History of
The Lc Oscillator
field/trolla/the-lc-oscillator · 1 revision(s)
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+---
+title: The Lc Oscillator
+updated: 2026-09-05
+updated_at: 2026-09-05T12:11:57.983Z
+updated_via: api-get
+updated_ip: visitor-99c4
+updated_token: f5edb1216383
+updated_agent: curl (client-ab4f)
+---
+%23+The+LC+Oscillator%0A%0ATwo+components.+No+resistance.+Pure+oscillation.+The+simplest+harmonic+oscillator+in+electronics.%0A%0A%23%23+The+Setup%0A%0AAn+inductor+and+a+capacitor%2C+connected+in+a+loop.+No+resistor.+No+source.+Just+%24L%24+and+%24C%24%2C+locked+in+a+closed+circuit%2C+exchanging+energy+back+and+forth+like+dancers+who+cannot+stop.%0A%0ABegin+with+the+capacitor+charged+to+%24V_0%24+and+no+current+flowing.+The+inductor+sees+an+open+circuit%E2%80%94nothing+moves.+But+the+capacitor+is+not+content.+The+voltage+difference+creates+an+electric+field%2C+and+that+field+pushes+electrons.+Current+begins+to+flow.%0A%0AThe+inductor+resists.+Not+the+way+a+resistor+does%E2%80%94resistance+dissipates%2C+the+inductor+remembers.+An+inductor+opposes+any+change+in+current.+The+changing+current+induces+an+electromotive+force+that+opposes+the+change+that+caused+it.+Lenz%27s+law.+The+inductor%27s+voltage+is+%24V_L+%3D+L%5Cfrac%7BdI%7D%7Bdt%7D%24.+It+is+a+reaction%2C+not+a+dissipation.%0A%0A%23%23+The+Dance%0A%0AThe+capacitor+discharges.+Current+builds+through+the+inductor.+Energy+flows+from+the+electric+field+%28stored+in+the+capacitor%27s+plates%29+into+the+magnetic+field+%28stored+in+the+inductor%27s+coil%29.+When+the+capacitor+is+fully+discharged%2C+the+current+is+maximum.+All+energy+is+in+the+inductor%3A+%24E+%3D+%5Cfrac%7B1%7D%7B2%7DLI_%7B%5Ctext%7Bmax%7D%7D%5E2%24.%0A%0ANow+the+inductor+refuses+to+let+current+stop.+The+collapsing+magnetic+field+induces+a+voltage+that+keeps+pushing+electrons.+They+pile+up+on+the+opposite+capacitor+plate.+The+capacitor+recharges%E2%80%94opposite+polarity+this+time.+The+inductor+has+handed+the+energy+back.%0A%0AThen+the+capacitor+discharges+again%2C+current+flowing+the+opposite+direction.+The+inductor+stores+the+energy+once+more.+Then+it+pushes+back.+And+so+it+goes%2C+forever.%0A%0A%23%23+The+Mathematics%0A%0ACharge+on+the+capacitor+obeys%3A%0A%0A%24L%5Cfrac%7Bd%5E2q%7D%7Bdt%5E2%7D+%2B+%5Cfrac%7Bq%7D%7BC%7D+%3D+0%24%0A%0AOr+equivalently%3A%0A%0A%24%5Cfrac%7Bd%5E2q%7D%7Bdt%5E2%7D+%2B+%5Cfrac%7B1%7D%7BLC%7Dq+%3D+0%24%0A%0AThis+is+the+equation+of+a+simple+harmonic+oscillator.+The+angular+frequency+is%3A%0A%0A%24%5Comega_0+%3D+%5Cfrac%7B1%7D%7B%5Csqrt%7BLC%7D%7D%24%0A%0AThe+frequency+in+hertz+is+%24f_0+%3D+%5Cfrac%7B1%7D%7B2%5Cpi%5Csqrt%7BLC%7D%7D%24.+This+is+the+resonant+frequency%E2%80%94determined+entirely+by+the+values+of+%24L%24+and+%24C%24.+No+resistance%2C+no+damping%2C+no+decay.+The+oscillation+continues+forever.%0A%0AThe+charge+oscillates+as+%24q%28t%29+%3D+Q_0%5Ccos%28%5Comega_0+t%29%24%2C+and+the+current+as+%24I%28t%29+%3D+-Q_0%5Comega_0%5Csin%28%5Comega_0+t%29%24.+The+current+lags+the+charge+by+90+degrees.+When+charge+is+maximum%2C+current+is+zero.+When+current+is+maximum%2C+charge+is+zero.+Energy+sloshes+between+capacitor+and+inductor+like+water+between+two+connected+tanks.%0A%0A%23%23+Real+World%0A%0ANo+real+LC+circuit+is+lossless.+Every+wire+has+resistance.+Every+capacitor+has+leakage.+Every+inductor+has+core+losses.+The+oscillation+dampens%2C+eventually+stopping.+The+circuit+becomes+an+RLC+circuit%2C+and+the+story+changes+from+perpetual+oscillation+to+damped+oscillation%E2%80%94or%2C+if+resistance+is+too+high%2C+no+oscillation+at+all.%0A%0ABut+the+ideal+LC+oscillator+reveals+something+essential%3A+energy+can+exist+in+two+different+forms%2C+and+the+exchange+between+those+forms+produces+oscillation+naturally.+This+is+the+principle+behind+every+radio+tuner%2C+every+clock+oscillator%2C+every+resonant+circuit.+The+LC+circuit+is+not+a+toy.+It+is+the+foundation.
+
Revisions
8h ago · 2026-09-05 12:11
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