Movatterモバイル変換

[0]ホーム
URL:

Jump to content
WikipediaThe Free Encyclopedia
Search

LC circuit

From Wikipedia, the free encyclopedia
(Redirected fromTank circuit)
Electrical "resonator" circuit, consisting of inductive and capacitive elements with no resistance
This articleneeds additional citations forverification. Please helpimprove this article byadding citations to reliable sources. Unsourced material may be challenged and removed.
Find sources: "LC circuit" – news ·newspapers ·books ·scholar ·JSTOR(March 2009) (Learn how and when to remove this message)
Linear analog
electronic filters

AnLC circuit, also called aresonant circuit,tank circuit, ortuned circuit, is anelectric circuit consisting of aninductor, represented by the letter L, and acapacitor, represented by the letter C, connected together. The circuit can act as an electricalresonator, an electrical analogue of atuning fork, storing energy oscillating at the circuit'sresonant frequency.

LC circuits are used either for generating signals at a particular frequency, or picking out a signal at a particular frequency from a more complex signal; this function is called abandpass filter. They are key components in many electronic devices, particularly radio equipment, used in circuits such asoscillators,filters,tuners andfrequency mixers.

An LC circuit is an idealized model since it assumes there is no dissipation of energy due toresistance. Any practical implementation of an LC circuit will always include loss resulting from small but non-zero resistance within the components and connecting wires. The purpose of an LC circuit is usually to oscillate with minimaldamping, so the resistance is made as low as possible. While no practical circuit is without losses, it is nonetheless instructive to study this ideal form of the circuit to gain understanding and physical intuition. For a circuit model incorporating resistance, seeRLC circuit.

Terminology

[edit]

The two-element LC circuit described above is the simplest type ofinductor-capacitor network (orLC network). It is also referred to as asecond order LC circuit[1][2] to distinguish it from more complicated (higher order) LC networks with more inductors and capacitors. Such LC networks with more than two reactances may have more than oneresonant frequency.

The order of the network is the order of therational function describing the network in thecomplex frequency variables. Generally, the order is equal to the number of L and C elements in the circuit and in any event cannot exceed this number.

Operation

[edit]
Animated diagram showing the operation of atuned circuit (LC circuit). The capacitor C stores energy in itselectric fieldE and the inductor L stores energy in itsmagnetic fieldB(green). The animation shows the circuit at progressive points in the oscillation. The oscillations are slowed down; in an actual tuned circuit the charge may oscillate back and forth thousands to billions of times per second.

An LC circuit, oscillating at its naturalresonant frequency, can storeelectrical energy. See the animation. A capacitor stores energy in theelectric field (E) between its plates, depending on thevoltage across it, and an inductor stores energy in itsmagnetic field (B), depending on thecurrent through it.

If an inductor is connected across a charged capacitor, the voltage across the capacitor will drive a current through the inductor, building up a magnetic field around it. The voltage across the capacitor falls to zero as the charge is used up by the current flow. At this point, the energy stored in the coil's magnetic field induces a voltage across the coil, because inductors oppose changes in current. This induced voltage causes a current to begin to recharge the capacitor with a voltage of opposite polarity to its original charge. Due toFaraday's law, theEMF which drives the current is caused by a decrease in the magnetic field, thus the energy required to charge the capacitor is extracted from the magnetic field. When the magnetic field is completely dissipated the current will stop and the charge will again be stored in the capacitor, with the opposite polarity as before. Then the cycle will begin again, with the current flowing in the opposite direction through the inductor.

The charge flows back and forth between the plates of the capacitor, through the inductor. The energy oscillates back and forth between the capacitor and the inductor until (if not replenished from an external circuit) internalresistance makes the oscillations die out. The tuned circuit's action, known mathematically as aharmonic oscillator, is similar to apendulum swinging back and forth, or water sloshing back and forth in a tank; for this reason the circuit is also called atank circuit.[3] Thenatural frequency (that is, the frequency at which it will oscillate when isolated from any other system, as described above) is determined by the capacitance and inductance values. In most applications the tuned circuit is part of a larger circuit which appliesalternating current to it, driving continuous oscillations. If the frequency of the applied current is the circuit's natural resonant frequency (natural frequencyf0{\displaystyle f_{0}\,} below),resonance will occur, and a small driving current can excite large amplitude oscillating voltages and currents. In typical tuned circuits in electronic equipment the oscillations are very fast, from thousands to billions of times per second.[citation needed]

Resonance effect

[edit]

Resonance occurs when an LC circuit is driven from an external source at an angular frequencyω0 at which the inductive and capacitivereactances are equal in magnitude. The frequency at which this equality holds for the particular circuit is called the resonant frequency. Theresonant frequency of the LC circuit is

ω0=1LC,{\displaystyle \omega _{0}={\frac {1}{\sqrt {LC}}},}

whereL is theinductance inhenries, andC is thecapacitance infarads. Theangular frequencyω0 has units ofradians per second.

