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Tutoring abilities

What if an environmental sensor could run for twenty years inside a bridge or a jet engine without ever changing a battery?

By capturing ambient light, thermal gradients, and mechanical vibrations, **micro-power energy harvesting** converts tiny trickle flows of wasted environmental energy into usable electrical power.

## Photovoltaic Harvesting: Light into Volts

A **photovoltaic (PV) cell** converts photon flux directly into electrical energy via the photoelectric effect, producing a power output proportional to light intensity and active surface area.

The available input power depends on **solar irradiance** $G$ (measured in $\text{W/m}^2$, where indoor lighting is roughly $5\text{ to }10\text{ W/m}^2$ and direct outdoor sun is $\approx 1000\text{ W/m}^2$).

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      color: #1e2945;
      background: #ffffff;
      padding: 12px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      min-height: 100%;
    }

    .controls {
      display: flex;
      gap: 8px;
      width: 100%;
      max-width: 480px;
      justify-content: center;
      margin-bottom: 8px;
    }

    .btn {
      flex: 1;
      padding: 8px 12px;
      border-radius: 8px;
      border: 1px solid #e2e8f0;
      background: #f8fafc;
      color: #1e2945;
      font-size: 14px;
      font-weight: 600;
      cursor: pointer;
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 6px;
      transition: background 0.2s, border-color 0.2s, box-shadow 0.2s;
    }

    .btn.active {
      background: #22b7ff;
      color: #ffffff;
      border-color: #22b7ff;
      box-shadow: 0 2px 8px rgba(34, 183, 255, 0.3);
    }

    .graph-container {
      width: 100%;
      max-width: 480px;
      position: relative;
    }

    svg {
      width: 100%;
      height: auto;
      display: block;
      overflow: visible;
    }

    .axis-line { stroke: #94a3b8; stroke-width: 1.5; }
    .grid-line { stroke: #f1f5f9; stroke-width: 1; }
    .axis-label { fill: #1e2945; font-size: 13px; font-weight: 600; }
    .tick-label { fill: #64748b; font-size: 11px; }

    .power-rect {
      fill: rgba(16, 185, 129, 0.18);
      stroke: #10b981;
      stroke-width: 1.5;
      stroke-dasharray: 4 2;
      transition: all 0.4s ease;
    }

    .iv-curve {
      fill: none;
      stroke: #22b7ff;
      stroke-width: 3.5;
      stroke-linecap: round;
      transition: d 0.4s ease;
    }

    .mpp-dot {
      fill: #ef4444;
      stroke: #ffffff;
      stroke-width: 2.5;
      transition: cx 0.4s ease, cy 0.4s ease;
    }

    .mpp-tag {
      font-size: 12px;
      font-weight: 700;
      fill: #ef4444;
      transition: x 0.4s ease, y 0.4s ease;
    }

    .badge-bar {
      display: flex;
      justify-content: space-between;
      width: 100%;
      max-width: 480px;
      margin-top: 6px;
      padding: 6px 12px;
      background: #f8fafc;
      border-radius: 8px;
      font-size: 13px;
      font-weight: 600;
    }

    .metric {
      display: flex;
      align-items: center;
      gap: 4px;
    }
    .metric-val {
      color: #10b981;
      font-weight: 700;
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</head>
<body>

  <div class="controls">
    <button id="btnIndoor" class="btn" onclick="setMode('indoor')">
      💡 Indoor <span style="font-size:11px;opacity:0.85;">(10 W/m²)</span>
    </button>
    <button id="btnOutdoor" class="btn active" onclick="setMode('outdoor')">
      ☀️ Outdoor <span style="font-size:11px;opacity:0.85;">(1000 W/m²)</span>
    </button>
  </div>

  <div class="graph-container">
    <svg viewBox="0 0 460 260">
      
      <line x1="50" y1="210" x2="430" y2="210" class="axis-line" />
      <line x1="50" y1="30" x2="50" y2="210" class="axis-line" />
      
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      <line x1="240" y1="30" x2="240" y2="210" class="grid-line" />
      <line x1="335" y1="30" x2="335" y2="210" class="grid-line" />
      <line x1="430" y1="30" x2="430" y2="210" class="grid-line" />

      <line x1="50" y1="165" x2="430" y2="165" class="grid-line" />
      <line x1="50" y1="120" x2="430" y2="120" class="grid-line" />
      <line x1="50" y1="75" x2="430" y2="75" class="grid-line" />

