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F. Electricity and Magnetism · Electricity and Magnetism

Designing, Building and Testing Mixed DC Circuits

F2.3 F2.2 A1.4 A1.5 A1.8 A1.13About 75 minutes
Learning goals
  • I can connect ammeters and voltmeters correctly and explain why each must have the resistance it does.
  • I can design a circuit, including choosing a resistor, to meet a stated requirement.
  • I can build and test real or simulated mixed DC circuits safely.
  • I can compare measured values with predictions using percent error and explain the differences.

Words to know

ammeter
A meter that measures current. It is connected in series and has a very small resistance.
voltmeter
A meter that measures potential difference. It is connected in parallel (across a component) and has a very large resistance.
short circuit
A path of very low resistance across a source, which allows a dangerously large current.
tolerance
The allowed range of a component's value, e.g., a 100 Ω ±5% resistor may be anywhere from 95 Ω to 105 Ω.
internal resistance
The small resistance inside a real battery, which makes its terminal voltage drop as it supplies more current.
percent error
|measured − predicted| ÷ predicted × 100%.

From paper to prototype

An engineer designing the lights on a bike, a phone charger or a model railway does three things: predict with the circuit laws, build (often in a simulator first), and test with meters. The predictions and the measurements rarely match perfectly. A good designer knows why, and by how much. In this lesson you will design circuits to meet a goal, measure them correctly and explain the differences.

Connecting meters correctly

  • An ammeter measures the current through a component, so the current must pass through the meter: connect it in series. To leave the circuit unchanged, an ideal ammeter has almost zero resistance.
  • A voltmeter compares the potential at two points, so connect it in parallel, across the component. To avoid stealing current, an ideal voltmeter has very large resistance (often millions of ohms).

Two classic mistakes:

  1. Ammeter in parallel (for example, straight across a battery). Its tiny resistance makes a short circuit: a huge current that can blow the meter's fuse, damage the meter and heat the battery.
  2. Voltmeter in series. Its huge resistance stops almost all current. The lamps go out, and the voltmeter reads close to the full battery voltage.
+12 VAR₁AR₂R₃Vmain-line ammeterammeterin branchvoltmeteracross R₃
Correct meter placement in a mixed circuit. The ammeters are in series: one in the main line (total current) and one in a branch (branch current). The voltmeter is connected across R₃, in parallel with it.

Why measurements differ from predictions

  • Tolerance: a resistor labelled 100 Ω with ±5% tolerance could be anywhere from 95 Ω to 105 Ω.
  • Internal resistance: a real battery has a small resistance inside it. When it supplies current I, some potential is lost inside, so the terminal voltage is less than its rated value: V_terminal = V_rated − Ir.
  • Wires and connections have a little resistance, especially loose clips.
  • Meters are not perfectly ideal and have limited precision.
  • Temperature: resistors and lamps change resistance as they heat up.

Compare with percent error = |measured − predicted| ÷ predicted × 100%. In a school lab, a few percent is normal.

Worked example

Example 1: Choosing a resistor for an LED

An LED needs about 2.0 V across it and should carry 20 mA. It will be powered by a 9.0 V battery with a resistor in series. Find the resistance needed. Standard resistors come in values such as 330 Ω and 390 Ω. Which should you choose, and what current will flow?

  1. By KVL, the resistor must drop 9.0 − 2.0 = 7.0 V.
  2. In series, the resistor carries the LED's current: I = 20 mA = 0.020 A.
  3. R = V ÷ I = 7.0 ÷ 0.020 = 350 Ω.
  4. Choose the next larger standard value, 390 Ω, so the current stays at or below the LED's limit. (330 Ω would give 7.0 ÷ 330 ≈ 21 mA, slightly over.)
  5. With 390 Ω: I = 7.0 ÷ 390 ≈ 0.018 A (18 mA). The resistor's power is P = VI = (7.0)(0.018) ≈ 0.13 W, so a common ¼ W (0.25 W) resistor is fine.

350 Ω is needed; use 390 Ω, giving about 18 mA.

+9.0 VR = ?LEDdrops 7.0 Vdrops 2.0 VKVL: 9.0 V = 7.0 V (resistor) + 2.0 V (LED)
The design for Example 1. The resistor "uses up" the extra potential difference so that the LED gets only the 2.0 V it needs. The small arrows show light leaving the LED.
Worked example

Example 2: Designing a mixed circuit

Two identical lamps are each rated 6.0 V, 0.50 A. You must run both at full brightness from a 12 V supply, with the lamps in parallel so that one can fail without the other going out. Design the circuit.

  1. Each lamp needs 6.0 V and draws 0.50 A, so its resistance is 6.0 ÷ 0.50 = 12 Ω.
  2. Lamps in parallel: both get the same 6.0 V; the total current is 0.50 + 0.50 = 1.0 A.
  3. A series resistor must drop the other 12 − 6.0 = 6.0 V while carrying 1.0 A: R = 6.0 ÷ 1.0 = 6.0 Ω.
  4. Its power: P = VI = (6.0)(1.0) = 6.0 W, so it needs a power rating of more than 6 W (it will get warm).
  5. Check: lamp pair = 12 ÷ 2 = 6.0 Ω; R_T = 6.0 + 6.0 = 12 Ω; I_T = 12 ÷ 12 = 1.0 A ✓.

A 6.0 Ω resistor (rated above 6 W) in series with the two lamps in parallel.