The equivalent frequency in units ofhertz is

f0=ω02π=12πLC.{\displaystyle f_{0}={\frac {\omega _{0}}{2\pi }}={\frac {1}{2\pi {\sqrt {LC}}}}.}

Applications

[edit]

The resonance effect of the LC circuit has many important applications in signal processing and communications systems.

  • The most common application of tank circuits istuning radio transmitters and receivers. For example, when tuning a radio to a particular station, the LC circuits are set at resonance for that particularcarrier frequency.
  • Aseries resonant circuit providesvoltage magnification.
  • Aparallel resonant circuit providescurrent magnification.
  • A parallel resonant circuit can be used as load impedance in output circuits of RF amplifiers. Due to high impedance, the gain of amplifier is maximum at resonant frequency.
  • Both parallel and series resonant circuits are used ininduction heating.

LC circuits behave as electronicresonators, which are a key component in many applications:

Time domain solution

[edit]

Kirchhoff's laws

[edit]

ByKirchhoff's voltage law, the voltageVC across the capacitor plus the voltageVL across the inductor must equal zero:

VC+VL=0.{\displaystyle V_{C}+V_{L}=0.}

Likewise, byKirchhoff's current law, the current through the capacitor equals the current through the inductor:

IC=IL.{\displaystyle I_{C}=I_{L}.}

From the constitutive relations for the circuit elements, we also know that

VL(t)=LdILdt,IC(t)=CdVCdt.{\displaystyle {\begin{aligned}V_{L}(t)&=L{\frac {\mathrm {d} I_{L}}{\mathrm {d} t}},\\I_{C}(t)&=C{\frac {\mathrm {d} V_{C}}{\mathrm {d} t}}.\end{aligned}}}

Differential equation

[edit]

Rearranging and substituting gives the second orderdifferential equation

d2dt2I(t)+1LCI(t)=0.{\displaystyle {\frac {\mathrm {d} ^{2}}{\mathrm {d} t^{2}}}I(t)+{\frac {1}{LC}}I(t)=0.}

The parameterω0, the resonantangular frequency, is defined as

ω0=1LC.{\displaystyle \omega _{0}={\frac {1}{\sqrt {LC}}}.}

Using this can simplify the differential equation:

d2dt2I(t)+ω02I(t)=0.{\displaystyle {\frac {\mathrm {d} ^{2}}{\mathrm {d} t^{2}}}I(t)+\omega _{0}^{2}I(t)=0.}

The associatedLaplace transform is

s2+ω02=0,{\displaystyle s^{2}+\omega _{0}^{2}=0,}

thus

s=±jω0,{\displaystyle s=\pm j\omega _{0},}

wherej is theimaginary unit.

Solution

[edit]

Thus, the complete solution to the differential equation is

I(t)=Ae+jω0t+Be−jω0t{\displaystyle I(t)=Ae^{+j\omega _{0}t}+Be^{-j\omega _{0}t}}

and can be solved forA andB by considering the initial conditions. Since the exponential iscomplex, the solution represents a sinusoidalalternating current. Since the electric currentI is a physical quantity, it must be real-valued. As a result, it can be shown that the constantsA andB must becomplex conjugates:

A=B∗.{\displaystyle A=B^{*}.}

Now let

A=I02e+jϕ.{\displaystyle A={\frac {I_{0}}{2}}e^{+j\phi }.}

Therefore,

B=I02e−jϕ.{\displaystyle B={\frac {I_{0}}{2}}e^{-j\phi }.}

Next, we can useEuler's formula to obtain a realsinusoid withamplitudeI0,angular frequencyω0 =⁠1/√LC⁠, andphase angleϕ{\displaystyle \phi }.

Thus, the resulting solution becomes

I(t)=I0cos⁡(ω0t+ϕ),{\displaystyle I(t)=I_{0}\cos \left(\omega _{0}t+\phi \right),}
VL(t)=LdIdt=−ω0LI0sin⁡(ω0t+ϕ).{\displaystyle V_{L}(t)=L{\frac {\mathrm {d} I}{\mathrm {d} t}}=-\omega _{0}LI_{0}\sin \left(\omega _{0}t+\phi \right).}

Initial conditions

[edit]

The initial conditions that would satisfy this result are

I(0)=I0cos⁡ϕ,{\displaystyle I(0)=I_{0}\cos \phi ,}
VL(0)=LdIdt|t=0=−ω0LI0sin⁡ϕ.{\displaystyle V_{L}(0)=L{\frac {\mathrm {d} I}{\mathrm {d} t}}{\Bigg |}_{t=0}=-\omega _{0}LI_{0}\sin \phi .}

Series circuit

[edit]
Series LC circuit

In the series configuration of the LC circuit, the inductor (L) and capacitor (C) are connected in series, as shown here. The total voltageV across the open terminals is simply the sum of the voltage across the inductor and the voltage across the capacitor. The currentI into the positive terminal of the circuit is equal to the current through both the capacitor and the inductor.

