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Oscillations are the periodic changes of any physical quantities.
Simple harmonic oscillations. The equation which describes the eigenmode of oscillations has the simple harmonic form:
wherex(t) – physical quantity, which makes the oscillations; А – oscillation amplitude; j0 – initial phase (phase constant). The time parameters of oscillations’ eigenmode are called eigen-parameters (or natural parameters) and are written down with index «0»: Cyclic eigenfrequency of oscillations (changing of oscillations’ phase per one second): w0 = 2p / Т 0 = 2p × n0, (2) n0 – eigenfrequency of oscillations (number of oscillations per one second); Т0 – eigenperiod of oscillations (minimal time interval of repeating of the value of oscillating quantity). In equation of oscillations (1) it is described both mechanical, and electromagnetic oscillations, therefore it is possible to set up correspondence of mechanical and electrical oscillations’ parameters:
Electromagnetic simple harmonic oscillations. There are three main parameters, which are changing during oscillations in oscillating LC-circuit: q (t) – charge of capacitor, uC (t) – voltage on capacitor, i (t) – current flawing through coil. They have identical eigenfrequency, but amplitudes and initial phases are different. Relation between these quantities represented in two definition: current:
Mathematic rules: Differentiation rule of harmonic function: 1. The multiplication constant (amplitude) is necessary to take out of the derivative sign. 2. Derivative of harmonic function has phase lead relative to own function on p / 2:
3. The result of derivation of harmonic function is necessary to multiply on derivative of a phase on time (on w0) (see example). Integration rule of harmonic function: 1. The multiplication constant (amplitude) is necessary to take out of the integral sign. 2. Integral from harmonic function has phase lag from own function on p / 2:
3. The result of integration of harmonic function is necessary to divide on derivative of a phase on time (on w0) (see example).
EXAMPLE OF PROBLEM SOLUTION
Example 1. The oscillation circuit consists of coil by inductance of L = 25 mH and capacitor. Current in circuit changes by the law i(t) = Im × cosω 0 t, where Im= 20 mА and ω 0 = 104 rad/ s. 1) To get the equation of changing during the time charge of capacitor and voltage on the capacitor and on the coil. 2) To define total energy of oscillations in circuit.
Solution: 1) From definition of the current we find expression of charge from current (as the integration – is the mathematical function, inverse to differentiation):
Let’s substitute the input equation of current oscillations in this expression and integrate. We get the equation of oscillations of charge of capacitor:
where Im – current oscillation amplitude; w0 – cyclic eigenfrequency of oscillations. In (1.1) we consider that integral from harmonic function has phase lag from own function on p / 2. From definition of the electrocapacity we find the expression of dependence of the voltage on capacitor from charge of its plate:
Let’s substitute the equation (1.1) in this expression we get the equation of oscillations of voltage on capacitor:
Electrocapacity can be found with the formula of oscillations’ cyclic eigenfrequency in the oscillating circuit:
where L – circuit inductance. Substituted in the formula (1.2) the perceived equation for С (1.3) finally we obtain:
From the 2nd Kirchoff’s rule voltage on the inductance coil equal to Back EMF: Let’s substitute the input equation of current oscillations in this expression and differentiate. We get the equation of oscillations of volktage on coil:
In (1.5) we consider that derivative of harmonic function has phase lead relative to own function on p / 2.
2. Total energy of oscillations in circuit equals the sum of energy of electric field in capacitor W E and energy of magnetic field in coil W М: W = W E + W М;
Substituting the expression for С from the formula (1.3), amplitude of oscillations of voltage on capacitor (1.2) and input equation of current oscillations, we get
Let's note, a total energy of oscillations has no time dependence, because absence of power loss. Let’s check, whether the right part of equation of amplitude (1.1) gives the unit of charge [ C ], of equation of amplitude (1.4) the unit of voltage [ V ] and the formulas of amplitude (5) the unit of energy [ J ].
Substituted numerical values, let’s write down the equation of changing q and uC with numerical coefficients and calculate the full energy of oscillations in circuit
Results:
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