\documentclass{article}
\begin{document}
\title{Single NPN Transistor Colpitts Oscillator}
\maketitle
\section{Circuit Theory}
A single NPN transistor uses common-base feedback from a capacitive divider. The base is biased by R1 and R2. The drawing and simulation are an educational simplified model.
\section{Oscillation Frequency}
\[
C_{eq}=\frac{C_1 C_2}{C_1+C_2}
\]
\[
f_0=\frac{1}{2\pi\sqrt{LC_{eq}}}
\]
For L = 10 mH and C1 = C2 = 10 nF, Ceq = 5 nF and the ideal frequency is approximately 22.508 kHz. Actual component tolerances, losses, loading, transistor parasitics and bias change a real circuit.
\section{Circuit Animation}
The waveform shows the AC component at the collector with its DC offset removed. The LC current display shows circulating AC current, not the total DC supply current. Playback uses slow educational time; the frequency measurement is the actual ideal LC resonance.
\begin{scienceanimation}
{
  "version": "1.0",
  "id": "single-tr-colpitts-rlc-001",
  "type": "electronic-circuit",
  "title": "Single-Transistor RLC Colpitts Oscillator",
  "duration": 10,
  "educationalModel": true,
  "parameters": {
    "VCC": 12.0,
    "frequencyHz": 22507.9,
    "startupTimeConstant": 1.25,
    "steadyOutputPeakV": 4.0
  },
  "components": [
    {
      "id": "VCC",
      "type": "dc-source",
      "value": "12V",
      "x": 70,
      "y": 110,
      "rotation": 0
    },
    {
      "id": "R1",
      "type": "resistor",
      "value": "47k",
      "x": 190,
      "y": 180,
      "rotation": 0
    },
    {
      "id": "R2",
      "type": "resistor",
      "value": "10k",
      "x": 190,
      "y": 450,
      "rotation": 0
    },
    {
      "id": "Q1",
      "type": "npn",
      "value": "Generic NPN",
      "x": 350,
      "y": 290,
      "rotation": 0
    },
    {
      "id": "RE",
      "type": "resistor",
      "value": "1k",
      "x": 350,
      "y": 460,
      "rotation": 0
    },
    {
      "id": "L1",
      "type": "inductor",
      "value": "10mH",
      "x": 350,
      "y": 145,
      "rotation": 0
    },
    {
      "id": "C1",
      "type": "capacitor",
      "value": "10nF",
      "x": 540,
      "y": 280,
      "rotation": 0
    },
    {
      "id": "C2",
      "type": "capacitor",
      "value": "10nF",
      "x": 540,
      "y": 460,
      "rotation": 0
    },
    {
      "id": "GND",
      "type": "ground",
      "x": 350,
      "y": 560,
      "rotation": 0
    },
    {
      "id": "OUT",
      "type": "probe",
      "label": "Vout",
      "x": 610,
      "y": 200,
      "rotation": 0
    },
    {
      "id": "CB",
      "type": "capacitor",
      "value": "100nF",
      "x": 100,
      "y": 450,
      "rotation": 0
    }
  ],
  "connections": [
    [
      "VCC.positive",
      "R1.1"
    ],
    [
      "VCC.positive",
      "L1.1"
    ],
    [
      "R1.2",
      "Q1.base"
    ],
    [
      "R2.1",
      "Q1.base"
    ],
    [
      "R2.2",
      "GND"
    ],
    [
      "Q1.emitter",
      "RE.1"
    ],
    [
      "RE.2",
      "GND"
    ],
    [
      "Q1.collector",
      "L1.2"
    ],
    [
      "Q1.collector",
      "OUT"
    ],
    [
      "Q1.collector",
      "C1.1"
    ],
    [
      "C1.2",
      "C2.1"
    ],
    [
      "C1.2",
      "Q1.emitter"
    ],
    [
      "C2.2",
      "GND"
    ],
    [
      "VCC.negative",
      "GND"
    ],
    [
      "Q1.base",
      "CB.1"
    ],
    [
      "CB.2",
      "GND"
    ]
  ],
  "simulation": {
    "model": "colpitts-oscillator",
    "masterClock": true,
    "resonator": {
      "L": "L1",
      "C1": "C1",
      "C2": "C2",
      "effectiveCapacitanceF": 5e-09,
      "resonantFrequencyHz": 22507.9
    },
    "startup": {
      "mode": "exponential-envelope",
      "initialAmplitude": 0.02,
      "finalAmplitude": 1.0,
      "timeConstant": 1.25
    },
    "waveforms": {
      "vout": "A(t)*sin(2*pi*f0*t)",
      "tankCurrent": "A(t)*sin(2*pi*f0*t-pi/2)",
      "capacitorState": "sin(2*pi*f0*t)"
    }
  },
  "display": {
    "showCurrentParticles": true,
    "particleDirectionFromCurrentSign": true,
    "particleSpeedFromCurrentMagnitude": true,
    "showNodeVoltages": true,
    "showComponentLabels": true,
    "showMeasurements": true,
    "showEnergyExchange": true,
    "showWaveform": true,
    "showTimeline": true,
    "controls": [
      "play",
      "pause",
      "restart",
      "seek",