V=VL+VC,I=IL=IC.{\displaystyle {\begin{aligned}V&=V_{L}+V_{C},\\I&=I_{L}=I_{C}.\end{aligned}}}

Resonance

[edit]

Inductive reactance XL=ωL {\displaystyle \ X_{\mathsf {L}}=\omega L\ } increases as frequency increases, whilecapacitive reactance XC=1 ωC  {\displaystyle \ X_{\mathsf {C}}={\frac {1}{\ \omega C\ }}\ } decreases with increase in frequency (defined here as a positive number). At one particular frequency, these two reactances are equal and the voltages across them are equal and opposite in sign; that frequency is called the resonant frequency f0  for the given circuit.

Hence, at resonance,

XL=XC ,ωL=1 ωC  .{\displaystyle {\begin{aligned}X_{\mathsf {L}}&=X_{\mathsf {C}}\ ,\\\omega L&={\frac {1}{\ \omega C\ }}~.\end{aligned}}}

Solving forω, we have

ω=ω0=1 LC  ,{\displaystyle \omega =\omega _{0}={\frac {1}{\ {\sqrt {LC\;}}\ }}\ ,}

which is defined as the resonant angular frequency of the circuit. Converting angular frequency (in radians per second) into frequency (inHertz), one has

f0=ω0 2π =1 2πLC  ,{\displaystyle f_{0}={\frac {\omega _{0}}{\ 2\pi \ }}={\frac {1}{\ 2\pi {\sqrt {LC\;}}\ }}\ ,}

and

XL0=XC0= L C{\displaystyle X_{{\mathsf {L}}0}=X_{{\mathsf {C}}0}={\sqrt {{\frac {\ L\ }{C}}\;}}}

atω0{\displaystyle \omega _{0}}.

In a series configuration,XC andXL cancel each other out. In real, rather than idealised, components, the current is opposed, mostly by the resistance of the coil windings. Thus, the current supplied to a series resonant circuit is maximal at resonance.

  • In the limit as f →f0  current is maximal. Circuit impedance is minimal. In this state, a circuit is called anacceptor circuit[4]
  • For  f  <  f0 ,  XL≪XC ;   hence, the circuit is capacitive.
  • For  f  >  f0 ,  XL ≫XC ;   hence, the circuit is inductive.

Impedance

[edit]

In the series configuration, resonance occurs when the complex electrical impedance of the circuit approaches zero.

First consider theimpedance of the series LC circuit. The total impedance is given by the sum of the inductive and capacitive impedances:

Z=ZL+ZC .{\displaystyle Z=Z_{\mathsf {L}}+Z_{\mathsf {C}}~.}

Writing the inductive impedance as ZL =jωL  and capacitive impedance as ZC =⁠1/ j ω C ⁠  and substituting gives

Z(ω)=jωL+1 jωC  .{\displaystyle Z(\omega )=j\omega L+{\frac {1}{\ j\omega C\ }}~.}

Writing this expression under a common denominator gives

Z(ω)=j( ω2LC−1 ωC) .{\displaystyle Z(\omega )=j\left({\frac {\ \omega ^{2}LC-1\ }{\omega C}}\right)~.}

Finally, defining the natural angular frequency as

ω0=1 LC  ,{\displaystyle \omega _{0}={\frac {1}{\ {\sqrt {LC\;}}\ }}\ ,}

the impedance becomes

Z(ω)=j L ( ω2−ω02 ω)=j ω0L (ω ω0 − ω0 ω)=j 1 ω0C (ω ω0 − ω0 ω) ,{\displaystyle Z(\omega )=j\ L\ \left({\frac {\ \omega ^{2}-\omega _{0}^{2}\ }{\omega }}\right)=j\ \omega _{0}L\ \left({\frac {\omega }{\ \omega _{0}\ }}-{\frac {\ \omega _{0}\ }{\omega }}\right)=j\ {\frac {1}{\ \omega _{0}C\ }}\left({\frac {\omega }{\ \omega _{0}\ }}-{\frac {\ \omega _{0}\ }{\omega }}\right)\ ,}

whereω0L {\displaystyle \,\omega _{0}L\ \,} gives the reactance of the inductor at resonance.

The numerator implies that in the limit as ω → ±ω0 , the total impedance Z  will be zero and otherwise non-zero. Therefore the series LC circuit, when connected in series with a load, will act as aband-pass filter having zero impedance at the resonant frequency of the LC circuit.

