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Circuits, AC and DC

AC and DC power, wire and wire gauge, series and parallel circuits, and measuring circuits.

  1. AC and DC
  2. Wire and Wire Gauge
  3. Series and Parallel Circuits
  4. Measuring Electric Circuits
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

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

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, Vpeak = VRMS × √2, and VRMS = Vpeak / √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).

Supporting detail: Addendum 1 — Why use RMS instead of the average voltage?.

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
WaveformSigned meanAC-component RMSTotal RMSMinimum / maximumAverage resistor power
0 V offset0 V1.414 V1.414 V−2 / +2 V20 mW
+1 V offset+1 V1.414 V1.732 V−1 / +3 V30 mW
+3 V offset+3 V1.414 V3.317 V+1 / +5 V110 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) = VDC + A sin(2πft), the signed mean is VDC, the sine component has RMS A/√2, and total RMS is √(VDC² + A²/2). In one ideal resistor, average power is VRMS²/R; “RMS power” is not the quantity being calculated.

Supporting detail: Addendum 2 — Why RMS answers the heating question.

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

Higher transmission voltage reduces current and line loss

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 Ploss = I2Rline. Halve the current and the loss becomes one quarter, if line resistance stays the same. Here Rline 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.

Supporting detail: Addendum 3 — A comparison with the same power and the same line.

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

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

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.

Engineering Fundamentals/Electrical Systems

Wire and Wire Gauge

Separate the conductor from its insulation, and understand what a wire gauge does and does not tell you.

The conductor carries current; insulation separates conductors

The conductor carries current; insulation separates conductors

The metal inside a wire is the conductor. Its resistance depends on material, length, cross-sectional area and temperature. The covering is insulation: it keeps the conductor from contacting other conductors or surrounding objects. The outside diameter includes that covering and does not by itself tell you how much metal is inside.

Solid wire has one solid conductor; stranded wire has several smaller strands forming a conductor. Insulation type and condition matter separately from the metal size. See Electrical Safety for use and inspection guidance.

A smaller AWG number means a thicker conductor

A smaller AWG number means a thicker conductor

American Wire Gauge (AWG) uses smaller numbers for larger conductor cross sections. For the same material, length and temperature, a thicker conductor has less resistance. See Adafruit’s wire-gauge explanation. Under comparable installation and temperature conditions, it can carry more current without exceeding its temperature limit.

Ampacity means the allowable current under specified conditions. Gauge alone is not an ampacity rating: material, insulation temperature rating, ambient temperature, bundling and installation all matter. Use the applicable manufacturer data and installation requirements for a real selection; this page is not a wire-sizing table. IEWC explains these ampacity factors. Choosing a wire involves voltage drop as well as allowable current.

Supporting detail: Addendum 4 — Why a long wire can matter even below its current limit.

Count the return wire: explore voltage drop and heating; Interactive; Practice

Count the return wire: explore voltage drop and heating

A two-conductor circuit carries current out and back. Use twice the one-way run for its wire resistance. At the same current, longer or thinner wire drops more voltage and wastes more power as heat. Voltage drop is a separate check from ampacity: get the current rating from the applicable code table or manufacturer for the insulation and installation.

Out: 50 ftReturn: 50 ftSourceLoadLoop = 30.48 mAWG 18 · 5 A
AWG 18, 50 ft one way, 5 A, 24 V source
Single wire: Ω / 1000 ft6.385
Single wire: Ω / km20.95
Loop resistance (Ω)0.6385
Voltage drop (V)3.192
Source voltage lost (%)13.3
Load voltage (V)20.81
Wire heating loss (W)15.96

Worked example: 50 ft out + 50 ft back = 30.48 m of copper. R = ρL/A = 0.6385 Ω. At 5 A, IR = 3.192 V and I²R = 15.96 W.

Voltage drop: baseline 3.192 V; now 6.385 V
Wire heating loss: baseline 15.96 W; now 63.85 W

Outline = baseline; filled = now. Each pair shares its own scale; compare within a pair. Loss is heating power, not wire temperature.

Doubling only current doubles the drop and quadruples heating loss. Doubling only length doubles both; a smaller AWG number lowers both at the same current.

This model assumes solid copper at 20 °C, equal outgoing and return wires, and DC/resistive behavior. Current is an input, not a prediction of how a particular load responds. Heating loss is not a temperature prediction; copper resistance rises with temperature (approx. 0.393%/°C near 20 °C). Connections and AC reactance are omitted.

Supporting detail: Addendum 5 — Definitions used to calculate every result.

Practice: Loop resistance · Voltage drop · Percent drop · Load voltage · Choose a gauge.

Engineering Fundamentals/Electrical Systems

Series and Parallel Circuits

Trace current paths, combine resistances, and explore how opening a switch changes voltage, current and power.

