Compound Direct Current Circuits
Compound DC circuits mix series and parallel elements, so the first task is topology: series elements share the same current; parallel branches share the same potential difference. Reduce valid resistor groups to an equivalent resistance, solve the source/load current, then expand backward. Topic 11.5 also includes nonideal batteries with internal resistance, terminal voltage, and correct ammeter/voltmeter connections—including the fact that nonideal meters alter the circuit they measure.
Topic sequence · 5 of 8
Learning objectives
Objective 1
Identify true series and parallel connections from circuit topology and calculate equivalent resistance for compound resistor networks.
Objective 2
Model a nonideal battery as an ideal emf in series with internal resistance and distinguish emf from loaded terminal potential difference.
Objective 3
Connect ideal and nonideal ammeters/voltmeters correctly and predict or calculate how meter resistance changes the circuit.
- Topic 11.2 circuit loops, nodes, branches, shorts, and schematic connectivity.
- Topic 11.3 Ohm’s law and resistance.
- Topic 11.4 electric power for energy-transfer comparisons.
Definitions first
Representations you must read and create
Representation 1
Redrawn circuit preserving nodes: use connectivity, not visual proximity, to decide whether two resistors are in series or parallel.
Representation 2
Reduction tree showing a parallel subgroup replaced by R_p, then a series group replaced by R_eq, followed by reverse expansion to recover branch currents and voltages.
Representation 3
Nonideal battery model: ideal emf source ℰ in series with internal resistance r and an external load.
Representation 4
Terminal-voltage-versus-current graph for a discharging nonideal battery: intercept ℰ and slope −r in the simple model.
Representation 5
Meter diagrams: ammeter inserted in series with the measured element; voltmeter connected across the two nodes whose potential difference is measured.
Core relationships and calculus
Variables: R Ω
Use: resistors that are truly in series and therefore carry the same current
Variables: R Ω
Use: resistors connected across the same two nodes
Variables: R Ω
Use: quick limiting check for any positive parallel resistor network
Variables: I A; ℰ V; R,r Ω
Use: simple nonideal battery with equivalent load resistance R_load in series with internal resistance r
Variables: V
Use: battery discharging through a load in the internal-resistance model
Variables: V
Use: same terminal voltage measured across the external equivalent load
Variables: R Ω
Use: ideal ammeter and voltmeter models
Equivalent-resistance reduction is algebraic, but it reflects conservation laws. Series elements carry the same current because charge cannot accumulate indefinitely at an intermediate point in steady state; parallel branches have the same potential difference because their endpoints are the same two nodes. The source with internal resistance follows from a loop energy equation: ℰ−Ir−IR_load=0. Later Topics 11.6 and 11.7 formalize these energy and charge constraints as Kirchhoff’s rules.
For series resistors carrying the same I, the total potential drop is ΔV=ΣΔV_i=IΣR_i, so R_eq=ΣR_i. For parallel resistors sharing the same ΔV, total current is I=ΣI_i=Σ(ΔV/R_i)=ΔVΣ(1/R_i), so 1/R_eq=Σ1/R_i. For a nonideal battery, model r in series with the load. Energy conservation around the loop gives ℰ=IR_load+Ir; therefore ΔV_terminal=IR_load=ℰ−Ir during discharge.
Unless otherwise stated, the current framework assumes batteries, wires, and meters are ideal. Circuits with batteries of different potential differences connected in parallel are outside the assessed scope. Unlike the algebra-based Physics 2 boundary, Physics C does not restrict nonideal meter effects to qualitative discussion, so finite meter resistance should be understood quantitatively when specified. Do not collapse a network with series/parallel formulas unless the topology truly matches those definitions; otherwise use Kirchhoff’s rules in later topics.
Physical examples
Example 1
If R₂ and R₃ connect between the same two nodes, they are parallel even if the drawing bends one branch around the page.
Example 2
Adding another positive resistor branch in parallel lowers R_eq because it creates an additional path for charge.