Parallel circuit

[edit]
Parallel LC circuit

When the inductor (L) and capacitor (C) are connected in parallel as shown here, the voltageV across the open terminals is equal to both the voltage across the inductor and the voltage across the capacitor. The total currentI flowing into the positive terminal of the circuit is equal to the sum of the current flowing through the inductor and the current flowing through the capacitor:

V=VL=VC ,I=IL+IC .{\displaystyle {\begin{aligned}V&=V_{\mathsf {L}}=V_{\mathsf {C}}\ ,\\I&=I_{\mathsf {L}}+I_{\mathsf {C}}~.\end{aligned}}}

Resonance

[edit]

WhenXL equalsXC, the two branch currents are equal and opposite. They cancel each other out to give minimal current in the main line (in principle, for a finite voltageV, there is zero current). Since total current in the main line is minimal, in this state the total impedance is maximal. There is also a larger current circulating in the loop formed by the capacitor and inductor. For a finite voltageV, this circulating current is finite, with value given by the respective voltage-current relationships of the capacitor and inductor. However, for a finite total currentI in the main line, in principle, the circulating current would be infinite. In reality, the circulating current in this case is limited by resistance in the circuit, particularly resistance in the inductor windings.

The resonant frequency is given by

f0=ω0 2π =1 2πLC  .{\displaystyle f_{0}={\frac {\omega _{0}}{\ 2\pi \ }}={\frac {1}{\ 2\pi {\sqrt {LC\;}}\ }}~.}

Any branch current is not minimal at resonance, but each is given separately by dividing source voltage (V) by reactance (Z). Hence  I =⁠ V /Z⁠ , as perOhm's law.

  • At  f0 , the line current is minimal. The total impedance is maximal. In this state a circuit is called arejector circuit.[5]
  • Below  f0 , the circuit is inductive.
  • Above  f0 , the circuit is capacitive.

Impedance

[edit]

The same analysis may be applied to the parallel LC circuit. The total impedance is then given by

Z= ZLZC ZL+ZC ,{\displaystyle Z={\frac {\ Z_{\mathsf {L}}Z_{\mathsf {C}}\ }{Z_{\mathsf {L}}+Z_{\mathsf {C}}}}\ ,}

and after substitution ofZL=j ω L andZC=⁠1/ j ω C ⁠ and simplification, gives

Z(ω)=−j⋅ωL ω2LC−1  .{\displaystyle Z(\omega )=-j\cdot {\frac {\omega L}{\ \omega ^{2}LC-1\ }}~.}

Using

ω0=1 LC  ,{\displaystyle \omega _{0}={\frac {1}{\ {\sqrt {LC\;}}\ }}\ ,}

it further simplifies to

Z(ω)=−j (1 C )(ω ω2−ω02 )=+j 1 ω0C(ω0ω−ωω0) =+j ω0L (ω0ω−ωω0)  .{\displaystyle Z(\omega )=-j\ \left({\frac {1}{\ C\ }}\right)\left({\frac {\omega }{\ \omega ^{2}-\omega _{0}^{2}\ }}\right)=+j\ {\frac {1}{\ \omega _{0}C\left({\tfrac {\omega _{0}}{\omega }}-{\tfrac {\omega }{\omega _{0}}}\right)\ }}=+j\ {\frac {\omega _{0}L}{\ \left({\tfrac {\omega _{0}}{\omega }}-{\tfrac {\omega }{\omega _{0}}}\right)\ }}~.}

Note that

limω→ω0Z(ω)=∞ ,{\displaystyle \lim _{\omega \to \omega _{0}}Z(\omega )=\infty \ ,}

but for all other values ofω the impedance is finite.

Thus, the parallel LC circuit connected in series with a load will act asband-stop filter having infinite impedance at the resonant frequency of the LC circuit, while the parallel LC circuit connected in parallel with a load will act asband-pass filter.

Laplace solution

[edit]

The LC circuit can be solved using theLaplace transform.

We begin by defining the relation between current and voltage across the capacitor and inductor in the usual way:

vC(t)=v(t) , {\displaystyle v_{\mathrm {C} }(t)=v(t)\ ,~}i(t)=C d vCdt , {\displaystyle i(t)=C\ {\frac {\mathrm {d} \ v_{\mathrm {C} }}{\mathrm {d} t}}\ ,~} and vL(t)=L d idt.{\displaystyle ~v_{\mathrm {L} }(t)=L\ {\frac {\mathrm {d} \ i}{\mathrm {d} t}}\;.}

Then by application of Kirchhoff's laws, we may arrive at the system's governing differential equations

vin(t)=vL(t)+vC(t)=L d idt+v=L C d2 vdt2+v.{\displaystyle v_{in}(t)=v_{\mathrm {L} }(t)+v_{\mathrm {C} }(t)=L\ {\frac {\mathrm {d} \ i}{\mathrm {d} t}}+v=L\ C\ {\frac {\mathrm {d} ^{2}\ v}{\mathrm {d} t^{2}}}+v\;.}

With initial conditions v(0)=v0 {\displaystyle \ v(0)=v_{0}\ } and i(0)=i0=C⋅v′(0)=C⋅v0′.{\displaystyle \ i(0)=i_{0}=C\cdot v'(0)=C\cdot v'_{0}\;.}

Making the following definitions,

ω0≡1 L C  {\displaystyle \omega _{0}\equiv {\frac {1}{\ {\sqrt {L\ C\ }}}}~} and f(t)≡ω02 vin(t){\displaystyle ~f(t)\equiv \omega _{0}^{2}\ v_{\mathrm {in} }(t)}

gives

f(t)= d2 v dt2+ω02 v.{\displaystyle f(t)={\frac {\ \mathrm {d} ^{2}\ v\ }{\mathrm {d} t^{2}}}+\omega _{0}^{2}\ v\;.}

Now we apply the Laplace transform.