      
      <text x="50" y="226" text-anchor="middle" class="tick-label">0</text>
      <text x="145" y="226" text-anchor="middle" class="tick-label">0.2</text>
      <text x="240" y="226" text-anchor="middle" class="tick-label">0.4</text>
      <text x="335" y="226" text-anchor="middle" class="tick-label">0.6</text>
      <text x="430" y="226" text-anchor="middle" class="tick-label" id="xMaxLabel">0.8V</text>

      <text x="42" y="214" text-anchor="end" class="tick-label">0</text>
      <text x="42" y="124" text-anchor="end" class="tick-label" id="yMidLabel">20</text>
      <text x="42" y="34" text-anchor="end" class="tick-label" id="yMaxLabel">40</text>

      
      <rect id="powerRect" class="power-rect" x="50" y="55" width="280" height="155" />

      
      <path id="ivCurve" class="iv-curve" d="" />

      
      <circle id="mppDot" class="mpp-dot" cx="330" cy="55" r="6" />
      <text id="mppTag" class="mpp-tag" x="330" y="42" text-anchor="middle">MPP</text>

      
      <text x="240" y="250" text-anchor="middle" class="axis-label">Voltage (V)</text>
      <text x="18" y="120" text-anchor="middle" class="axis-label" transform="rotate(-90 18 120)" id="yAxisTitle">Current (mA/cm²)</text>
    </svg>
  </div>

  <div class="badge-bar">
    <div class="metric"><span>V<sub>mp</sub>:</span> <span id="vmpVal" class="metric-val">0.50 V</span></div>
    <div class="metric"><span>I<sub>mp</sub>:</span> <span id="impVal" class="metric-val">32.0 mA</span></div>
    <div class="metric"><span>P<sub>max</sub>:</span> <span id="pmaxVal" class="metric-val">16.0 mW</span></div>
  </div>

  <script>
    const data = {
      outdoor: {
        voc: 0.62,
        isc: 36.0,
        vmp: 0.50,
        imp: 32.0,
        pmax: '16.0 mW/cm²',
        vmpStr: '0.50 V',
        impStr: '32.0 mA/cm²',
        yMid: '20',
        yMax: '40',
        yUnit: 'Current (mA/cm²)',
        yScaleMax: 40,
        xScaleMax: 0.8
      },
      indoor: {
        voc: 0.45,
        isc: 0.35,
        vmp: 0.36,
        imp: 0.30,
        pmax: '108 µW/cm²',
        vmpStr: '0.36 V',
        impStr: '300 µA/cm²',
        yMid: '0.2',
        yMax: '0.4',
        yUnit: 'Current (mA/cm²)',
        yScaleMax: 0.4,
        xScaleMax: 0.8
      }
    };

    function vToX(v, xMax) {
      return 50 + (v / xMax) * 380;
    }

    function iToY(i, yMax) {
      return 210 - (i / yMax) * 180;
    }

    function calculateCurvePath(modeKey) {
      const d = data[modeKey];
      const pts = [];
      const steps = 40;
      const I0 = d.isc * Math.exp(-d.voc / 0.045);
      
      for (let s = 0; s <= steps; s++) {
        const v = (s / steps) * d.voc;
        const current = Math.max(0, d.isc - I0 * (Math.exp(v / 0.045) - 1));
        const x = vToX(v, d.xScaleMax);
        const y = iToY(current, d.yScaleMax);
        pts.push(`${s === 0 ? 'M' : 'L'} ${x.toFixed(1)} ${y.toFixed(1)}`);
      }
      return pts.join(' ');
    }

    function setMode(mode) {
      document.getElementById('btnIndoor').classList.toggle('active', mode === 'indoor');
      document.getElementById('btnOutdoor').classList.toggle('active', mode === 'outdoor');

      const d = data[mode];
      const xMp = vToX(d.vmp, d.xScaleMax);
      const yMp = iToY(d.imp, d.yScaleMax);
      const originX = 50;
      const originY = 210;

      // Update Power Rect
      const rect = document.getElementById('powerRect');
      rect.setAttribute('x', originX);
      rect.setAttribute('y', yMp);
      rect.setAttribute('width', xMp - originX);
      rect.setAttribute('height', originY - yMp);

      // Update Curve
      document.getElementById('ivCurve').setAttribute('d', calculateCurvePath(mode));

