Rlc circuits and differential equations1
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The document describes deriving a differential equation to model the behavior of an RLC circuit. It provides the component values for an RLC circuit that was designed and built. Through applying Kirchhoff's voltage law and differentiating the equation, a second order differential equation is derived. The parameters are then substituted into the equation to solve for the natural response of the underdamped circuit. The derived differential equation solution is compared to simulations and measurements from an oscilloscope.
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- 2. Designed and built RLC circuit to test response time of current
- 3. Derive the constant coefficient differential equation Resistance (R) = 643.108 Ω Inductor (L) = 9.74 × 10^-3 H Capacitor (C) = 9.42 × 10^-8 F
- 4. Kirchhoff’s Voltage Law (KVL) The sum of voltage drops across the elements of a series circuit is equal to applied voltage.
- 5. Voltage Drop per Circuit Elements Inductor Resistor Capacitor
- 6. Substitute and Differentiation Differentiate both sides for equation for current Substitute dQ/dt for I in original equation and arrive at second order differential equation Current is the rate of change of charge (in Coulombs) with respect to time Substitute voltage drops into equation KVL )(tVVVV CLR )( 1 tVQ C RI dt dI L )(tI dt dQ )( 1 '" tVQ C RQLQ )(' 1 '" tVI C RILI
- 7. General forms of equation Which is the same as: Divide by L: 0)( 1)()( 2 2 ti LCdt tdi L R dt tid 0)( )( 2 )( 2 02 2 ti dt tdi dt tid
- 8. Calculate Parameters for Substitution 0 1 ''' I LC I L R I 5.660272 L R 92 0 100899.1 1 LC 0100899.1'5.66027'' 9 III 0100899.15.66027 92 RR 0)( 1)()( 2 2 ti LCdt tdi L R dt tid
- 9. Solve Characteristic Equation for R 62.32 76.33013 62.3276.33013 )1(2 )100899.1)(1(45.66027 5.66027 2 4 0)100899.1(5.66027 92 2 92 iR iR R a acb bR RR
- 10. Select Natural Response of Circuit Three forms: Over-damped: Two negative real roots Under-damped: Two distinct complex roots Critically-damped: Two real, distinct roots
- 11. Underdamped Circuit Response is a decaying exponential that oscillates t d etCtCti d ))cos()sin(()( 21
- 12. Substitute Parameters into Underdamped Equation t t d etCtCti L CR etCtCti d d 62.32 21 2 0 22 0 0 0 21 ))57.23cos()57.23sin(()( 57.231 30339999997452.0 2 62.32 76.33013 ))cos()sin(()(
- 13. LTSpice vs Oscilloscope t etCtCti 62.32 21 ))57.23cos()57.23sin(()(