L⁡[ f(t) ]=L⁡[  d2 v dt2+ω02 v ],{\displaystyle \operatorname {\mathcal {L}} \left[\ f(t)\ \right]=\operatorname {\mathcal {L}} \left[\ {\frac {\ \mathrm {d} ^{2}\ v\ }{\mathrm {d} t^{2}}}+\omega _{0}^{2}\ v\ \right]\,,}
F(s)=s2 V(s)−s v0−v0′+ω02 V(s).{\displaystyle F(s)=s^{2}\ V(s)-s\ v_{0}-v'_{0}+\omega _{0}^{2}\ V(s)\;.}

The Laplace transform has turned our differential equation into an algebraic equation. Solving forV in thes domain (frequency domain) is much simpler viz.

V(s)= s v0+v0′+F(s) s2+ω02{\displaystyle V(s)={\frac {\ s\ v_{0}+v'_{0}+F(s)\ }{s^{2}+\omega _{0}^{2}}}}
V(s)= s v0s2+ω02+v0′s2+ω02+F(s) s2+ω02,{\displaystyle V(s)={\frac {\ s\ v_{0}}{s^{2}+\omega _{0}^{2}}}+{\frac {v'_{0}}{s^{2}+\omega _{0}^{2}}}+{\frac {F(s)\ }{s^{2}+\omega _{0}^{2}}}\,,}

Which can be transformed back to the time domain via the inverse Laplace transform:

v(t)=L−1⁡[ V(s) ]{\displaystyle v(t)=\operatorname {\mathcal {L}} ^{-1}\left[\ V(s)\ \right]}
v(t)=L−1⁡[  s v0s2+ω02+v0′s2+ω02+F(s) s2+ω02 ],{\displaystyle v(t)=\operatorname {\mathcal {L}} ^{-1}\left[\ {\frac {\ s\ v_{0}}{s^{2}+\omega _{0}^{2}}}+{\frac {v'_{0}}{s^{2}+\omega _{0}^{2}}}+{\frac {F(s)\ }{s^{2}+\omega _{0}^{2}}}\ \right],}

For the second summand, an equivalent fraction ofω0{\displaystyle \omega _{0}} is needed:

v(t)=v0L−1⁡[ ss2+ω02 ]+v0′L−1⁡[ ω0ω0(s2+ω02) ]+L−1⁡[ F(s) s2+ω02 ],{\displaystyle v(t)=v_{0}\operatorname {\mathcal {L}} ^{-1}\left[\ {\frac {s}{s^{2}+\omega _{0}^{2}}}\ \right]+v'_{0}\operatorname {\mathcal {L}} ^{-1}\left[\ {\frac {\omega _{0}}{\omega _{0}(s^{2}+\omega _{0}^{2})}}\ \right]+\operatorname {\mathcal {L}} ^{-1}\left[\ {\frac {F(s)\ }{s^{2}+\omega _{0}^{2}}}\ \right],}

For the second summand, an equivalent fraction ofω0{\displaystyle \omega _{0}} is needed:

v(t)=v0L−1⁡[ ss2+ω02 ]+v0′ω0L−1⁡[ ω0(s2+ω02) ]+L−1⁡[ F(s) s2+ω02 ],{\displaystyle v(t)=v_{0}\operatorname {\mathcal {L}} ^{-1}\left[\ {\frac {s}{s^{2}+\omega _{0}^{2}}}\ \right]+{\frac {v'_{0}}{\omega _{0}}}\operatorname {\mathcal {L}} ^{-1}\left[\ {\frac {\omega _{0}}{(s^{2}+\omega _{0}^{2})}}\ \right]+\operatorname {\mathcal {L}} ^{-1}\left[\ {\frac {F(s)\ }{s^{2}+\omega _{0}^{2}}}\ \right],}
v(t)=v0cos⁡(ω0 t)+v0′ ω0  sin⁡(ω0 t)+L−1⁡[ F(s) s2+ω02  ]{\displaystyle v(t)=v_{0}\cos(\omega _{0}\ t)+{\frac {v'_{0}}{\ \omega _{0}\ }}\ \sin(\omega _{0}\ t)+\operatorname {\mathcal {L}} ^{-1}\left[\ {\frac {F(s)}{\ s^{2}+\omega _{0}^{2}\ }}\ \right]}

