Lecture 22: Energy and Power

What this lecture covers

Up to this point the course has focused on speed: switching times, waveforms, and frequency response. This lecture turns to energy and power, the factor that actually determines the size and weight of battery-powered devices like cell phones and laptops. It builds, from first principles, the tools needed to compute how much energy a digital circuit consumes when it holds a value (standby) and when it switches (active use).

After watching, you can compute the average power dissipated by a MOSFET inverter driven by a periodic input, split that power into its standby and dynamic components, and use a chip's gate count, clock frequency, capacitance, and supply voltage to estimate total power consumption — the same calculation manufacturers use to publish power-per-gate specifications.

Key ideas

• Clock as a time-discretizing abstraction: a square-wave clock lets a sender and receiver agree on when to sample a signal, extending the course's earlier idea of discretizing values (into 0s and 1s) to discretizing time as well.
• Standby power: the power a circuit draws simply from being connected between the supply and ground through resistors, independent of switching.
• Dynamic (active-use) power: the extra power consumed because a capacitor is repeatedly charged and discharged as the circuit switches.
• Energy split during charging: charging a capacitor to VS through a resistor draws C*VS^2 of energy from the source, but only 1/2*C*VS^2 ends up stored on the capacitor — the rest is dissipated in the resistor, regardless of the resistor's value.
• Full discharge dissipates the rest: when a charged capacitor fully discharges through a resistor, all its stored energy (1/2*C*VS^2) is dissipated in that resistor, again independent of the resistor's size.
• Average dynamic power: for a capacitor charged and discharged once per cycle, the average power is C*VS^2*F, where F is the switching frequency — so higher frequency, larger capacitance, or higher supply voltage all increase power draw.
• Inverter power formula: for an inverter with load resistor RL, on-resistance RON, supply VS, load capacitance C, and frequency F (with T1 = T2), average power is approximately VS^2/(2*(RL+RON)) + C*VS^2*F*RL^2, simplifying to VS^2/(2*RL) + C*VS^2*F when RL >> RON.
• Real fabrication numbers: a cited IBM copper-08 process supports around 70-80 million gates per chip at supply voltages of 0.7-1.3 V, with a published power figure of about 6 nanowatts per megahertz per gate — numbers the lecture uses to motivate why low voltage and careful power management matter.

Walkthrough

Review: clocks and discretizing time (1:02)

The lecture opens with a recap of the op-amp oscillator built the previous week, which produces a square wave by repeatedly charging a capacitor and flipping output state. It uses this square wave as a clock: a sender and receiver in a digital system need a shared time base to know when to sample a signal as a 1 or a 0, since a wire only carries voltage levels, not explicit bits. Applying the same clock to both sender and receiver, and agreeing to sample on the rising edge, resolves this ambiguity. This is framed as discretizing time, paralleling the earlier discretization of value into 0s and 1s.

Why energy and power matter (8:10)

The lecture argues that battery size, not the electronics or antenna, is the main driver of handheld device size and weight, which is why early wireless phones were confined to cars with room for a large battery. It cites a real fabrication process (IBM copper-08 / Blue Logic) capable of roughly 70-80 million gates per chip, running at supply voltages of 0.7-1.3 V with a published power dissipation of about 6 nanowatts per megahertz per gate — numbers the lecture promises to let you convert into whole-chip power estimates by the end of the session.

Energy in a simple resistor circuit (17:19)

As a warm-up, the lecture computes power and energy for a voltage source driving a resistor: power is V^2/R, and energy dissipated over time T is power multiplied by time. It then maps this onto the inverter's two states — MOSFET on (current flows through RL and RON, giving power VS^2/(RL+RON)) and MOSFET off (zero current, zero power) — establishing the baseline for standby-style power.

Charging and discharging a capacitor (20:21)

Using a circuit with two switches, two resistors, and a capacitor that alternately charges through R1 and discharges through R2, the lecture derives the energy supplied by the source while charging: integrating instantaneous power gives C*VS^2*(1 - e^(-T1/(R1*C))), which approaches C*VS^2 when the charging time T1 is much larger than the time constant R1*C. Since the capacitor only ends up storing 1/2*C*VS^2, the resistor must dissipate the other half — a result that holds regardless of the resistor's value.

Total energy per cycle and average power (37:30)

The discharge half of the cycle is shown to dissipate the capacitor's full stored energy, 1/2*C*VS^2, in the discharging resistor whenever the discharge time is long compared to its time constant. Adding the charging and discharging losses gives a total of C*VS^2 dissipated per cycle, and dividing by the period T (or multiplying by frequency F) gives an average dynamic power of C*VS^2*F.

Applying the result to the inverter (42:35)

Returning to the original inverter circuit, the lecture states the combined average power formula, VS^2/(2*(RL+RON)) + C*VS^2*F*RL^2, simplifying to VS^2/(2*RL) + C*VS^2*F when RL dominates RON. This separates cleanly into standby power (current flowing from supply to ground through the resistors) and dynamic power (from charging and discharging the load capacitance), matching the two components identified earlier.

Live demonstration and closing numbers (47:42)

A bench demonstration drives an inverter at increasing frequency while a meter shows power draw rising as the switching rate increases, visually confirming the frequency-dependent term. The lecture closes with example numbers — 10^8 gates, 1 GHz frequency, 0.1 femtofarad capacitance, 5 V supply, 10 kiloohm load resistor — and invites you to compute the resulting chip power yourself, previewing that a naive design would draw an impractically large amount of power.

Before you watch

• Be comfortable with RC charging and discharging transients (exponential voltage and current expressions), since the derivation relies on them directly.
• Recall the inverter circuit model with a MOSFET switch, load resistor RL, on-resistance RON, and load capacitance C from earlier lectures in this course.
• Know the basic capacitor energy formula, 1/2*C*V^2, before the lecture uses it to compare source-supplied energy against stored energy.

Check your understanding

1. Why does charging a capacitor to VS through a resistor always dissipate 1/2*C*VS^2 in the resistor, no matter what the resistor's value is?
2. What are the two distinct components of average power in the inverter's power formula, and what physical process does each one correspond to?
3. Why does the average dynamic power of a switched-capacitor circuit increase with switching frequency F?
4. How does using a clock to discretize time solve the ambiguity a receiver faces when reading a sequence of 1s and 0s off a wire?
5. Given a chip's gate count, clock frequency, load capacitance, and supply voltage, how would you estimate its total dynamic power consumption?