DC Circuits

Articles 1 to 8 introduced the parts: resistors, capacitors, coils, diodes, transistors and transformers, and article 8 wrapped that series up. This article starts a new sequence about what happens when you connect them. AC Circuits and RCL Circuits follow as articles 10 and 11.

1. A short recap

• Current (I, in amperes, A) is the flow rate of charge. Voltage (V, in volts, V) is the energy per coulomb between two points. Resistance (R, in ohms, Ω) opposes current. Current is drawn in the direction positive charge would move, out of a source's + terminal. (Recap: current, voltage and resistance.)
• Ohm's law: V = IR, so I = V/R and R = V/I. Article 3 also introduced series and parallel resistors and the voltage divider. (Recap: resistors in circuits.)
• Capacitors store charge, coils (inductors) store magnetic energy, and both take time to charge through a resistor. (Recap: RC charging; the RL time constant.)
• Diodes, transistors and transformers turn mains AC into the steady DC that most electronics run on. (Recap: a linear power supply.)

2. What DC is

Direct current (DC) is the flow of electric charge in only one direction. It is what you get once a circuit connected to a constant-voltage source settles.^[1] Alternating current (AC), by contrast, periodically reverses direction, like the mains in your home.^[1] Strictly, DC need only keep one direction; its size can vary. The rectified pulses of article 6 are one-way but far from steady.^[2]

Common sources of DC:

• Batteries, which make DC from a chemical reaction.^[2]
• Solar cells and panels, which generate DC; an inverter is needed to turn it into AC for the grid.^[3]
• DC power supplies, from phone chargers to bench supplies, which use a rectifier to convert AC to DC.^[2] (Recap: smoothing the result.)

On a schematic, a cell is drawn as one long line and one short line; the long line is the positive terminal. A battery, several cells in series, is drawn with more pairs of lines.^[4] Power is often shown instead with a small arrow or bar labeled with its voltage, such as "+5V" or "VCC".^[4] Multimeters mark their DC ranges with a solid line over a dashed line (⎓) and their AC ranges with a wavy line (~).^[5]

3. What a circuit is

A circuit is a closed loop: a path that starts and ends at the same place. Current flows only when a conductive path runs from the source's positive terminal, through something useful, and back to its negative terminal.^[6] Every circuit has four ingredients:

• a source, such as a battery, which supplies the push;
• a load, the part that does the work, such as a lamp or LED;^[6]
• conductors, the wires that join them (article 2); and
• usually a switch, which opens or closes the loop on demand.

Two faults matter from day one (Figure 1). An open circuit has a break somewhere, so the loop is incomplete and nothing flows. Nothing gets hurt, but nothing works. A short circuit is a path straight from + to − with no load to limit the current. The source then delivers as much current as it can, which can burn wires, damage the source or drain the battery.^[6]

Ground, common and reference

Voltage is always measured between two points, so circuits pick one point to call 0 V and measure everything else from it. In a battery circuit that is usually the negative terminal.^[6] Schematics call this point ground (GND) or common, draw it with one to three short flat lines, and treat every ground symbol as connected to every other even when no wire is drawn.^[4] A multimeter's black "COM" (common) jack is almost always connected to this point.^[7] Do not confuse it with earth ground, a real connection to the soil used in building wiring for safety.^[8] A flashlight's "ground" touches no earth at all.

4. Reading a schematic

A schematic shows how parts are connected, not where they sit on a board. Figure 2 shows the symbols used in this article. Some rules for reading one:^[4]

• Lines are wires. Everything joined by lines without a part in between is one electrical point, called a node or net.
• A dot where lines meet means they are connected. Lines that cross without a dot are just passing by.
• Nets with the same label, such as GND or 5V, are connected even if no wire is drawn between them.
• Resistors are zigzags in American drawings and plain rectangles in international ones.