Example 3
A battery with internal resistance has terminal voltage equal to its emf only when current is zero; under load, the internal drop Ir reduces the available terminal voltage during discharge.
Example 4
An ideal ammeter placed in parallel would create a zero-resistance bypass—a short-circuit model—not a harmless measurement.
Example 5
An ideal voltmeter placed in series behaves like an open circuit because its resistance is infinite.
Example 6
A finite-resistance voltmeter in parallel with a component changes the effective resistance of that branch and therefore loads the circuit.
Worked reasoning
Exam reasoning patterns
Reasoning pattern 1
Topology first: mark nodes and paths before applying series/parallel formulas; visual arrangement alone is unreliable.
Reasoning pattern 2
Reduction then expansion: solve R_eq and total current first, then walk backward using same-current and same-voltage rules to recover branch quantities.
Reasoning pattern 3
Sanity checks: series R_eq exceeds each positive component; parallel R_eq is less than the smallest positive branch resistance.
Reasoning pattern 4
Nonideal source: terminal voltage equals ℰ only at zero current; during discharge it falls as current and internal drop Ir increase.
Reasoning pattern 5
Meter placement: ammeter in series, voltmeter in parallel; ideal R_A=0 and R_V→∞. Finite meter resistance changes the circuit.
Reasoning pattern 6
Symbolic comparisons: keep r and load resistance symbolic long enough to expose ratios and limiting cases rather than substituting numbers too early.
Deeper calculus and limiting-case check
A nonideal voltmeter of finite resistance R_V connected across a resistor R does not merely “read” the original voltage; it forms the parallel combination R||R_V and changes the branch resistance. A nonideal ammeter with resistance R_A adds series resistance. These loading effects vanish in the ideal limits R_V→∞ and R_A→0. Terminal-voltage data can also reveal battery parameters: plotting ΔV_terminal versus I for discharge gives intercept ℰ and slope −r.
Model-selection workflow: decide what the problem is before choosing an equation
For Compound Direct Current Circuits, the fastest reliable route is not equation hunting. Start by identifying the nodes, branches, and circuit state, encode it with the circuit schematic/graph, choose the Kirchhoff/constitutive or transient equation, and then test the ideal-element and topology assumption. This sequence prevents a mathematically correct manipulation from being attached to the wrong physical model.
Define the physical object or system
Name what is being modeled, what is outside the system, the coordinate or orientation convention, and the interval or geometry over which the statement applies. Treat series connection and parallel connection as physical quantities first and symbols second.
Choose a representation
Build Redrawn circuit preserving nodes: use connectivity, not visual proximity, to decide whether two resistors are in series or parallel.. If the prompt supplies Reduction tree showing a parallel subgroup replaced by R_p, then a series group replaced by R_eq, followed by reverse expansion to recover branch currents and voltages., translate it into the same sign, axis, orientation, or reference convention before combining information.
Write the governing relationship
A useful starting relationship is R_eq,series = ΣR_i. State what the variables mean and why this relation—not merely a familiar formula—matches the physical situation.
Check conditions and limits
Unless otherwise stated, the current framework assumes batteries, wires, and meters are ideal. Circuits with batteries of different potential differences connected in parallel are outside the assessed scope. Unlike the algebra-based Physics 2 boundary, Physics C does not restrict nonideal meter effects to qualitative discussion, so finite meter resistance should be understood quantitatively when specified. Do not collapse a network with series/parallel formulas unless the topology truly matches those definitions; otherwise use Kirchhoff’s rules in later topics. After solving, test units, sign or direction, zero/large-parameter limits, and whether the answer remains consistent with the original representation.
