Circuit RLC

A electrokinetic circuit RLC in is a linear circuit containing a electrical resistance, a winds (Inductance) and a condensing (capacity).

There exist two types of circuits parallel RLC series or , according to the interconnection of the three types of components. The behavior of a circuit RLC is generally described by a differential equation of the second order (where circuits RL or circuits RC behave like first order circuits).

Using a generating of signals, one can inject into the circuit of the Oscillation S and observe in certain cases a Résonance, characterized by an increase in the current (when the selected entry signal corresponds to the own pulsation of the circuit, calculable starting from the differential equation which governs it).

Circuit RLC in series

Circuit subjected to a level of tension

If a circuit RLC series is subjected to a level of tension $E\; \backslash ,$, the law of the meshs imposes the relation:

$E\; =\; u\_C\; +\; L\; \backslash \; frac\; \{di\}\; \{dt\}\; +\; R\_ti$

By introducing the relation characteristic of the condenser:

$i\_C\; =\; I\; =\; C\; \backslash \; frac\; \{du\_C\}\; \{dt\}$

one obtains the differential equation of the second order:

$E\; =\; u\_C\; +\; LLC\; \backslash \; frac\; \{d^2u\_c\}\; \{dt^2\}\; +\; R\_tC\; \backslash \; frac\; \{du\_c\}\; \{dt\}$

With:

• E the electromotive force of the generator, in V;
• u_C the tension at the boundaries of the condenser, in V;
• L the Inductance of the reel, in H;
• I the intensity of the electric current in the circuit, in has;
• Q the electric Charge of the condenser, in C;
• C the electric Capacity of the condenser, in F;
• R_t the total resistance of the circuit, in Ω;
• T the Time in S

In the case of a mode without losses, i.e. for $R\; =\; 0\; \backslash ,$, one obtains a solution putting itself in the form:

$u\_c\; =\; E\; \backslash \; cos\; (\backslash \; frac\; \{2\; \backslash \; pi\; T\}\; \{T\_0\}\; +\; \backslash \; phi)$

$T\_0\; =\; 2\; \backslash \; pi\; \backslash \; sqrt\; \{LLC\}$

With:

• T₀ the period of oscillation, in seconds;
• φ the phase in the beginning (generally selected such as φ = 0)

Circuit subjected to a sinusoidal tension

The Transformation complexes applied to the various tensions makes it possible to write the law of the meshs in the form:

$\backslash \; underline\; U\_G\; =\; \backslash \; underline\; U\_C\; +\; \backslash \; underline\; U\_L\; +\; \backslash \; underline\; U\_R\; \backslash ,$

maybe, by introducing the impedance S complexes:

$\backslash \; underline\; U\_G\; =\; \backslash \; frac\; \{-\; \backslash \; mathbf\; \{J\}\}\; \{C\; \backslash \; Omega\}\; \backslash \; underline\; I\; +\; \backslash \; mathbf\; \{J\}\; L\; \backslash \; Omega\; \backslash \; underline\; I\; +R\_t\; \backslash \; underline\; I\; =\; \backslash \; left\; \backslash \; underline\; I$

The angular frequency of Résonance in intensity of such a circuit ω₀ is given by:

$\backslash \; omega\_0\; =\; \backslash \; frac\; \{1\}\; \{\backslash \; sqrt\; \{LLC\}\}$

For this frequency the relation above becomes:

$\backslash \; underline\; U\_G\; =\; \backslash \; underline\; U\_R=\; R\_t\; \backslash \; underline\; I\; \backslash ,$, and one a: $\backslash \; underline\; U\_L\; =\; -\; \backslash \; underline\; U\_C\; =\; \backslash \; frac\; \{\backslash \; mathbf\; \{J\}\}\; \{R\_t\}\; \backslash \; sqrt\; \{\backslash \; frac\; \{L\}\; \{C\}\}\; \backslash \; cdot\; \backslash \; underline\; U\_G\; \backslash ,$

Circuit RLC in parallel

$i\_R\; =\; \backslash \; frac\; \{U\}\; \{R\}$
$\backslash \; frac\; \{di\_L\}\; \{dt\}\; =\; \backslash \; frac\; \{U\}\; \{L\}$
$i\_C\; =\; \backslash \; frac\; \{dq\}\; \{dt\}\; =\; C\; \backslash ,\; \backslash \; frac\; \{of\; the\}\; \{dt\}$
because $\backslash ,\; Q\; =\; C\; *\; U$
$\backslash ,\; I\; =\; i\_R\; +\; i\_L\; +\; i\_C$
$\backslash \; Rightarrow\; \backslash \; frac\; \{di\}\; \{dt\}\; =\; C\; \backslash \; \backslash \; frac\; \{d^2u\}\; \{dt^2\}\; +\; \backslash \; frac\; \{1\}\; \{R\}\; \backslash \; frac\; \{of\; the\}\; \{dt\}\; +\; \backslash \; frac\; \{1\}\; \{L\}\; \backslash \; U$

Caution: The branch C is in short-circuit: one cannot connect has, B directly at the boundaries of a generator E, it is necessary to add a resistance to him.

The two initial conditions are:

• $\backslash ,\; i\_\; \{L0\}$ guard its value before the powering (because inductance is opposed to the variation of the current)
• $\backslash ,\; q\_0$ guard its value before the powering $\backslash \; Rightarrow\; u\_0\; =\; q\_0/C$.

Circuit subjected to a sinusoidal tension

The Transformation complexes applied to the various intensities gives:

$\backslash \; underline\; I\; =\; \backslash \; underline\; I\_R\; +\; \backslash \; underline\; I\_L\; +\; \backslash \; underline\; I\_C\; \backslash ,$

maybe, by introducing the impedance S complexes:

$\backslash \; underline\; I\; =\; \backslash \; frac\; \{1\}\; \{R\}\; \backslash \; underline\; U\; +\; \backslash \; frac\; \{1\}\; \{\backslash \; mathbf\; \{J\}\; L\; \backslash \; Omega\}\; \backslash \; underline\; U\; +\; \backslash \; mathbf\; \{J\}\; C\; \backslash \; Omega\; \backslash \; underline\; U$ $=\; \backslash \; left\; +\; \backslash \; mathbf\; \{J\}\; (C\; \backslash \; Omega\; \backslash \; frac\; \{1\}\; \{L\; \backslash \; Omega\})\; \backslash \; right\; \backslash \; underline\; U$

The angular frequency of Résonance in intensity of such a circuit ω₀ is given by:

$\backslash \; omega\_0\; =\; \backslash \; frac\; \{1\}\; \{\backslash \; sqrt\; \{LLC\}\}$

For this frequency the relation above becomes:

$\backslash \; underline\; I\; =\; \backslash \; underline\; I\_R\; =\; \backslash \; frac\; \{1\}\; \{R\}\; \backslash \; underline\; U\; \backslash ,$, and one a: $\backslash \; underline\; I\_C\; =\; -\; \backslash \; underline\; I\_L\; =\; \backslash \; mathbf\; \{J\}\; \backslash \; sqrt\; \{\backslash \; frac\; \{C\}\; \{L\}\}\; \backslash \; cdot\; \backslash \; underline\; U\; \backslash ,$

See too

• Circuit RC
• Circuit RL

External bonds

• a video explanatory on resonance in a circuit RLC

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