5. Series circuits

Parts are in series when they are connected end to end, so there is only one path.^[9] Three rules follow:

• The current is the same everywhere. There is nowhere else for it to go.^[10]
• The voltages add. The separate drops across the parts add up to the source voltage.^[9]
• The resistances add: R_total = R₁ + R₂ + R₃ + …^[10]

Worked example (Figure 3, left; all values calculated). A 12 V supply feeds 100 Ω, 220 Ω and 680 Ω in series:

1. Total resistance: 100 + 220 + 680 = 1,000 Ω.
2. Current: 12 V ÷ 1,000 Ω = 0.012 A, or 12 mA, through every part.
3. Drops: 12 mA × 100 Ω = 1.2 V; × 220 Ω = 2.64 V; × 680 Ω = 8.16 V.
4. Check: 1.2 + 2.64 + 8.16 = 12.00 V. The biggest resistor takes the biggest share.

A series circuit has a weakness: if any one part opens, everything stops, because there is only one path.^[9]

6. Parallel circuits

Parts are in parallel when each is connected across the same two points, giving several paths.^[9] The rules mirror the series ones:

• The voltage is the same across every branch.^[9]
• The currents add. The branch currents add up to the total from the source.^[10]
• The reciprocals add: 1/R_total = 1/R₁ + 1/R₂ + 1/R₃ + … The total is always less than the smallest branch, because each branch adds another path.^[10]

For exactly two resistors, the product-over-sum shortcut is quicker:^[11]

R_total = (R₁ × R₂) / (R₁ + R₂)

For three or more, combine two at a time, or use the reciprocal formula.^[11]

Worked example (Figure 3, right; all values calculated). A 6 V supply feeds 100 Ω and 150 Ω in parallel:

1. Each branch gets the full 6 V: 6 ÷ 100 = 60 mA and 6 ÷ 150 = 40 mA.
2. Total current: 60 + 40 = 100 mA.
3. Product over sum: (100 × 150) ÷ (100 + 150) = 15,000 ÷ 250 = 60 Ω. Check: 6 V ÷ 60 Ω = 100 mA.
4. Add a 300 Ω branch and the reciprocal formula gives 1/100 + 1/150 + 1/300 = 6/300, so 50 Ω.

An open branch stops only its own current; the others carry on.^[9] That is why a car's headlights and radio are wired in parallel.^[10] A shorted branch, though, draws a huge current from the source, which affects everything.^[9]

7. Series-parallel circuits

Most real circuits mix the two. The method is to find groups that are purely series or purely parallel, replace each with its equivalent resistance, and repeat until one resistor is left. Then work backward through the steps to find each part's voltage and current.^[12]

Worked example (Figure 4; all values calculated). A 12 V supply feeds R1 = 2 kΩ (2,000 Ω) in series with a parallel pair, R2 = 3 kΩ and R3 = 6 kΩ.

1. Reduce the parallel pair. (3 × 6) ÷ (3 + 6) = 2 kΩ.
2. Reduce the series chain. 2 + 2 = 4 kΩ total.
3. Total current. 12 V ÷ 4 kΩ = 3 mA, all of which flows through R1.
4. Work backward. R1 drops 3 mA × 2 kΩ = 6 V. The pair, worth 2 kΩ, drops the other 6 V.
5. Split the current. R2 carries 6 V ÷ 3 kΩ = 2 mA; R3 carries 6 V ÷ 6 kΩ = 1 mA.

The commonest mistake is mixing values from different parts. Ohm's law only works when V, I and R all belong to the same part or group, between the same two points.^[9]

8. Kirchhoff's laws

Gustav Kirchhoff's two rules work in any circuit, even ones that will not reduce to series and parallel groups. They are the conservation of charge and of energy applied to circuits.^[13]

• Kirchhoff's current law (KCL), the junction rule: the total current flowing into a junction equals the total flowing out. Charge cannot pile up at a point.^[13]
• Kirchhoff's voltage law (KVL), the loop rule: going around any closed loop, the rises and drops in voltage add to zero. Each coulomb gains energy from the source and gives it all up before it returns.^[13]

Worked example (Figure 5; calculated), checking section 7's answers:

• KCL at the junction above R2 and R3: 3 mA flows in and 2 mA + 1 mA = 3 mA flows out.
• KVL around the outer loop through R3: start at the battery's − terminal and go clockwise: +12 V across the battery, −6 V across R1, −6 V across R3. Total: 12 − 6 − 6 = 0.
• KVL around the small loop through R2 and R3: −6 V down R2, +6 V back up R3: total 0, which is why parallel parts share one voltage.