The final term is dependent on the exact form of the input voltage. Two common cases are theHeaviside step function and asine wave. For a Heaviside step function we get

vin(t)=M u(t),{\displaystyle v_{\mathrm {in} }(t)=M\ u(t)\,,}
L−1⁡[ ω02Vin(s) s2+ω02  ] = L−1⁡[ ω02 M 1 s (s2+ω02)  ] = M (1−cos⁡(ω0 t)),{\displaystyle \operatorname {\mathcal {L}} ^{-1}\left[\ \omega _{0}^{2}{\frac {V_{\mathrm {in} }(s)}{\ s^{2}+\omega _{0}^{2}\ }}\ \right]~=~\operatorname {\mathcal {L}} ^{-1}\left[\ \omega _{0}^{2}\ M\ {\frac {1}{\ s\ (s^{2}+\omega _{0}^{2})\ }}\ \right]~=~M\ {\Bigl (}1-\cos(\omega _{0}\ t){\Bigr )}\,,}
v(t)=v0 cos⁡(ω0 t)+v0′ω0 sin⁡(ω0 t)+M (1−cos⁡(ω0 t)).{\displaystyle v(t)=v_{0}\ \cos(\omega _{0}\ t)+{\frac {v'_{0}}{\omega _{0}}}\ \sin(\omega _{0}\ t)+M\ {\Bigl (}1-\cos(\omega _{0}\ t){\Bigr )}\;.}

For the case of a sinusoidal function as input we get:

vin(t)=U sin⁡(ωf t)⇒Vin(s)= U ωf  s2+ωf2 {\displaystyle v_{\mathrm {in} }(t)=U\ \sin(\omega _{\mathrm {f} }\ t)\Rightarrow V_{\mathrm {in} }(s)={\frac {\ U\ \omega _{\mathrm {f} }\ }{\ s^{2}+\omega _{\mathrm {f} }^{2}\ }}\,}

whereU{\displaystyle U} is the amplitude andωf{\displaystyle \omega _{f}} the frequency of the applied function.

L−1⁡[ ω02 1 s2+ω02  U ωf s2+ωf2  ]{\displaystyle \operatorname {\mathcal {L}} ^{-1}\left[\ \omega _{0}^{2}\ {\frac {1}{\ s^{2}+\omega _{0}^{2}\ }}\ {\frac {U\ \omega _{\mathrm {f} }}{\ s^{2}+\omega _{\mathrm {f} }^{2}\ }}\ \right]}

Using the partial fraction method:

L−1⁡[ ω02 U ωf1 s2+ω02  1 s2+ωf2  ]=L−1⁡[ ω02 U ωfA+Bs s2+ω02  +C+Ds s2+ωf2  ]{\displaystyle \operatorname {\mathcal {L}} ^{-1}\left[\ \omega _{0}^{2}\ U\ \omega _{\mathrm {f} }{\frac {1}{\ s^{2}+\omega _{0}^{2}\ }}\ {\frac {1}{\ s^{2}+\omega _{\mathrm {f} }^{2}\ }}\ \right]=\operatorname {\mathcal {L}} ^{-1}\left[\ \omega _{0}^{2}\ U\ \omega _{\mathrm {f} }{\frac {A+Bs}{\ s^{2}+\omega _{0}^{2}\ }}\ +{\frac {C+Ds}{\ s^{2}+\omega _{\mathrm {f} }^{2}\ }}\ \right]}

Simplifiying on both sides

1=(A+Bs)( s2+ωf2 )+(C+Ds)( s2+ω02 ){\displaystyle 1=(A+Bs)(\ s^{2}+\omega _{\mathrm {f} }^{2}\ )+(C+Ds)(\ s^{2}+\omega _{0}^{2}\ )}
1=(A s2+ A ωf2 + B s3+ B ωf2 )+(C s2+ C ω02 + D s3+ D sω02 ){\displaystyle 1=(A\ s^{2}+\ A\ \omega _{\mathrm {f} }^{2}\ +\ B\ s^{3}+\ B\ \omega _{\mathrm {f} }^{2}\ )+(C\ s^{2}+\ C\ \omega _{0}^{2}\ +\ D\ s^{3}+\ D\ s\omega _{0}^{2}\ )}
1=s3(B + D )+s2(A + C)+s(B ωf2+ D ω02)+(A ωf2 + C ω02){\displaystyle 1=s^{3}(B\ +\ D\ )+s^{2}(A\ +\ C)+s(B\ \omega _{\mathrm {f} }^{2}+\ D\ \omega _{0}^{2})+(A\ \omega _{\mathrm {f} }^{2}\ +\ C\ \omega _{0}^{2})}

We solve the equation for A, B and C:

