Energy Packets · Pulsed to Average — Voltage Regulation

Along the way

1
Power is joules per second. Instantaneous power \(P\) is how fast energy is being delivered. At constant \(P\) for time \(t\), energy is the rectangle \(E = P\,t\) (area under the power curve).
2
One packet. A pulse of height \(P_{\mathrm{hi}}\) lasting \(T_{\mathrm{on}}\) carries \(E_{\mathrm{pkt}} = P_{\mathrm{hi}}\,T_{\mathrm{on}}\). Taller and narrower, or shorter and wider — equal area means equal joules.
3
From packet to average. Repeat the pulse every period \(T\). The average power over a period is \(P_{\mathrm{avg}} = E_{\mathrm{pkt}}/T = P_{\mathrm{hi}}\,D\), where duty \(D = T_{\mathrm{on}}/T\). That is the “pulsed → average” step.
4
Pulse train vs steady line. Continuous delivery at constant \(P_{\mathrm{avg}}\) puts the same energy into each full period as one pulse packet (\(E = P_{\mathrm{avg}}\,T\)). On the energy chart both share the same long-term slope — the average rate.
5
Why this matters next. Part 2 “made” a lower voltage by burning power in a resistor. Packet thinking is different: move energy in chunks, then smooth later. Real converters open and close paths to do that — that story comes after this page.
6
Faster rate, smaller packets. Raise the pulse rate \(f = 1/T\) while holding \(P_{\mathrm{avg}}\) fixed: each \(E_{\mathrm{pkt}}\) shrinks. Same average power, finer grain in time.

Ideal rectangular pulses only — no switch model, no inductor, no regulator IC. The lab is pure \(P(t)\) and energy.

Next: devices and topologies that actually route those packets — without using a series resistor as the “converter.”

Match the areas · pulsed vs average

Duty from areas: \(D = P_{\mathrm{avg}}/P_{\mathrm{hi}}\) · packet \(E_{\mathrm{pkt}} = P_{\mathrm{hi}}T_{\mathrm{on}}\) · steady line at \(P_{\mathrm{avg}}\)

Average power \(P_{\mathrm{avg}}\) (W) Pulse height \(P_{\mathrm{hi}}\) (W) Pulse rate \(f\) (kHz)

The lab keeps the areas matched: \(D = P_{\mathrm{avg}}/P_{\mathrm{hi}}\) (never above 100% on). Raise \(P_{\mathrm{hi}}\) → taller, narrower pulses. Raise \(f\) → more packets in the fixed time window; each packet is smaller.

Instantaneous power · shaded pulse area = energy in that packet

Accumulated energy · steps (pulsed) vs ramp (steady) · same average rate

P_avg—

P_hi—

Duty D—

T_on—

Period T—

E_packet—

\[ E_{\mathrm{pkt}} = P_{\mathrm{hi}}\,T_{\mathrm{on}} \]

→ —

\[ P_{\mathrm{avg}} = E_{\mathrm{pkt}}/T = P_{\mathrm{hi}}\,D \]

→ —

\[ D = T_{\mathrm{on}}/T = P_{\mathrm{avg}}/P_{\mathrm{hi}} \]

→ —

Over one full period, continuous delivery at \(P_{\mathrm{avg}}\) also delivers \(E = P_{\mathrm{avg}}\,T\) — the same joules as one pulse packet. Mid-window the step curve can sit slightly above or below the ramp; over many periods the averages match.

Try it: (1) Raise \(P_{\mathrm{hi}}\) — pulses get taller and thinner; the average line holds. (2) Raise \(f\) — more, smaller packets in the same window; average rate unchanged. (3) Drop \(P_{\mathrm{hi}}\) toward \(P_{\mathrm{avg}}\) — duty approaches 100% and the pulse train looks like the steady line.

Analog Intuition

Energy packets · pulsed to average

Match the areas · pulsed vs average