Pick a direction around each loop and keep to it: a part counts as a drop when you travel in the direction its current flows, and as a rise when you travel against it. If a guessed current direction is wrong, the answer simply comes out negative.^[13]

9. Voltage and current dividers

A voltage divider is two resistors in series; the voltage across each is a fixed fraction of the total, in proportion to its resistance.^[14] V_out = V_in × R₂ / (R₁ + R₂) (Recap: the voltage divider.)

A current divider is its parallel twin: the branches split the total current in fixed proportions, with the smaller resistance taking more.^[14] For any branch,

I_branch = I_total × R_total / R_branch^[14]

In section 6's example, the 100 Ω branch gets 100 mA × 60 / 100 = 60 mA (calculated), as before.

10. Power and power budgets

Power, in watts (W), is the rate of energy use. In a DC circuit, P = VI = I²R = V²/R, where the last two forms come from combining P = VI with Ohm's law.^[15] In any circuit, the powers of the parts add up to the power the source delivers.^[9]

In section 7's example (calculated): R1 dissipates 6 V × 3 mA = 18 mW, R2 6 V × 2 mA = 12 mW and R3 6 mW. The total, 36 mW, equals 12 V × 3 mA from the battery. In section 5's example, the 680 Ω resistor dissipates 8.16 V × 12 mA ≈ 98 mW, comfortably inside a quarter-watt (250 mW) rating. (Recap: choosing a wattage.)

A power budget applies the same idea before you build. List every load, add up the currents, and size the supply, wires and fuse to suit. Suppose a 5 V project has these loads, all running at once and so in parallel:

• an LED at 10 mA, as in article 3's example;
• the 5 V G5V-1 relay coil from article 7, rated 30 mA;^[16] and
• a circuit board you have measured at 20 mA.

By KCL the supply must provide 10 + 30 + 20 = 60 mA, and the power is 5 V × 0.06 A = 0.3 W (calculated). A 5 V supply rated 1 A covers it with a wide margin. For battery power, the same 60 mA feeds the runtime estimate in the next section.

11. Real sources

Internal resistance

A real battery behaves like a perfect voltage source in series with a small internal resistance, r, which comes from the materials inside it. The perfect source's voltage is called the emf (electromotive force). Under load, the voltage at the terminals is V_terminal = emf − Ir, so it sags as current rises. Internal resistance also increases as a battery runs down.^[17] A fresh Energizer E91 AA alkaline cell has 150 to 300 mΩ (thousandths of an ohm).^[18] Drawing 0.5 A loses 0.075 to 0.15 V inside it, and a dead short across it would be limited to roughly 1.5 ÷ 0.3 = 5 A to 1.5 ÷ 0.15 = 10 A (all calculated). (Figure 6.)

This explains a common trap. An old battery can show a healthy voltage on a meter, because the meter draws almost no current, yet collapse under a real load. Check batteries while they are loaded.^[19]​^[17]

Batteries in series and parallel

• In series, voltages add and so do internal resistances.^[17] The amp-hour capacity stays the same as one cell's.^[19]
• In parallel, the voltage stays the same, the internal resistance falls, and the capacity adds.^[17]​^[20] A laptop pack of four 3.6 V lithium-ion cells in series and two in parallel ("4s2p") gives 14.4 V and doubles the capacity from 2,400 to 4,800 mAh.^[20]

Never mix mismatched cells. Battery University advises using the same type, voltage and capacity, and never mixing makes and sizes. In series, a weak cell empties first and then, in its words, "gets hammered" by the stronger ones.^[20] In parallel, cells at different voltages drive large currents through each other,^[19] and a shorted cell is drained by the healthy ones, a fire hazard.^[20]

Capacity in mAh and Wh

Capacity is the charge a battery can deliver, in amp-hours (Ah) or milliamp-hours (mAh). One amp-hour supplies 1 A for an hour, or 0.5 A for two hours, in the ideal case.^[19] Multiply by voltage to get the stored energy in watt-hours (Wh); a 3.6 V, 3,400 mAh cell holds 12.24 Wh.^[20] Watt-hours let you compare packs of different voltages.