A+C=0⇒C=−A{\displaystyle A+C=0\Rightarrow C=-A}
A ωf2 + C ω02=1⇒A ωf2 − A ω02=1{\displaystyle A\ \omega _{\mathrm {f} }^{2}\ +\ C\ \omega _{0}^{2}=1\Rightarrow A\ \omega _{\mathrm {f} }^{2}\ -\ A\ \omega _{0}^{2}=1}
⇒A =1(ωf2 −ω02){\displaystyle \Rightarrow A\ ={\frac {1}{(\omega _{\mathrm {f} }^{2}\ -\omega _{0}^{2})}}}
⇒C =−1(ωf2 −ω02){\displaystyle \Rightarrow C\ =-{\frac {1}{(\omega _{\mathrm {f} }^{2}\ -\omega _{0}^{2})}}}
B+C=0{\displaystyle B+C=0}
B ωf2+ D ω02=0⇒B ωf2− B ω02=0⇒B (ωf2−ω02)=0{\displaystyle B\ \omega _{\mathrm {f} }^{2}+\ D\ \omega _{0}^{2}=0\Rightarrow B\ \omega _{\mathrm {f} }^{2}-\ B\ \omega _{0}^{2}=0\Rightarrow B\ (\omega _{\mathrm {f} }^{2}-\omega _{0}^{2})=0}
⇒B=0, D=0{\displaystyle \Rightarrow B=0,\ D=0}

Substitute the values of A, B and C:

L−1⁡[ ω02 U ωf1(ωf2 −ω02) s2+ω02 +−1(ωf2 −ω02) s2+ωf2  ]{\displaystyle \operatorname {\mathcal {L}} ^{-1}\left[\ \omega _{0}^{2}\ U\ \omega _{\mathrm {f} }{\frac {\frac {1}{(\omega _{\mathrm {f} }^{2}\ -\omega _{0}^{2})}}{\ s^{2}+\omega _{0}^{2}\ }}+{\frac {-{\frac {1}{(\omega _{\mathrm {f} }^{2}\ -\omega _{0}^{2})}}}{\ s^{2}+\omega _{\mathrm {f} }^{2}\ }}\ \right]}

Isolating the constant and using equivalent fractions to adjust for lack of numerator:

 ω02 Uωf  ωf2−ω02 L−1⁡[(ω0ω0(s2+ω02)−ωfωf(s2+ωf2))]{\displaystyle {\frac {\ \omega _{0}^{2}\ U\omega _{\mathrm {f} }\ }{\ \omega _{\mathrm {f} }^{2}-\omega _{0}^{2}\ }}\operatorname {\mathcal {L}} ^{-1}\left[\left({\frac {\omega _{0}}{\omega _{0}(s^{2}+\omega _{0}^{2})}}-{\frac {\omega _{f}}{\omega _{f}(s^{2}+\omega _{f}^{2})}}\right)\right]\,}

Performing the reverse Laplace transform on each summands:

 ω02 Uωf  ωf2−ω02 (L−1⁡[ 1ω0ω0(s2+ω02)]−L−1⁡[1ωf ωf (s2+ωf2)]){\displaystyle {\frac {\ \omega _{0}^{2}\ U\omega _{\mathrm {f} }\ }{\ \omega _{\mathrm {f} }^{2}-\omega _{0}^{2}\ }}\left(\operatorname {\mathcal {L}} ^{-1}\left[\ {\frac {1}{\omega _{0}}}{\frac {\omega _{0}}{(s^{2}+\omega _{0}^{2})}}\right]-\operatorname {\mathcal {L}} ^{-1}\left[{\frac {1}{\omega _{\mathrm {f} }\ }}{\frac {\omega _{\mathrm {f} }\ }{(s^{2}+\omega _{f}^{2})}}\right]\right)\,}
 ω02 Uωf  ωf2−ω02 (1ω0L−1⁡[ω0(s2+ω02)]−1ωf L−1⁡[ωf (s2+ωf2)]){\displaystyle {\frac {\ \omega _{0}^{2}\ U\omega _{\mathrm {f} }\ }{\ \omega _{\mathrm {f} }^{2}-\omega _{0}^{2}\ }}\left({\frac {1}{\omega _{0}}}\operatorname {\mathcal {L}} ^{-1}\left[{\frac {\omega _{0}}{(s^{2}+\omega _{0}^{2})}}\right]-{\frac {1}{\omega _{\mathrm {f} }\ }}\operatorname {\mathcal {L}} ^{-1}\left[{\frac {\omega _{\mathrm {f} }\ }{(s^{2}+\omega _{f}^{2})}}\right]\right)\,}
vin(t)= ω02 U ωf ωf2−ω02(1ω0 sin⁡(ω0 t)−1 ωf  sin⁡(ωf t)),{\displaystyle v_{\mathrm {in} }(t)={\frac {\ \omega _{0}^{2}\ U\ \omega _{\mathrm {f} }\ }{\omega _{\mathrm {f} }^{2}-\omega _{0}^{2}}}\left({\frac {1}{\omega _{0}}}\ \sin(\omega _{0}\ t)-{\frac {1}{\ \omega _{\mathrm {f} }\ }}\ \sin(\omega _{\mathrm {f} }\ t)\right)\;,}

