2005/03/16 | CLR电路[CLR Circuit]
类别(Ω〖物理〗) | 评论(0) | 阅读(55) | 发表于 15:33
Given an electric circuit composed of an inductor with inductance L, a resistor with resistance R, and a capacitor with capacitance C, the Kirchhoff loop rule requires that the sum of the changes in potential around the circuit must be zero, so
(1)

where I is the current through the inductor,resistor and Q is the charge on the capacitor, and t is the elapsed time. Differentiating gives
(2)

which can be rewritten
(3)

Now define the variables
(4)

(5)

to write the differential equation in the standard form
(6)

Here is the differential operator.
As in any simple harmonic motion,there are three classes of solution depending on the sign of : underdamped, critically damped, and overdamped. For an underdamped circuit,
(7)

which is equivalent to
(8)

and the simple harmonic motion solution is
(9)

where
(10)

and


For a critically damped circuit
(11)

which is equivalent to
(12)

and the solution is
(13)

where


For an overdamped circuit,
(14)

or
(15)

The solution is
(16)

where

giving


and the constants are given by


For a sinusoidally driven CLR circuit, define
(17)

(18)

(19)

(20)

(21)

where is the permittivity of free space. The equations are
(22)

(23)

(24)

(25)

The problem can be solved by solving the differential equation. However, it is much simpler to assume a harmonic solution
(26)

Use complex impedance to find the solution of this type, if it exists


(27)

So the harmonic solution is
(28)

where
(29)

(30)

Dividing by CL,
(30)


(32)

(33)

The homogeneous solution will be underdamped, critically, or overdamped depending on the initial conditions. However, it will be exponentially decaying, so the steady state solution is given be the solution above. The maximum of
(34)

occurs at
(35)

Furthermore, at ,, so
(36)



]
(37)

but
(38)

so





(39)

Both and are maximum at , so
(40)

To find the half-power points, solve
(41)

(42)

(43)

(44)

(45)


(46)

The full width at half maximum is

The amplitude decay time is
(48)

The energy stored in the system is
(49)

(50)

so
(51)


(52)

(53)

Q is a minimum when

(54)

(55)

Note that near resonance,
(56)
0

评论Comments

日志分类
首页[1408]
∑〖数学〗[349]
Ω〖物理〗[357]
¤〖天文〗[343]
℃〖化学〗[359]