Real capacity depends on how hard you draw. Energizer's chart for the E91 shows about 3,000 mAh at a 25 mA drain but only about 1,500 mAh at 500 mA (read from the chart).^[18] Powered from AA cells in series, which keep one cell's capacity, the 60 mA budget above would last roughly 2,500 ÷ 60 ≈ 40 hours, taking the chart's figure of about 2,500 mAh at 100 mA (calculated, and only an estimate).^[18]

12. Thevenin's theorem, briefly

Thevenin's theorem says that any linear circuit of sources and resistors, however complicated, looks to a load exactly like one voltage source in series with one resistor. That is the same picture as a battery with internal resistance. To find the two values:^[21]

1. Remove the load and measure or calculate the voltage across the open terminals. This is the Thevenin voltage.
2. Replace each voltage source with a plain wire and find the resistance between the terminals. This is the Thevenin resistance.

Take article 3's divider: two 10 kΩ resistors on 9 V. With no load the output is 4.5 V. With the battery replaced by a wire, the two resistors appear in parallel from the output: 5 kΩ. So the divider acts like a 4.5 V source behind 5 kΩ (Figure 7). Attach a 10 kΩ load and the output is 4.5 × 10 / (5 + 10) = 3 V (calculated), the same sag that article 3 found the long way. A divider is a weak source, which is why it is used for signals, not power.

13. Capacitors and coils in DC circuits

When a DC circuit containing a capacitor or coil is switched on, there is a brief transient before it settles.

• A capacitor charging through a resistor reaches 63.2 percent of its final voltage in one time constant, τ = RC, then 63.2 percent of what remains in each further τ.^[22] Once charged, no current flows, so in a settled DC circuit a capacitor acts as an open circuit.^[22]
• A coil's current rises toward V/R with the time constant τ = L/R, again reaching about 63 percent in one τ. The back-EMF falls to zero as the current levels off,^[23] so in a settled DC circuit a coil acts as a short circuit, limited only by its wire's resistance.

For example, 10 kΩ and 100 µF give τ = 10,000 × 0.0001 = 1 second, and after five time constants (5 s) the capacitor is within 1 percent of its final voltage (both calculated). Two warnings carry over: a charged capacitor can hold its voltage after the power is off, and a coil kicks back a voltage spike when its current is interrupted. (Recap: stored charge; inductive kickback.) When the source itself keeps changing, the transients never settle; that is the subject of article 10.

14. Measuring DC circuits

A multimeter measures the three quantities in three different ways (Figure 9).

Voltage: in parallel. Touch the probes across the part, with the circuit powered. A voltmeter goes in parallel because parallel parts share the same voltage.^[24]

Current: in series. All the current must pass through the meter, so the circuit has to be broken and the meter inserted into the gap.^[24] Fluke's procedure: remove power, move the red lead to the A or mA jack, break the circuit, connect the leads across the break, then restore power.^[5]

Resistance: with the power off. Turn off the circuit and discharge any capacitors before measuring.^[26] Other parts connected in parallel with the one you are testing change the reading, usually lowering it, so for a reliable value remove the part or lift one end.^[26]

15. Troubleshooting, step by step

When a simple DC circuit does not work, the most likely cause is an open circuit, a broken connection or loose wire. Look for shorts too.^[6] Work in order:

1. Stop if anything is hot. If a part suddenly gets hot or burns out, turn off the power and look for a short.^[6]
2. Check the source under load. Measure the supply or battery with the circuit switched on, since a weak battery can read normally with no load.^[19]
3. Follow the voltage. With the black probe on ground, step the red probe along the path from the source toward the load, and find where the voltage stops getting through. Across an open break or open switch you will read the full supply voltage, because all of it is dropped across the gap;^[8] across a closed, healthy switch or a good wire, close to zero (Ohm's law: almost no resistance, almost no drop).
4. Confirm with the power off. Disconnect power and check the suspect wire, fuse or part for continuity or resistance.^[26]
5. Fix the cause, not just the part. Resistors fail open more often than shorted, and they almost never fail unless overheated or abused,^[9] so a burned resistor usually means something else drew too much current. Find that before powering up again.