Using initial conditions in the Laplace solution:

v(t)=v0cos⁡(ω0 t)+v0′ω0  sin⁡(ω0 t)+ω02 U ωf ωf2−ω02 (1ω0 sin⁡(ω0 t)−1 ωf  sin⁡(ωf t)).{\displaystyle v(t)=v_{0}\cos(\omega _{0}\ t)+{\frac {v'_{0}}{\omega _{0}\ }}\ \sin(\omega _{0}\ t)+{\frac {\omega _{0}^{2}\ U\ \omega _{\mathrm {f} }}{\ \omega _{\mathrm {f} }^{2}-\omega _{0}^{2}\ }}\left({\frac {1}{\omega _{0}}}\ \sin(\omega _{0}\ t)-{\frac {1}{\ \omega _{\mathrm {f} }\ }}\ \sin(\omega _{\mathrm {f} }\ t)\right)\;.}

History

[edit]

The first evidence that a capacitor and inductor could produce electrical oscillations was discovered in 1826 by French scientistFelix Savary.[6][7] He found that when aLeyden jar was discharged through a wire wound around an iron needle, sometimes the needle was left magnetized in one direction and sometimes in the opposite direction. He correctly deduced that this was caused by a damped oscillating discharge current in the wire, which reversed the magnetization of the needle back and forth until it was too small to have an effect, leaving the needle magnetized in a random direction. American physicistJoseph Henry repeated Savary's experiment in 1842 and came to the same conclusion, apparently independently.[8][9]

Irish scientistWilliam Thomson (Lord Kelvin) in 1853 showed mathematically that the discharge of a Leyden jar through an inductance should be oscillatory, and derived its resonant frequency.[6][8][9] British radio researcherOliver Lodge, by discharging a large battery of Leyden jars through a long wire, created a tuned circuit with its resonant frequency in the audio range, which produced a musical tone from the spark when it was discharged.[8] In 1857, German physicistBerend Wilhelm Feddersen photographed the spark produced by a resonant Leyden jar circuit in a rotating mirror, providing visible evidence of the oscillations.[6][8][9] In 1868, Scottish physicistJames Clerk Maxwell calculated the effect of applying an alternating current to a circuit with inductance and capacitance, showing that the response is maximum at the resonant frequency.[6] The first example of an electricalresonance curve was published in 1887 by German physicistHeinrich Hertz in his pioneering paper on the discovery of radio waves, showing the length of spark obtainable from his spark-gap LC resonator detectors as a function of frequency.[6]

One of the first demonstrations ofresonance between tuned circuits was Lodge's "syntonic jars" experiment around 1889.[6][8] He placed two resonant circuits next to each other, each consisting of a Leyden jar connected to an adjustable one-turn coil with a spark gap. When a high voltage from an induction coil was applied to one tuned circuit, creating sparks and thus oscillating currents, sparks were excited in the other tuned circuit only when the circuits were adjusted to resonance. Lodge and some English scientists preferred the term "syntony" for this effect, but the term "resonance" eventually stuck.[6] The first practical use for LC circuits was in the 1890s inspark-gap radio transmitters to allow the receiver and transmitter to be tuned to the same frequency. The first patent for a radio system that allowed tuning was filed by Lodge in 1897, although the first practical systems were invented in 1900 by Italian radio pioneerGuglielmo Marconi.[6]

See also

[edit]

References

[edit]
  1. ^Makarov, Sergey N.; Ludwig, Reinhold; Bitar, Stephen J. (2016).Practical Electrical Engineering. Springer. pp. X-483.ISBN 9783319211732.
  2. ^Dorf, Richard C.; Svoboda, James A. (2010).Introduction to Electric Circuits, 8th Ed. John Wiley and Sons. p. 368.ISBN 9780470521571.
  3. ^Rao, B. Visvesvara; et al. (2012).Electronic Circuit Analysis. India: Pearson Education India. p. 13.6.ISBN 978-9332511743.
  4. ^"What is an acceptor circuit?".qsstudy.com. Physics. ].
  5. ^"rejector circuit".Oxford Dictionaries. English. Archived fromthe original on September 20, 2018. Retrieved2018-09-20.
  6. ^abcdefghBlanchard, Julian (October 1941)."The History of Electrical Resonance".Bell System Technical Journal.20 (4). U.S.: American Telephone & Telegraph Co.:415–433.doi:10.1002/j.1538-7305.1941.tb03608.x.S2CID 51669988. Retrieved2011-03-29.
  7. ^Savary, Felix (1827). "Memoirs sur l'Aimentation".Annales de Chimie et de Physique.34. Paris: Masson:5–37.
  8. ^abcdeKimball, Arthur Lalanne (1917).A College Text-book of Physics (2nd ed.). New York: Henry Hold. pp. 516–517.
  9. ^abcHuurdeman, Anton A. (2003).The Worldwide History of Telecommunications. U.S.: Wiley-IEEE. pp. 199–200.ISBN 0-471-20505-2.

External links

[edit]
The WikibookCircuit Idea has a page on the topic of:How do We Create Sinusoidal Oscillations?
Authority control databases: NationalEdit this at Wikidata
Retrieved from "https://en.wikipedia.org/w/index.php?title=LC_circuit&oldid=1279486102"
Categories:
Hidden categories:
[8]ページ先頭
©2009-2025 Movatter.jp