      "speed"
    ],
    "speeds": [
      0.25,
      0.5,
      1,
      2,
      4
    ]
  },
  "measurements": [
    {
      "id": "VB",
      "label": "Base Voltage",
      "source": "Q1.base",
      "unit": "V"
    },
    {
      "id": "VE",
      "label": "Emitter Voltage",
      "source": "Q1.emitter",
      "unit": "V"
    },
    {
      "id": "VC",
      "label": "Collector Voltage",
      "source": "Q1.collector",
      "unit": "V"
    },
    {
      "id": "IC",
      "label": "Collector Current",
      "source": "Q1.collectorCurrent",
      "unit": "mA"
    },
    {
      "id": "VOUT",
      "label": "Output",
      "source": "OUT",
      "unit": "V"
    },
    {
      "id": "FREQ",
      "label": "Oscillation Frequency",
      "value": 22507.9,
      "unit": "Hz"
    }
  ],
  "graphs": [
    {
      "id": "graph-vout",
      "title": "Output Voltage Vout",
      "x": "time",
      "y": "vout",
      "xUnit": "s",
      "yUnit": "V",
      "cursor": true,
      "followTimeline": true
    },
    {
      "id": "graph-tank-current",
      "title": "Resonant Tank Current",
      "x": "time",
      "y": "tankCurrent",
      "xUnit": "s",
      "yUnit": "normalized",
      "cursor": true,
      "followTimeline": true
    }
  ],
  "currentPaths": [
    {
      "id": "collector-current",
      "path": [
        "VCC.positive",
        "L1.1",
        "L1.2",
        "Q1.collector",
        "Q1.emitter",
        "RE.1",
        "RE.2",
        "GND"
      ],
      "signal": "collectorCurrent",
      "particles": true
    },
    {
      "id": "tank-current",
      "path": [
        "L1.2",
        "C1.1",
        "C1.2",
        "C2.1",
        "C2.2",
        "GND"
      ],
      "signal": "tankCurrent",
      "particles": true,
      "bidirectional": true
    }
  ],
  "steps": [
    {
      "time": 0.0,
      "title": "Power ON",
      "description": "The 12 V supply is applied and the transistor bias network begins establishing the operating point."
    },
    {
      "time": 0.8,
      "title": "Transistor Bias",
      "description": "Q1 enters its active operating region. Small noise and transient components appear in the collector circuit."
    },
    {
      "time": 1.8,
      "title": "Resonance Begins",
      "description": "The LC network emphasizes the component near its resonant frequency. Energy begins alternating between capacitor electric fields and the inductor magnetic field."
    },
    {
      "time": 3.0,
      "title": "Positive Feedback",
      "description": "The capacitive divider returns part of the resonant signal with the phase relationship needed to reinforce oscillation."
    },
    {
      "time": 4.5,
      "title": "Oscillation Build-Up",
      "description": "The transistor supplies energy to compensate circuit losses, so the output envelope increases."
    },
    {
      "time": 6.5,
      "title": "Current Reversal",
      "description": "Tank current reverses every half cycle. Current particles reverse direction while capacitor polarity and inductor current change continuously."
    },
    {
      "time": 8.0,
      "title": "Steady-State Oscillation",
      "description": "Losses and transistor nonlinearity limit the amplitude, producing an approximately constant sinusoidal output."
    }
  ],
  "analysis": {
    "equations": [
      "Ceq=C1*C2/(C1+C2)",
      "f0=1/(2*pi*sqrt(L*Ceq))",
      "A(t)=Amax*(1-exp(-t/tau_g))",
      "Vout(t)=A(t)*sin(2*pi*f0*t)"
    ],
    "notes": [
      "The animation uses a simplified educational oscillator model.",
      "The resonant-frequency calculation neglects transistor parasitic capacitance, inductor resistance, loading, and component tolerance.",
      "A practical circuit may require bias and coupling-value adjustment to guarantee startup and stable amplitude.",
      "CB provides the AC ground needed by the common-base feedback model. Vout is the collector AC component after removing the DC bias."
    ]
  },
  "viewport": {
    "width": 690,
    "height": 610
  }
}
\end{scienceanimation}
\section{Energy Exchange}
\[
E_C=\frac{1}{2}C_{eq}v_{ac}^{2},\qquad E_L=\frac{1}{2}Li_L^2
\]
Electric and magnetic energy alternate during a cycle. The transistor replaces losses. A bounded envelope represents start-up and amplitude limiting; this is not a transistor-level SPICE simulation.
\end{document}