16. Safety

Short-circuit current. A short is limited only by what resistance is left in the loop, mostly the source's own internal resistance.^[9] A battery with 10 V and 0.2 Ω inside can drive 50 A into a short.^[19] OPTIMA's REDTOP 34 car battery is rated at 800 cold cranking amps (CCA),^[27] meaning it can deliver 800 A for 30 seconds at 0 °F while holding at least 7.2 V.^[28] In one reported case, a mechanic's gold ring touched a 12 V car battery's positive terminal while his wrench bridged the ring to the negative; the burn around his finger needed a skin graft. The authors advise removing all jewelry and covering the live terminal.^[29]

Lithium cells. Lithium-ion cells store a great deal of energy; an internal short can heat a cell into thermal runaway, which external protection cannot stop once it starts.^[30] Unbranded 18650 cells sold for vaping are among the most accident-prone.^[30] Keep loose cells in a case, away from coins, keys and other metal,^[31] and keep spare-battery terminals covered, as the FAA requires on flights: retail packaging, tape or a battery case.^[32]

Fuses. A fuse is a short piece of wire designed to melt and open the circuit when current is excessive. It goes in series with what it protects, on the "hot" side, not the grounded side.^[33] Its job is to protect the wiring: one fuse protects the wiring of a series battery bank, while parallel banks need a fuse on each battery as well, in case one shorts internally.^[19] Fuses are rated in amps and in volts; slow-blow types ride through brief surges.^[33] Never fit a bigger fuse to stop one blowing; find out why it blew.

Wire gauge. Thin wire has more resistance, so it heats more at a given current; a wire's safe current is its ampacity, set mainly by how hot its insulation can get. In one table derived from the U.S. National Electrical Code, 14 AWG copper in free air is rated 25 to 35 A and 10 AWG 40 to 55 A, depending on the insulation.^[33] (AWG, American Wire Gauge, runs backward: smaller numbers are thicker.) Wire in bundles or enclosures is rated lower, so check the rating for your wire and use. A fuse must blow before its wire overheats.

The whole story in brief

• DC flows one way. A circuit is a closed loop of source, load and conductors; an open stops everything, and a short is dangerous.
• Series: one current; voltages and resistances add. Parallel: one voltage; currents and reciprocals add. Reduce mixed circuits step by step, then work back.
• KCL: current in equals current out. KVL: voltages around a loop sum to zero.
• Real batteries have internal resistance and lose capacity at high drain; never mix mismatched cells.
• Once settled, a capacitor acts open and a coil acts short.
• Measure voltage in parallel, current in series, resistance with the power off. Fuse batteries and size wires for the current.

Coming up in this section: 10 AC Circuits and 11 RCL Circuits.

Sources

All links were checked in October 2026. External links open in a new tab.

