Power, efficiency and DC power systems

Circuit analysis is often taught as a way of finding voltages and currents, but in practice the question is frequently about power: how much reaches the load, how much is wasted as heat, and how to deliver large amounts of it efficiently. This lesson applies the tools of the course to DC power systems. We distinguish maximum power transfer from maximum efficiency, model batteries and their internal resistance, see why data centers are moving to higher distribution voltages inside the rack, compare linear and switching regulators, and combine the efficiencies of a chain of converters into the figure that decides an electricity bill.

Maximum power or maximum efficiency?

Lesson 6 showed that a source with Thévenin equivalent \(V_{th}\), \(R_{th}\) delivers maximum power to a load when \(R_{L} = R_{th}\). The efficiency of power delivery, the fraction of the source's power that reaches the load, is

At maximum power transfer, \(\eta = 50\%\): half the power is wasted in the source. That is acceptable, even desirable, in communications and sensing, where the signal is weak and the power involved tiny: an antenna, a microphone or a radio-frequency amplifier is matched to its load to extract every possible microwatt. It is unacceptable in power systems, where efficiency is everything. Utilities, batteries and power supplies are designed with \(R_{th} \ll R_{L}\), so that efficiency is high and the voltage stays nearly constant as the load changes.

Batteries as circuit elements

A battery is well modeled for many purposes by its Thévenin equivalent: an open-circuit voltage (which depends on its state of charge and temperature) in series with an internal resistance. Under load, the terminal voltage falls by the voltage across the internal resistance, and the internal resistance dissipates power as heat inside the battery.

Internal resistance rises as a battery ages, gets cold or discharges, which is why a weak car battery may show a normal voltage when idle and then collapse when the starter motor draws hundreds of amperes. Measuring internal resistance (or its AC counterpart, impedance) is the standard way to assess the health of UPS batteries (DCE-340).

Cells are combined in series to raise voltage and in parallel to raise capacity (ELEC-110, Lesson 8). For \(n\) identical cells in series, the open-circuit voltages and internal resistances add; for \(m\) identical strings in parallel, the internal resistance divides by \(m\).

Why distribution voltage matters

For a given power, current is inversely proportional to voltage, and conductor loss is proportional to current squared. Raising the distribution voltage by a factor of four cuts the current by four and the conductor loss by sixteen. This simple consequence of \(P = VI\) and \(P = I^{2}R\) explains why power grids transmit at hundreds of kilovolts, and why server racks are moving from 12 V to 48 V internal distribution as their power rises.

R_path = 0.5e-3        # total resistance of the rack power path (busbar, connectors), ohms
print("rack kW   bus V   current A   path loss W   loss %")
for kw in [12, 40]:
    for v in [12, 48, 380]:
        i = kw * 1000 / v
        loss = i**2 * R_path
        print(f"{kw:>7} {v:>7} {i:>11.0f} {loss:>13.1f} {100 * loss / (kw * 1000):>8.2f}")

rack kW   bus V   current A   path loss W   loss %
     12      12        1000         500.0     4.17
     12      48         250          31.2     0.26
     12     380          32           0.5     0.00
     40      12        3333        5555.6    13.89
     40      48         833         347.2     0.87
     40     380         105           5.5     0.01

At 12 kW, a 12 V bus must carry 1,000 A and loses 4% of the power in just half a milliohm of busbar and connectors. At 48 V the loss is 0.26%. For AI racks of 40 kW and more, 12 V distribution would be impractical, which is why the Open Compute Project's Open Rack designs moved to 48 V, and why the industry is now developing distribution at several hundred volts DC (such as ±400 V) for racks approaching a megawatt.

Linear and switching regulators

Electronic loads need precisely regulated voltages, such as 3.3 V, 1.8 V or under 1 V for a processor core, derived from a higher supply. Two approaches exist:

• A linear regulator acts as an automatically adjusted series resistance: a transistor drops the excess voltage, controlled by an op amp feedback loop comparing the output with a reference (Lessons 8 and 9). The input and output currents are almost equal, so its efficiency is at best \(V_{out}/V_{in}\). It is simple, quiet and cheap, but wasteful when the voltage difference is large.
• A switching regulator, such as the buck converter, switches a transistor on and off rapidly and uses an inductor and capacitor to smooth the result (ELE-470). Ideally it wastes nothing; real switching regulators achieve 85% to 97% efficiency across a wide range of voltages.

Efficiency chains

Power from the utility reaches a processor through a chain of conversions, and the overall efficiency is the product of the stage efficiencies. Every percentage point lost becomes heat that the cooling system must also remove.

In the US, the Department of Energy sets minimum efficiency requirements for external power supplies and distribution transformers, and ENERGY STAR specifications cover servers and UPS systems.