accretion and appearance angle
The assets beyond the two apparatus are begin by demography the magnitudes of the aloft expressions:
G_L = | H_L(s) | = \left|\frac{V_L(s)}{V_{in}(s)}\right| = \frac{\omega L}{\sqrt{R^2 + \left(\omega L\right)^2}}
and
G_R = | H_R(s) | = \left|\frac{V_R(s)}{V_{in}(s)}\right| = \frac{R}{\sqrt{R^2 + \left(\omega L\right)^2}},
and the appearance angles are:
\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).
edit Phasor notation
These expressions calm may be commissioned into the accepted announcement for the phasor apery the output:
V_L = G_{L}V_{in} e^{j \phi_L}
V_R = G_{R}V_{in}e^{j \phi_R}.
edit Actuation Response
The actuation acknowledgment for anniversary voltage is the changed Laplace transform of the agnate alteration function. It represents the acknowledgment of the ambit to an ascribe voltage consisting of an actuation or Dirac basin function.
The actuation acknowledgment for the inductor voltage is
h_L(t) = \delta(t)- { R \over L} e^{-tR / L} u(t) = \delta(t) - { 1 \over \tau} e^{-t / \tau} u(t)
where u(t) is the Heaviside footfall action and
\tau = { L \over R}
is the time constant.
Similarly, the actuation acknowledgment for the resistor voltage is
h_R(t) = {R \over L} e^{-tR / L} u(t) = { 1 \over \tau} e^{-t / \tau} u(t)
edit Aught ascribe acknowledgment (ZIR)
The Aught ascribe response, aswell alleged the accustomed response, of an RL ambit describes the behavior of the ambit afterwards it has accomplished connected voltages and currents and is broken from any ability source. It is alleged the zero-input acknowledgment because it requires no input.
The ZIR of an RL ambit is:
i(t) = i(0)e^{-(R/L) t} = i(0)e^{-t/ \tau} \!\ .
edit Abundance area considerations
These are abundance area expressions. Assay of them will appearance which frequencies the circuits (or filters) canyon and reject. This assay rests on a application of what happens to these assets as the abundance becomes actual ample and actual small.
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 achievement is taken beyond the inductor, top frequencies are anesthetized and low frequencies are attenuated (rejected). Thus, the ambit behaves as a high-pass filter. If, though, the achievement is taken beyond the resistor, top frequencies are alone and low frequencies are passed. In this configuration, the ambit behaves as a low-pass filter. Compare this with the behaviour of the resistor achievement in an RC circuit, area the about-face is the case.
The ambit of frequencies that the clarify passes is alleged its bandwidth. The point at which the clarify attenuates the arresting to bisected its unfiltered ability is termed its blow frequency. This requires that the accretion of the ambit be bargain to
G_L = G_R = \frac{1}{\sqrt{2}}.
Solving the aloft blueprint yields
\omega_{c} = \frac{R}{L}rad/s
or
f_c = \frac{R}{2\pi L}Hz
which is the abundance that the clarify will abate to bisected its aboriginal power.
Clearly, the phases aswell depend on frequency, although this aftereffect is beneath absorbing about than the accretion 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 appearance with the arresting voltage while the inductor voltage leads it by 90°. As abundance increases, the resistor voltage comes to accept a 90° lag about to the arresting and the inductor voltage comes to be in-phase with the signal.
edit Time area considerations
This area relies on ability of e, the accustomed logarithmic constant.
The a lot of aboveboard way to acquire the time area behaviour is to use the Laplace transforms of the expressions for VL and VR accustomed above. This finer transforms j\omega \to s. Assuming a footfall ascribe (i.e. Vin = 0 afore t = 0 and again Vin = V afterwards):
V_{in}(s) = V\frac{1}{s}
V_L(s) = V\frac{sL}{R + sL}\frac{1}{s}
and
V_R(s) = V\frac{R}{R + sL}\frac{1}{s}.
Inductor voltage step-response.
Resistor voltage step-response.
Partial fractions expansions and the changed Laplace transform yield:
\,\!V_L(t) = Ve^{-tR/L}
\,\!V_R(t) = V\left(1 - e^{-tR/L}\right).
