Practical Coilgun Design 

Inductors

Damped OscillatorsAn oscillator is anything that has a rythmic periodic response. A damped oscillator has a response that fades away over time. Examples include a swinging pendulum, a bobbing weight on a spring, and also a resistor  inductor  capacitor (RLC) circuit. The interactive RLC simulation is nice, but what equations drive it? Is there a better approach than a numerical simulation? What component values make it underdamped? Overdamped? When is it a criticallydamped oscillator? Mathemeticians have completely solved the equations that govern a damped oscillator. My web page shows their results, and explains how to compute frequency and criticallydamped parts values. EquationsSuppose you have an RLC circuit, which has a resistor + inductor + capacitor in series. When the switch closes at time t=0 the capacitor will discharge into a series resistor and inductor. What is the voltage V and current I as a function of time? where: and V= initial voltage What Does This Mean?The above equation is the current for a damped sine wave. It represents a sine wave of maximum amplitude (V/BL) multiplied by a damping factor of an exponential decay. The resulting time variation is an oscillation bounded by a decaying envelope. Critical DampingWe can use these equations to discover when the energy dies out smoothly (overdamped) or rings (underdamped). Look at the term under the square root sign, which can be simplified to: R^{2}C^{2}4LC
Resonant FrequencyThe equations above will tell us the value as a function of time, but what we really want to know is the frequency of oscillation. An LC circuit will oscillate at an angular frequency of: To convert radians/sec to frequency f in Hertz, simply divide by 2p to get this: To convert Hz to time period T, use the reciprocal of frequency: ExamplesQuestion: What is a coilgun's firing period if it has an RLC circuit where R=0 and C=10,000 µF and L=100 µH? Answer: The firing period is only the first halfwave of the oscillation. Question: What resistance will make this coilgun critically damped? Answer: Solve R^{2}C^{2}=4LC for resistance R to get: 



Last Update 20121110
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