      // Update MPP Dot & Label
      const dot = document.getElementById('mppDot');
      dot.setAttribute('cx', xMp);
      dot.setAttribute('cy', yMp);

      const tag = document.getElementById('mppTag');
      tag.setAttribute('x', xMp);
      tag.setAttribute('y', yMp - 12);

      // Update Axes Labels
      document.getElementById('yMidLabel').textContent = d.yMid;
      document.getElementById('yMaxLabel').textContent = d.yMax;
      document.getElementById('yAxisTitle').textContent = d.yUnit;

      // Update Metrics
      document.getElementById('vmpVal').textContent = d.vmpStr;
      document.getElementById('impVal').textContent = d.impStr;
      document.getElementById('pmaxVal').textContent = d.pmax;
    }

    // Init
    setMode('outdoor');
  </script>
</body>
</html>

The electrical power generated at the maximum power point is given by: $$P_{\text{out}} = G \cdot A \cdot \eta$$ where $A$ is the cell surface area in $\text{m}^2$ and $\eta$ is the cell conversion efficiency (typically $15\text{--}22\%$ for monocrystalline silicon).

Let's calculate the harvesting power for an indoor sensor using a small $4\text{ cm} \times 5\text{ cm}$ monocrystalline cell ($A = 0.002\text{ m}^2$) under office lighting ($G = 8\text{ W/m}^2$) with efficiency $\eta = 0.18$.

$$P_{\text{out}} = 8\text{ W/m}^2 \times 0.002\text{ m}^2 \times 0.18 = 0.00288\text{ W} = 2.88\text{ mW}$$ This provides plenty of energy for a low-power Bluetooth beacon duty-cycling at $50\,\mu\text{W}$.

What if there is no light, but a pipe or machine casing runs hot?

## Thermoelectric Harvesting: Heat Flow to Electricity

In 1821, Thomas Johann Seebeck discovered that a temperature difference between two dissimilar electrical conductors produces a voltage — known today as the **Seebeck effect**.

A **thermoelectric generator (TEG)** stacks dozens of p-type and n-type semiconductor couples electrically in series and thermally in parallel between hot and cold ceramic plates.

<!DOCTYPE html>
<html lang="en">
<head>
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  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <style>
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    html, body {
      width: 100%;
      height: 100%;
      background: #ffffff;
      font-family: system-ui, -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, sans-serif;
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      flex-direction: column;
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      display: flex;
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      gap: 10px;
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    /* SVG Canvas */
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      width: 100%;
      height: auto;
      max-height: 400px;
      display: block;
      overflow: visible;
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      background: #f1f5f9;
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      border-radius: 20px;
      border: 1px solid #cbd5e1;
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      font-size: 13px;
      font-weight: 600;
      color: #475569;
      border-radius: 16px;
      cursor: pointer;
      transition: background-color 0.2s ease, color 0.2s ease;
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      background: #e2e8f0;
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    /* Animations */
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      to { stroke-dashoffset: 0; }
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    @keyframes carrierDrift {
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      75% { opacity: 1; }
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    /* State visibility */
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The open-circuit voltage $V_{\text{oc}}$ generated across $N$ thermocouple pairs with a Seebeck coefficient $\alpha$ ($\,\mu\text{V/K}$) subject to a temperature difference $\Delta T = T_{\text{hot}} - T_{\text{cold}}$ is: $$V_{\text{oc}} = N \cdot \alpha \cdot \Delta T$$

Because the TEG has an internal electrical resistance $R_{\text{int}}$, the **Maximum Power Transfer Theorem** states that maximum power is extracted when the load resistance equals the internal resistance ($R_L = R_{\text{int}}$).

Under matched load conditions ($V_L = V_{\text{oc}} / 2$), the maximum harvestable power is: $$P_{\text{max}} = \frac{V_{\text{oc}}^2}{4 R_{\text{int}}}$$

Consider a 127-couple TEG module ($N = 127$, $\alpha = 200\,\mu\text{V/K}$) mounted on an industrial motor casing where $\Delta T = 15\text{ K}$ and internal resistance $R_{\text{int}} = 2.5\,\Omega$.

First calculate the open-circuit voltage: $$V_{\text{oc}} = 127 \times (200 \times 10^{-6}\text{ V/K}) \times 15\text{ K} = 0.381\text{ V}$$

Then calculate the peak output power under matched impedance: $$P_{\text{max}} = \frac{(0.381\text{ V})^2}{4 \times 2.5\,\Omega} = \frac{0.14516}{10} = 14.52\text{ mW}$$

What if an environment has neither excess light nor heat, but experiences constant mechanical shake or footsteps?