Thus, the voltage beyond the inductor tends appear 0 as time passes, while the voltage beyond the resistor tends appear V, as apparent in the figures. This is in befitting with the automatic point that the inductor will alone accept a voltage beyond as continued as the accepted in the ambit is alteration — as the ambit alcove its steady-state, there is no added accepted change and ultimately no inductor voltage.
These equations appearance that a alternation RL ambit has a time constant, usually denoted τ = L / R getting the time it takes the voltage beyond the basic to either abatement (across L) or acceleration (across R) to aural 1 / e of its final value. That is, τ is the time it takes VL to ability V(1 / e) and VR to ability V(1 − 1 / e).
The amount of change is a apportioned \left(1 - \frac{1}{e}\right) per τ. Thus, in traveling from t = Nτ to t = (N + 1)τ, the votage will accept confused about 63% of the way from its akin at t = Nτ against its final value. So the voltage beyond L will accept alone to about 37% afterwards τ, and about to aught (0.7%) afterwards about 5τ. Kirchhoff's voltage law implies that the voltage beyond the resistor will acceleration at the aforementioned rate. If the voltage antecedent is again replaced with a short-circuit, the voltage beyond R drops exponentially with t from V appear 0. R will be absolved to about 37% afterwards τ, and about absolutely absolved (0.7%) afterwards about 5τ. Note that the current, I, in the ambit behaves as the voltage beyond R does, via Ohm's Law.
The adjournment in the rise/fall time of the ambit is in this case acquired by the back-EMF from the inductor which, as the accepted abounding through it tries to change, prevents the accepted (and appropriately the voltage beyond the resistor) from ascent or falling abundant faster than the time-constant of the circuit. Back all affairs accept some self-inductance and resistance, all circuits accept a time constant. As a result, if the ability accumulation is switched on, the accepted does not anon ability its steady-state value, V / R. The acceleration instead takes several time-constants to complete. If this were not the case, and the accepted were to ability steady-state immediately, acutely able anterior electric fields would be generated by the aciculate change in the alluring acreage — this would advance to breakdown of the air in the ambit and electric arcing, apparently damaging apparatus (and users).
These after-effects may aswell be acquired by analytic the cogwheel blueprint anecdotic the circuit:
V_{in} = IR + L\frac{dI}{dt},
and
\,\!V_R = V_{in} - V_L.
The aboriginal blueprint is apparent by application an amalgam agency and yields the accepted which have to be differentiated to accord VL; the additional blueprint is straightforward. The solutions are absolutely the aforementioned as those acquired via Laplace transforms.
Parallel circuit
The alongside RL ambit is about of beneath absorption than the alternation ambit unless fed by a accepted source. This is abundantly because the achievement voltage Vout is according to the ascribe voltage Vin — as a result, this ambit does not act as a clarify for a voltage ascribe signal.
With circuitous impedances:
I_R = \frac{V_{in}}{R}
and
\,\!I_L = \frac{V_{in}}{j\omega L} = -\frac{jV_{in}}{\omega L}.
This shows that the inductor lags the resistor (and source) accepted by 90°.
The alongside ambit is apparent on the achievement of abounding amplifier circuits, and is acclimated to abstract the amplifier from capacitive loading furnishings at top frequencies. Because of the appearance about-face alien by capacitance, some amplifiers become ambiguous at actual top frequencies, and tend to oscillate. This affects complete superior and basic activity (especially the transistors), and is to be avoided.
With circuitous impedances:
I_R = \frac{V_{in}}{R}
and
\,\!I_L = \frac{V_{in}}{j\omega L} = -\frac{jV_{in}}{\omega L}.
This shows that the inductor lags the resistor (and source) accepted by 90°.
The alongside ambit is apparent on the achievement of abounding amplifier circuits, and is acclimated to abstract the amplifier from capacitive loading furnishings at top frequencies. Because of the appearance about-face alien by capacitance, some amplifiers become ambiguous at actual top frequencies, and tend to oscillate. This affects complete superior and basic activity (especially the transistors), and is to be avoided.
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