Series has one path; parallel has separate branches

Series has one path; parallel has separate branches

In a series circuit, the same current passes through every element. The voltage drops around the loop add up to the source voltage. One break stops current everywhere in that loop.

In a parallel circuit, each branch connects to the same two supply nodes (a node is an electrically connected junction), so each complete branch has the same voltage. The branch currents add up to the supply current. A break in one branch leaves the other branches working, provided the source and shared wiring remain intact.

Comparison of two resistors in one series loop and two resistors in separate parallel branches across a DC source.

Equal resistance gives equal drops in series and equal currents in parallel. Different resistances make the drops or currents different. The formulas for each individual resistor are on Current, Voltage, and Resistance.

Add resistances in series; add their reciprocals in parallel

Add resistances in series; add their reciprocals in parallel

Equivalent resistance is the single resistance that would draw the same total current at the supply terminals.

  • Series: Req = R1 + R2 + R3.
  • Parallel: 1/Req = 1/R1 + 1/R2 + 1/R3. A reciprocal is 1 ÷ R: one divided by the resistance. Include only connected branches.
  • Two parallel resistors: Req = R1R2 / (R1 + R2).

Check the result: series resistance is greater than any one positive resistor; parallel resistance is smaller than the smallest connected resistor when two or more branches are connected. An open circuit has no conducting path and is represented by infinite equivalent resistance.

Supporting detail: Addendum 6 — Why the parallel formula uses reciprocals.

Change a resistor or open a switch and predict what changes; Interactive

Change a resistor or open a switch and predict what changes

Use this ideal DC model to compare two or three fixed resistors. Before moving a control, predict which current, voltage and power will change. Switches are ideal, wires have zero resistance and the source holds its set voltage.

Static example: a 15 V source and 30 Ω and 60 Ω resistors, with both switches closed.

The same resistors in two connections
ConnectionEquivalent resistanceTotal currentR1: current / voltage / powerR2: current / voltage / power
Series90 Ω0.1667 A0.1667 A / 5 V / 0.8333 W0.1667 A / 10 V / 1.667 W
Parallel20 Ω0.75 A0.5 A / 15 V / 7.5 W0.25 A / 15 V / 3.75 W

Opening either series switch makes both resistor currents and both resistor voltage drops zero. Opening the R1 parallel switch makes R1 current and power zero while R2 still draws 0.25 A. The voltage across the whole open branch remains 15 V; it falls across the open switch, not the resistor.

The live table reports voltage across each resistor, excluding its switch. In an open series loop the source voltage is across the break or breaks. With several open switches, this ideal model does not determine how that voltage divides between the breaks.

Power explains the bulb comparison, within the model; Practice

Power explains the bulb comparison, within the model

For fixed resistors in series, the larger resistance dissipates more power because both carry the same current. In parallel, the smaller resistance dissipates more power because both have the same voltage. Check the power column rather than assuming the same bulb wins in both arrangements.

Greater electrical power is a useful brightness comparison for otherwise comparable incandescent bulbs. Real filaments change resistance as they heat, and LED lamps contain additional circuitry. The fixed-resistor model is not a universal lamp-brightness predictor.

Build another circuit in PhET's Circuit Construction Kit: DC – Virtual Lab. Try an unequal pair of resistors, predict the effect of a break, and then test your prediction.

Add a series-current problem to practice

Engineering Fundamentals/Electrical Systems

Measuring Electric Circuits

Connect voltage, current, resistance and continuity measurements to the part of the circuit they describe.

Voltage is measured across; current is measured through

Voltage is measured across; current is measured through

A voltmeter measures the potential difference between two points. Connect it across the component, in parallel. A series ammeter becomes part of the current path: open the path and insert it in series with the element whose current you want.

A DC source supplies an ammeter and resistor in series. A voltmeter connects across the resistor, on two separate parallel leads.

A meter set for current has a low-resistance input; connecting that input across a source makes a short circuit. Read Electrical Safety before making physical measurements. The diagrams here explain connections, not a procedure for working on mains wiring.

Supporting detail: Addendum 7 — Real meters change the circuit a little.

Place both voltage probes, predict the sign, then reveal; Interactive; Practice

Place both voltage probes, predict the sign, then reveal

A DC voltmeter reports an ordered difference: V(red) − V(black). The two probes must touch the two named points you want to compare. Reversing them reverses the sign; putting both on the same conductor gives zero.

Static example: an ideal 12 V source supplies 30 Ω and 60 Ω resistors in series. Each resistor has an ideal switch immediately before it.