## Piezoelectric Harvesting: Kinetic Strain to Charge

Discovered in 1880 by Pierre and Jacques Curie, the **piezoelectric effect** generates an electric polarization charge when a crystal or ceramic material experiences mechanical strain.

A cantilever beam with a piezoelectric element (such as PZT ceramic) flexes with ambient vibration, creating a mechanical charge: $$Q = d_{33} \cdot F$$ where $d_{33}$ is the piezoelectric charge coefficient (in $\text{C/N}$) and $F$ is the dynamic force applied.

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Electrically, a piezoelectric element acts as an alternating current source in parallel with an internal capacitance $C_p$. The electrostatic energy stored per vibration deflection cycle is: $$E_{\text{cycle}} = \frac{1}{2} C_p V_{\text{pk}}^2 = \frac{Q^2}{2 C_p}$$

If the structure vibrates at resonance frequency $f$, the total raw kinetic power generated is: $$P_{\text{piezo}} = E_{\text{cycle}} \cdot f = \frac{1}{2} C_p V_{\text{pk}}^2 f$$

Let's calculate power from a piezoelectric cantilever with internal capacitance $C_p = 25\text{ nF}$ vibrating at $f = 60\text{ Hz}$ producing peak open-circuit voltage spikes of $V_{\text{pk}} = 12\text{ V}$.

$$P_{\text{piezo}} = 0.5 \times (25 \times 10^{-9}\text{ F}) \times (12\text{ V})^2 \times 60\text{ Hz} = 1.8 \times 10^{-6} \times 60 = 108\,\mu\text{W}$$

Why can't we simply wire these raw transducers straight into a standard $3.3\text{ V}$ microcontroller?

## Power Conditioning and Rectification Losses

Transducers produce raw outputs incompatible with digital logic: TEGs output sub-$400\text{ mV}$ DC, piezoelectric elements output high-voltage AC spikes, and PV cells drift with changing lux levels.

A **power conditioning circuit** combines rectification, an ultra-low-power boost converter with Maximum Power Point Tracking (MPPT), and an energy storage element like a supercapacitor.

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A standard diode bridge rectifier consumes two forward diode drops ($2 V_F \approx 2 \times 0.6\text{ V} = 1.2\text{ V}$), which could wipe out $50\text{--}90\%$ of micro-power harvested from low-voltage transducers.

Dedicated energy harvesting PMICs (Power Management ICs) instead use **active MOSFET rectification** and synchronous boost topologies with cold-start charge pumps that start operating from inputs as low as $20\text{ mV}$.

The overall net usable continuous power delivered to your storage capacitor ($P_{\text{stored}}$) accounts for conversion efficiency $\eta_{\text{conv}}$ and PMIC quiescent power loss $P_Q$: $$P_{\text{stored}} = (P_{\text{gen}} \cdot \eta_{\text{conv}}) - P_Q$$

A common mistake is multiplying a transducer's open-circuit voltage directly by its short-circuit current ($P = V_{\text{oc}} \cdot I_{\text{sc}}$) to estimate power.

At $V_{\text{oc}}$ the current is zero, and at $I_{\text{sc}}$ the terminal voltage is zero — actual peak power delivers only about $25\%$ to $75\%$ of that product depending on the transducer's **fill factor**.

When designing an energy-harvesting node, match your sensor sleep interval so the average continuous power consumed is always less than $P_{\text{stored}}$.

electronics

Energy Harvesting Transducers and Power Conditioning. Calculate micro-power energy harvesting generation from photovoltaic cells, piezoelectric elements, and TEGs.

AQA A-Level Physics: Electrical Energy & Power Revision Notes

Isaac Physics: EMF, Internal Resistance and Maximum Power Transfer

Introduction to Energy Harvesting Transducers and Power Conditioning Circuits

Teaching this 12th-grade lesson successfully requires bridging foundational A-Level circuit equations ($P = VI = I^2 R = \frac{V^2}{R}$ and $E = Pt$) with practical micro-power electronic systems. Emphasise order-of-magnitude conversions (mW, $\mu$W, nW, and $\mu$A) as these scale factors represent real-world Internet of Things (IoT) hardware constraints. Use the Maximum Power Transfer Theorem ($R_L = R_{int}$) as the unifying analytical tool across all three transducer types (PV, Piezo, TEG) to consolidate student mastery of internal resistance and load matching.