Worked example

Example 3: Prediction versus measurement

A student predicts a total current of 3.0 A for a circuit but measures 2.85 A. The battery's terminal voltage, measured while the circuit runs, is 11.4 V instead of the rated 12.0 V. Find the percent error and suggest the main cause.

  1. Percent error = |2.85 − 3.0| ÷ 3.0 × 100% = 0.15 ÷ 3.0 × 100% = 5.0%.
  2. The terminal voltage is 12.0 − 11.4 = 0.6 V lower than rated. That is 0.6 ÷ 12.0 = 5% lower, which matches the 5% drop in current.
  3. So the main cause is the battery's internal resistance: r = 0.6 V ÷ 2.85 A ≈ 0.21 Ω.

5.0% error, mainly from the battery's internal resistance.

Just for fun

Why do ammeters and voltmeters never get along at parties?

The ammeter insists on getting in line with everyone, and the voltmeter only ever wants to stand alongside you.

Hands-on activity

Build, measure and compare

Option A (computer): use the free PhET "Circuit Construction Kit: DC" simulation. Option B (school or kit): a breadboard, 2 to 4 AA cells in a holder, assorted resistors (for example 100 Ω, 220 Ω, 330 Ω), and a digital multimeter. Grown-up note for Option B: use only small AA cells, never a wall outlet or car battery; never connect a wire or an ammeter straight across the battery; disconnect the battery whenever you change the circuit; if anything gets hot, disconnect it at once.

  1. Design a mixed circuit with one resistor in series with a parallel pair. Draw the diagram and predict I_T, every branch current and every potential difference.
  2. Build it. Measure each value, moving the ammeter into each position in turn (in series), and the voltmeter across each part (in parallel).
  3. Record predicted and measured values in a table with a percent-error column.
  4. Measure the battery's terminal voltage with the circuit off and on. Use the difference to estimate its internal resistance.
  5. Write two sentences explaining your largest percent error.

What to look for: branch currents that add to the main-line current (KCL); potential differences around each loop that add to the terminal voltage (KVL); percent errors of a few percent in a real circuit. In the simulation, errors are near zero unless you turn on battery resistance.

Your turn

1

How must an ammeter and a voltmeter be connected, and what resistance should each ideally have?

2

Why must an ammeter have very low resistance?

3

An LED needs 2.2 V and at most 15 mA. It runs from a 5.0 V USB supply with a series resistor. Find the resistance needed. Should you round up or down to a standard value?

4

A student predicts 0.50 A and measures 0.47 A. Find the percent error.

5

A student connects a voltmeter in series with a lamp and a battery. What happens, and why?

6

A 9.0 V battery must supply 1.5 A to a parallel pair of 3.0 Ω resistors plus one resistor R in series with the pair. Find R.

More practice: warm-up

1

Draw (or describe) the symbols for an ammeter and a voltmeter.

2

What is a short circuit?

3

A resistor is labelled 100 Ω ±5%. What range of resistance could it have?

4

Why should you disconnect the battery before changing a circuit?

5

Give the formula for percent error.

More practice: on your own

1

A 100 Ω ±5% resistor is connected to exactly 10.0 V. What range of current is possible?

2

Two lamps rated 12 V, 0.50 A must run in parallel at full brightness from an 18 V supply. Find the series resistor needed and its power.

3

A student predicts 6.0 V across a resistor and measures 5.8 V. Find the percent error and give two possible reasons.

4

A battery's terminal voltage is 9.0 V with no current and 8.4 V when it supplies 0.30 A. Estimate its internal resistance.

5

An ammeter in the main line reads 2.4 A. An ammeter in one of two parallel branches reads 0.9 A. What should an ammeter in the other branch read? Which law did you use?

Challenge

1

Two 10 kΩ resistors are in series across 12 V. (a) What should a voltmeter across one of them read? (b) A cheap voltmeter with a resistance of only 10 kΩ is used. What does it actually read? (c) What does this show?

2

Using only resistors of 4.0 Ω (as many as you need), design a circuit for a 6.0 V battery that draws exactly 1.0 A, with at least one parallel part. Show it works.

3

Three identical LEDs (2.0 V each at 20 mA) are wired in series with one resistor across a 12 V supply. Find the resistor needed, choose between 270 Ω and 330 Ω, and find the actual current.

Check your understanding

Quiz

1.

How should a voltmeter be connected to measure the potential difference across a lamp?

2.

What happens if an ammeter is connected directly across a battery?

3.

An LED needs 3.0 V at 0.020 A from a 5.0 V supply. What series resistance is needed?

4.

A predicted current is 2.0 A and the measured current is 1.9 A. What is the percent error?

5.

Why is the terminal voltage of a real battery lower when it supplies current?

6.

Why should an ideal voltmeter have a very high resistance?

For parents and teachers

Big idea: Expectation F2.3 asks students to design and build real or simulated mixed DC circuits and explain how they work. This lesson adds the practical side: correct meter use, choosing components, safety, and comparing predictions with measurements (scientific investigation skills A1.4, A1.5, A1.8, A1.13).

How to help: The free PhET simulation runs in any browser and is a fine substitute for a lab kit. If you do use real parts, use only AA cells; check that your teen disconnects the battery before rewiring and never puts an ammeter or a bare wire across the battery.

Common difficulties:

  • Placing meters backwards (ammeter in parallel, voltmeter in series). Ask: "Does the meter need the current to go through it, or does it compare two points?"
  • Rounding a design resistor down instead of up (rounding down lets too much current through).
  • Expecting measurements to match predictions exactly. A few percent is normal; the skill is explaining why.
  • Forgetting to convert mA to A before dividing.