Compound Direct Current (DC) Circuits
Compound DC circuits combine branches, nodes, series elements, and parallel elements. Analyze them from electrical topology: use equivalent resistance only for genuine series or parallel groups, and connect meters according to what they measure.
Compound Direct Current (DC) Circuits: quick answer
Compound DC circuits combine branches, nodes, series elements, and parallel elements. Analyze them from electrical topology: use equivalent resistance only for genuine series or parallel groups, and connect meters according to what they measure. Compound circuits are topology problems before they are algebra problems. The essential skills are identifying true series/parallel subsets, tracking how source current divides among branches, and connecting meters so the measurement minimally disturbs the ideal circuit.
Do not infer series or parallel from visual placement. Ammeters belong in series and are modeled ideally with zero resistance; voltmeters belong in parallel and are modeled ideally with infinite resistance. Nonideal-meter effects are qualitative only.
What you should be able to explain and do
Use these as closed-note retrieval prompts before looking at worked examples.
Reduce only genuine series and parallel resistor groups.
Use node structure to compare branch potential differences and currents.
Connect ideal ammeters in series and ideal voltmeters in parallel.
Predict qualitatively how a nonideal meter changes a measurement without requiring a full nonideal-meter calculation.
Key terms you need for this lesson
Definitions stay beside the circuit model so unfamiliar terminology does not break the reasoning chain.
compound circuit
compound circuit is a circuit containing both series and parallel relationships.
branch
branch is a conducting path between nodes that carries a branch current.
equivalent resistance
equivalent resistance is a single resistance that reproduces the terminal current–potential-difference behavior of a resistor network.
ammeter
ammeter is a current-measuring device connected in series with the element whose current is measured.
voltmeter
voltmeter is a potential-difference measuring device connected in parallel across the two points being compared.
ideal meter
ideal meter is an ammeter modeled with zero resistance or a voltmeter modeled with infinite resistance so the circuit is not disturbed.
Know what each relationship means and when it applies
The relationship, variables, topology condition, and physical anchor belong together.
| Relationship / model | Variables & units | When to use it | Physical anchor |
|---|---|---|---|
R_series = ΣR_i | R_series equivalent resistance (Ω); R_i component resistances (Ω) | components carrying the same current with no branching between them | 3 Ω and 5 Ω in series give 8 Ω |
1/R_parallel = Σ(1/R_i) | R_parallel equivalent resistance (Ω); R_i branch resistances (Ω) | resistors connected across the same two nodes | two identical resistors in parallel have half the resistance of either one |
I_source = ΔV_source/R_eq | I_source source current (A); ΔV_source source potential difference (V); R_eq equivalent resistance (Ω) | ideal source driving a reduced resistive network | reduce valid subgroups first, then determine total source current |
What a strong circuit diagram or graph must show
Translate the same physical model among topology, arrows, graphs, equations, and energy/charge reasoning.
Node-labeled schematic
elements sharing both nodes are parallel.
Current map
source current divides at junctions and recombines later.
Equivalent-circuit sequence
reduce one valid subgroup at a time while preserving terminal behavior.
Meter diagram
ammeter inserted into the current path; voltmeter bridged across two points.
Change one circuit condition and inspect what responds
Make a prediction before moving a control, then explain the output using topology, conservation, or the governing relationship.
Two examples with the model choice shown
The solution exposes the circuit reasoning pathway rather than only the arithmetic.
Mixed network
Situation: A 6 Ω resistor is in series with a parallel pair of 6 Ω and 3 Ω resistors across a 12 V source.
Reasoning: Parallel pair: 1/Rp=1/6+1/3=1/2, so Rp=2 Ω. Total R=6+2=8 Ω.
Source current is 12/8=1.5 A; the parallel section has 3.0 V across each branch.
Meter connection
Situation: A student wants the current through a lamp and the potential difference across it.
Reasoning: Current must pass through the ammeter, so place it in series. The voltmeter compares the lamp terminals, so place it in parallel.
Ideal ammeter resistance is 0; ideal voltmeter resistance is infinite.
Recognize the physics before reaching for an equation
Use the cue to choose the model, connect it to nearby ideas, then reject any method whose assumptions do not match the situation.
What should trigger Topic 11.5?
Use series/parallel reduction when the circuit can be simplified by identifying elements that share the same current path or the same two nodes.
How this idea fits the course
Equivalent-resistance reasoning reduces networks before Kirchhoff loop/junction methods are needed for circuits that cannot be simplified directly.
Scope check before solving
Do not infer series or parallel from visual placement. Ammeters belong in series and are modeled ideally with zero resistance; voltmeters belong in parallel and are modeled ideally with infinite resistance. Nonideal-meter effects are qualitative only.
Turn this topic into measurable evidence
Choose variables and meters that test the intended relationship without accidentally changing the circuit model.
Investigation target
Build a mixed resistor network and compare measured branch currents with values predicted from topology.
Reasoning errors to eliminate
Repair the circuit model before doing more arithmetic.
Six things you should be able to do before moving on
Use these as closed-note transfer prompts.
- I can identify the circuit topology before applying series/parallel formulas.
- I can define every electrical quantity and give its SI unit.
- I can distinguish charge conservation at junctions from energy conservation around loops.
- I can interpret current, potential, resistance, and power from a graph or schematic.
- I can place ideal meters correctly and predict qualitative loading effects of nonideal meters.
- I can design or critique a circuit experiment and justify a claim from measured evidence.
Practice is kept on its dedicated pages
Finish the lesson first. Then open the separate topic bank when you want exam-style practice without replacing the teaching content above.
42 multiple-choice questions
Use the exact-topic MCQ bank to test model selection, representations, calculations and boundaries after learning the concept.
Open 42 Topic MCQs4 multipart free-response tasks
Use the exact-topic FRQ set for mathematical, representation, experimental and qualitative/quantitative reasoning.
Open 4 Topic FRQsTopic 11.5 FAQs
How do I know two resistors are in series?
They carry the same current and the node between them has no branching path.
How do I know two resistors are in parallel?
Both terminals of one resistor connect to the same two nodes as both terminals of the other.
Where does an ammeter go?
In series with the element whose current is being measured.
Where does a voltmeter go?
In parallel across the two points whose potential difference is being measured.
What resistance does an ideal ammeter have?
Zero resistance.
What resistance does an ideal voltmeter have?
Infinite resistance.