Signed readings from the same ideal circuit
Circuit stateRed pointBlack pointReadingWhy
S1 openSupply +After S1+12 VThe single open switch takes the source voltage; no current flows.
Both closedAfter S1After R1+4 VCurrent is 12/(30+60) = 0.1333 A; R1 drops 4 V.
Both closed, probes reversedAfter R1After S1−4 VThe two point potentials are unchanged; only the ordered subtraction reverses.
S2 openAfter R1After S2+12 VThe upstream side stays at +12 V and the downstream side returns to 0 V.
Any supported stateOne named pointThe same point0 VA point cannot have a potential difference from itself.

With two open switches, the conductor between them floats in this ideal model. Its voltage is not assigned. The interactive intentionally offers no two-open state.

This is a low-voltage ideal-circuit model with an infinite-impedance voltmeter. It does not authorize live work, mains measurements, or current-mode probing. Fluke's DC-voltage guide explains the V and COM inputs and the negative sign from reversed DC probes; its 381 meter manual shows that voltage probes touch the two desired test points.

Practice choosing probe points Practice open-switch readings

A shunt or clamp lets you measure current indirectly

A shunt or clamp lets you measure current indirectly

A shunt is a known small resistance in the current path. Measure the voltage across it and use the relationship in Current, Voltage, and Resistance to find current. The shunt is in series; the voltage measurement across it is in parallel. Many ammeters contain a shunt internally. An installed shunt avoids opening the circuit each time you read it; adding one still requires changing the path.

A clamp meter senses the magnetic field around a conductor without breaking the circuit. Clamp around one conductor, not the outgoing and returning conductors together: their fields can cancel. The instrument must support the type of current being measured; an AC-only clamp does not measure steady DC.

Resistance and continuity are measured with the circuit unpowered

Resistance and continuity are measured with the circuit unpowered

An ohmmeter applies its own small test signal. Use it on an unpowered circuit, with stored energy discharged as described in Electrical Safety. Other connected paths can change the result; isolating the component lets you measure its resistance instead of the whole network.

Continuity is a resistance-threshold test. The beep means the measured resistance is below that meter's threshold. It does not prove exactly zero resistance, identify the intended path, or establish that a connection can carry its working current. The threshold comes from the meter's manual.

A multimeter combines voltage, current and resistance functions, often with continuity too. Select the function, AC or DC mode, range and lead sockets for the measurement. In a schematic, follow the actual connection points rather than the physical positions of the symbols.

Supporting detail

Addendum

Additional evidence, standards context, and background referenced from the main text.

  1. Engineering Fundamentals/Electrical Systems/AC and DC

    Addendum 1. 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.

    Back to reference 1

  2. Engineering Fundamentals/Electrical Systems/AC and DC

    Addendum 2. 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.

    Back to reference 2

  3. Engineering Fundamentals/Electrical Systems/AC and DC

    Addendum 3. 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.

    Back to reference 3

  4. Engineering Fundamentals/Electrical Systems/Wire and Wire Gauge

    Addendum 4. Why a long wire can matter even below its current limit

    A wire can stay within its temperature limit yet drop too much voltage for its load. Length increases resistance, and both voltage drop and heating follow from that resistance. The worked cord example in Current, Voltage, and Resistance shows the calculation.

    Back to reference 4

  5. Engineering Fundamentals/Electrical Systems/Wire and Wire Gauge

    Addendum 5. Definitions used to calculate every result

    For AWG number n, diameter d = 0.127 × 92(36 − n)/39 mm. Convert d to meters, then A = πd²/4 in m². Using annealed-copper resistivity ρ = 1.724 × 10−8 Ω·m at 20 °C, R = ρL/A. One foot is exactly 0.3048 m.

    With one-way run ℓ, L = 2ℓ and Rloop = ρ(2ℓ)/A. Vdrop = IRloop; percent drop = 100Vdrop/Vsource; Vload = Vsource − Vdrop; wire loss P = I²Rloop. A two-wire single-phase resistive approximation uses the same return path, but this model does not include AC effects.

    Back to reference 5

  6. Engineering Fundamentals/Electrical Systems/Series and Parallel Circuits

    Addendum 6. Why the parallel formula uses reciprocals

    Each branch draws the supply voltage divided by its resistance. Add those branch currents, then divide by the common voltage. The result is a sum of 1/R terms. This reciprocal of resistance is called conductance: parallel branches add ways for current to flow.

    Back to reference 6

  7. Engineering Fundamentals/Electrical Systems/Measuring Electric Circuits

    Addendum 7. Real meters change the circuit a little

    An ideal voltmeter draws no current. A real one has high but finite input resistance. An ideal ammeter drops no voltage. A real one has some resistance and a voltage drop, sometimes called burden voltage. The practice problems use ideal meters unless stated otherwise.

    Back to reference 7