CourseWiki/Engineering Fundamentals/Electrical Systems

AC and DC

Current direction, sine-wave voltage, transmission losses, and conversion between AC and DC.

DC flows one way; AC reverses direction

Direct current (DC) flows in one direction. Its magnitude can change without making it AC. Alternating current (AC) reverses direction repeatedly. A battery supplies DC; a wall receptacle supplies AC.

Electronics generally run from a DC supply. That does not mean a transistor cannot handle an AC signal: transistors can amplify changing signals and act as switches. The supply and the signal are different parts of the story.

Start with Current, Voltage, and Resistance for the quantities and equations used here. For measurement, see Measuring Electric Circuits.

A 120 V sine wave peaks near positive and negative 170 V

The nominal 120 V of a US receptacle is an RMS value: root mean square. For a sine wave, V_peak = V_RMS × √2, and V_RMS = V_peak / √2. Thus 120 × √2 = 169.7 V. The instantaneous voltage swings between about +170 V and −170 V, or about 340 V peak to peak.

One cycle of a sine wave: positive peak 169.7 V, negative peak minus 169.7 V, and RMS reference 120 V. RMS is not the waveform's average voltage.

Frequency counts complete cycles per second, in hertz (Hz). A 60 Hz waveform completes 60 whole cycles per second; a 50 Hz waveform completes 50. A cycle returns to the same phase, rather than ending at the opposite peak. US household mains is 60 Hz; many other countries use 50 Hz (DOE, Electricity 101).

Why use RMS instead of the average voltage?

A symmetrical sine wave averages to zero over a whole cycle, yet it still heats a resistor. RMS is the DC voltage that would produce the same average heating in that resistance. Square the instantaneous voltage, average those squares over a cycle, then take the square root. The square-root-of-two conversion applies to a sine wave; do not apply it to an arbitrary inverter waveform.

Compare signed mean, RMS, and ideal-resistor heating; Interactive; Practice

Compare signed mean, RMS, and ideal-resistor heating

A sine can have a signed average of zero over one cycle and still produce heating. Add a DC offset and the signed mean moves with the waveform, while total RMS combines the offset and the changing component. Predict first, then compare the same sine centered on zero, shifted by an offset, kept above zero, or reduced to pure DC.

Print / static comparison: use the analytic waveform v(t) = offset + amplitude × sin(2πft) over one complete cycle. The resistor is ideal and fixed at 100 Ω. These are low illustrative voltages, not mains or wiring examples.

The same 2 V sine amplitude with different DC offsets
Waveform	Signed mean	AC-component RMS	Total RMS	Minimum / maximum	Average resistor power
0 V offset	0 V	1.414 V	1.414 V	−2 / +2 V	20 mW
+1 V offset	+1 V	1.414 V	1.732 V	−1 / +3 V	30 mW
+3 V offset	+3 V	1.414 V	3.317 V	+1 / +5 V	110 mW

The centered sine has zero signed mean but nonzero RMS and average power. The +3 V offset keeps the total waveform positive even though it still contains a sine component. A constant pure-DC value has RMS equal to its magnitude and has no unique oscillation period.

The live explorer evaluates one complete ideal sine cycle analytically; the drawn points only illustrate the equation. For v(t) = V_DC + A sin(2πft), the signed mean is V_DC, the sine component has RMS A/√2, and total RMS is √(V_DC² + A²/2). In one ideal resistor, average power is V_RMS²/R; “RMS power” is not the quantity being calculated.

Why RMS answers the heating question

Analog Devices' RMS explanation shows why ordinary signed or absolute averages do not predict heating for arbitrary waveforms. Its power note distinguishes average power from the RMS value of a power waveform: RMS voltage or current can calculate meaningful average power in the stated load. This classroom explorer uses a purely resistive fixed load; it does not model impedance, power factor, temperature, a meter's coupling mode, or an actual circuit.

Practice zero-mean RMS Practice RMS with offset Practice average resistor power

Higher transmission voltage reduces current and line loss

For the same transmitted power, a higher voltage allows a lower current. The line's resistive heating is P_loss = I²R_line. Halve the current and the loss becomes one quarter, if line resistance stays the same. Here R_line includes the complete current path.

Transformers made it practical to step AC voltage up for transmission and down for use. That advantage helped the AC systems associated with Westinghouse and Tesla prevail over Edison's early DC distribution system in the “war of the currents.” It was not one inventor winning every part of electrical engineering.

A comparison with the same power and the same line

In an ideal DC or unity-power-factor comparison, transmitting 8,000 W at 400 V requires 20 A. At 4,000 V it requires 2 A. If the complete line resistance is 0.5 Ω, the losses are 200 W and 2 W. Ten times the voltage gives one hundredth the resistive loss. This simplified comparison omits converter losses and voltage regulation; it is not a design for a real transmission line. For AC loads with a phase shift, see Power and Efficiency.

The historical advantage belongs to the equipment available at the time. Modern converters also make high-voltage DC transmission practical.

A rectifier is one stage of an AC-to-DC power supply

A rectifier makes current flow in one direction from an AC input. Its output can still pulsate. Filtering smooths the output, and regulation controls the output voltage. A complete power supply may include all these functions as well as isolation.

Linear supply: commonly uses a line-frequency transformer, rectifier, filter, and linear regulator. The regulator dissipates excess voltage as heat.
Switch-mode supply: switches current rapidly and uses energy-storage components to convert voltage efficiently. An AC-input version also rectifies its input. Not every switch-mode converter provides isolation.

Power-supply manufacturer ARCH gives illustrative typical efficiency categories: linear supplies about 50–60%, conventional switching supplies 75–85%, and modern high-efficiency designs 90–95%.

Both lose some input energy as heat. Efficiency depends on the design, input voltage and load; compare published efficiency curves at the intended operating point rather than assuming one percentage for a whole category. See Power and Efficiency for the input/output calculation.

An inverter makes AC from DC; the waveform matters

A pure-sine inverter produces an output close to a sine wave. A modified-sine inverter produces a stepped approximation. Equipment may run differently, make more noise or heat, or fail to operate on a stepped waveform. Check the equipment manufacturer's compatibility guidance; equal RMS voltages do not make the waveforms interchangeable.

For grid-tied PV inverters, efficiency is typically above 95% under most operating conditions, according to Sandia PVPMC’s CEC Inverter Test Protocol. Efficiency depends on AC output power, DC voltage and sometimes temperature.

Every real conversion loses some energy. Several conversions in a row multiply their efficiencies rather than adding them. Much of what we use today runs on DC internally, even when it plugs into AC. Phones, computers, and microcontroller electronics bring the instructor's closing question into focus: who gets the last laugh?

Motors, generation, and energy storage make use of both forms of electrical power.

Appears in

Circuits, AC and DC DET 403 DET