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LCR Circuits

# LCR Circuits - LCR Circuits Akshaya Annavajhala Department...

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LCR Circuits Akshaya Annavajhala Department of Physics, Case Western Reserve University Cleveland, OH 44016 Abstract: Filler Introduction and Theory: The picture to the left shows a circuit with an inductor (L), a resistor (R), and a capacitor (C), and a voltage source (V), whose properties can be controlled by the two switches S1 and S2. When S1 is closed and S2 is left open, a voltage is placed across the circuit (excluding L, as S2 is left open), charging the capacitor. When S2 is closed and S1 is left open, A circuit consisting only of C, R, and L is formed, thus creating an LCR circuit. In this circuit, the capacitor discharges across the resistor and inductor, oscillating back and forth as the charge is flipped until all of the energy is dissipated across the resistor. This action is illustrated in the following figure: As one can see, this oscillatory motion is very similar to that of a mass on a spring, or a mechanical oscillator. In a mathematical sense, this electronic oscillator and its mechanical analogue are the same, as is evident from the following formulas: 2 2 0 d Q dQ Q L R dt C dt + + = 2 2 0 d y dy m b ky dt dt + + = The analogy is thus made very clear: L is the same as m , which is the inertial factor in the equation, and b , which is the damping force, translates into R , while the inverse capacitance is the analogue of the spring constant k , or the restorative force. Using many different types of solution methods, one can come up with a solution to the equation as: 2 0 1 Q=Q sin( ) 2 t R e t LC L τ ϕ - - + where Q 0 is the initial charge and phi is the phase shift of the sine wave. Because Q = CV, the following equation, which is far more helpful to us, is also true:

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2 0 1 V =V sin( ) 2 t c c R e t LC L τ ϕ - - + The figure to the left illustrates the forced, damped oscillator circuit built, with the forcing coming from the function generator in the schematic. The equation relating L, R, and C is
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LCR Circuits - LCR Circuits Akshaya Annavajhala Department...

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