1. OpenStax, College Physics 2e, section 20.5 "Alternating Current versus Direct Current": https://openstax.org/books/college-physics-2e/pages/20-5-alternating-current-versus-direct-current
2. SparkFun Learn, "Alternating Current (AC) vs. Direct Current (DC)": https://learn.sparkfun.com/tutorials/alternating-current-ac-vs-direct-current-dc/all
3. U.S. Department of Energy, "Solar Integration: Inverters and Grid Services Basics": https://www.energy.gov/cmei/systems/solar-integration-inverters-and-grid-services-basics
4. SparkFun Learn, "How to Read a Schematic": https://learn.sparkfun.com/tutorials/how-to-read-a-schematic/all
5. Fluke, 15B+/17B+/18B+ Digital Multimeters Users Manual (PDF): https://assets.fluke.com/manuals/151718__umeng0000.pdf
6. SparkFun Learn, "What is a Circuit?": https://learn.sparkfun.com/tutorials/what-is-a-circuit/all
7. SparkFun Learn, "How to Use a Multimeter": https://learn.sparkfun.com/tutorials/how-to-use-a-multimeter/all
8. Tony R. Kuphaldt, Lessons In Electric Circuits, Volume I (DC), chapter 3, "Electrical Safety": https://www.ibiblio.org/kuphaldt/electricCircuits/DC/DC_3.html
9. Tony R. Kuphaldt, Lessons In Electric Circuits, Volume I (DC), chapter 5, "Series and Parallel Circuits": https://www.ibiblio.org/kuphaldt/electricCircuits/DC/DC_5.html
10. OpenStax, College Physics 2e, section 21.1 "Resistors in Series and Parallel": https://openstax.org/books/college-physics-2e/pages/21-1-resistors-in-series-and-parallel
11. SparkFun Learn, "Series and Parallel Circuits": https://learn.sparkfun.com/tutorials/series-and-parallel-circuits/all
12. Tony R. Kuphaldt, Lessons In Electric Circuits, Volume I (DC), chapter 7, "Series-Parallel Combination Circuits": https://www.ibiblio.org/kuphaldt/electricCircuits/DC/DC_7.html
13. OpenStax, College Physics 2e, section 21.3 "Kirchhoff's Rules": https://openstax.org/books/college-physics-2e/pages/21-3-kirchhoffs-rules
14. Tony R. Kuphaldt, Lessons In Electric Circuits, Volume I (DC), chapter 6, "Divider Circuits and Kirchhoff's Laws": https://www.ibiblio.org/kuphaldt/electricCircuits/DC/DC_6.html
15. OpenStax, College Physics 2e, section 20.4 "Electric Power and Energy": https://openstax.org/books/college-physics-2e/pages/20-4-electric-power-and-energy
16. Omron, G5V-1 low signal relay data sheet (PDF): https://omronfs.omron.com/en_US/ecb/products/pdf/en-g5v_1.pdf
17. OpenStax, College Physics 2e, section 21.2 "Electromotive Force: Terminal Voltage": https://openstax.org/books/college-physics-2e/pages/21-2-electromotive-force-terminal-voltage
18. Energizer, E91 AA alkaline battery product data sheet (PDF): https://data.energizer.com/pdfs/e91.pdf
19. Tony R. Kuphaldt, Lessons In Electric Circuits, Volume I (DC), chapter 11, "Batteries and Power Systems": https://www.ibiblio.org/kuphaldt/electricCircuits/DC/DC_11.html
20. Battery University, "BU-302: Series and Parallel Battery Configurations": https://batteryuniversity.com/article/bu-302-series-and-parallel-battery-configurations
21. Tony R. Kuphaldt, Lessons In Electric Circuits, Volume I (DC), chapter 10, "DC Network Analysis" (Thevenin's Theorem): https://www.ibiblio.org/kuphaldt/electricCircuits/DC/DC_10.html
22. OpenStax, College Physics 2e, section 21.6 "DC Circuits Containing Resistors and Capacitors": https://openstax.org/books/college-physics-2e/pages/21-6-dc-circuits-containing-resistors-and-capacitors
23. OpenStax, University Physics Volume 2, section 14.4 "RL Circuits": https://openstax.org/books/university-physics-volume-2/pages/14-4-rl-circuits
24. OpenStax, College Physics 2e, section 21.4 "DC Voltmeters and Ammeters": https://openstax.org/books/college-physics-2e/pages/21-4-dc-voltmeters-and-ammeters
25. Fluke, "Choosing the correct fuse for your tester": https://www.fluke.com/en/learn/blog/digital-multimeters/choosing-the-correct-fuse-for-your-tester
26. Fluke, "How to Measure Resistance with a Digital Multimeter": https://www.fluke.com/en-us/learn/blog/digital-multimeters/how-to-measure-resistance
27. OPTIMA Batteries, REDTOP 34 automotive starting battery, product specifications: https://www.optimabatteries.com/products/redtop-34
28. OPTIMA Batteries, "What does Cold Cranking Amps (CCA) mean?": https://www.optimabatteries.com/experience/blog/what-does-cold-cranking-amps-(cca)-mean
29. P. A. Sibley and K. A. Godwin, "Electrothermal ring burn from a car battery," Orthopedics 36(8), 2013, abstract (PubMed): https://pubmed.ncbi.nlm.nih.gov/23937760/
30. Battery University, "BU-304a: Safety Concerns with Li-ion": https://batteryuniversity.com/article/bu-304a-safety-concerns-with-li-ion
31. U.S. Food and Drug Administration, Center for Tobacco Products, "5 Tips to Help Avoid 'Vape' Battery Explosions" (PDF infographic): https://www.fda.gov/media/104399/download
32. Federal Aviation Administration, "PackSafe: Lithium Batteries": https://www.faa.gov/hazmat/packsafe/lithium-batteries
33. Tony R. Kuphaldt, Lessons In Electric Circuits, Volume I (DC), chapter 12, "Physics of Conductors and Insulators" (ampacity and fuses): https://www.ibiblio.org/kuphaldt/electricCircuits/DC/DC_12.html