Representation → equation → calculus → physics chain
| Stage | What to write or inspect | What proves the step is physically valid |
|---|---|---|
| Representation | Redrawn circuit preserving nodes: use connectivity, not visual proximity, to decide whether two resistors are in series or parallel. | Every arrow, sign, axis, node, surface, path, or reference has an explicit meaning. The diagram must contain enough information to determine the relevant components or constraints. |
| Primary model | R_eq,series = ΣR_i | resistors that are truly in series and therefore carry the same current. Preserve the variables and units long enough to see dependence before inserting numerical values. |
| Second relationship | 1/R_eq,parallel = Σ(1/R_i) | resistors connected across the same two nodes. Combine relationships only after confirming they describe the same system, interval, frame, or geometry. |
| Calculus operation | Equivalent-resistance reduction is algebraic, but it reflects conservation laws. Series elements carry the same current because charge cannot accumulate indefinitely at an intermediate point in steady state; parallel branches have the same potential difference because their endpoints are the same two nodes. The source with internal resistance follows from a loop energy equation: ℰ−Ir−IR_load=0. Later Topics 11.6 and 11.7 formalize these energy and charge constraints as Kirchhoff’s rules. | State what the derivative, integral, differential equation, or limiting process represents physically; do not treat calculus notation as an algebraic decoration. |
| Physical check | Return to the original words and representation. | The final sign, direction, units, endpoint behavior, and functional dependence must all agree. A correct-looking expression that violates one of these checks needs repair. |
Advanced synthesis: how to handle an unfamiliar version of this topic
An emf ℰ=12.0 V with internal resistance r=0.50 Ω supplies R₁=2.0 Ω in series with a parallel pair R₂=6.0 Ω and R₃=3.0 Ω. First reduce the parallel pair: 1/R_p=1/6+1/3=1/2, so R_p=2.0 Ω. The external load is R_load=R₁+R_p=4.0 Ω. Including internal resistance, R_total=4.50 Ω and source current I=12.0/4.50=2.667 A. The terminal voltage is ℰ−Ir=12.0−(2.667)(0.50)=10.667 V, which also equals I R_load. The drop across R₁ is 5.333 V, leaving 5.333 V across both parallel branches. Therefore I₂=5.333/6=0.889 A and I₃=5.333/3=1.778 A; their sum equals the source current, providing a charge-conservation check. To deepen this example, change one assumption or parameter, predict the effect before recomputing, and then use the symbolic dependence to verify the prediction.
FRQ-quality response architecture for Topic 11.5
Mathematical routines
Start from the governing law or definition, keep the derivation symbolic, identify the condition that permits any simplification, and finish with units plus a limiting-case check. For this topic, the relationship R_eq,series = ΣR_i should appear only when its physical conditions have been established.
Translation between representations
Move deliberately between Redrawn circuit preserving nodes: use connectivity, not visual proximity, to decide whether two resistors are in series or parallel. and Reduction tree showing a parallel subgroup replaced by R_p, then a series group replaced by R_eq, followed by reverse expansion to recover branch currents and voltages.. Explain what a slope, area, direction, curvature, intercept, node value, field orientation, or other feature means instead of merely redrawing it.
Experimental design and analysis
Turn the topic into a measurable relationship. State the independent and dependent variables, important controls, repeated/ranged trials, and the graph or transformed graph whose slope/intercept would test the model or extract a physical parameter.
Qualitative/quantitative translation
Make the comparison or direction claim first, justify it through the model, derive the symbolic dependence independently, and then reconcile the derivation with the original qualitative claim.
Do not let an equation stand in for an explanation. A complete response names what changes, identifies the governing physical relationship, follows the causal chain, and states the requested outcome explicitly.
Experiment and data-analysis upgrade
Build a safe low-voltage compound resistor network. Before measuring, calculate R_eq, source current, branch currents, and branch voltages. Measure with an ammeter in series and voltmeter in parallel. Then intentionally add known series resistance to model a nonideal ammeter or a finite shunt resistance to model a nonideal voltmeter and compare the altered readings with predictions. If available, characterize a battery by measuring terminal voltage at several load currents and fit ΔV_terminal=ℰ−Ir. Include resistor tolerance, meter resistance, battery heating, contact resistance, and uncertainty.
Operationalize the variables
Say exactly what instrument or observable supplies each measured quantity. Distinguish a directly measured variable from a quantity calculated later from those measurements.
Predict the graph before collecting data
Use the governing relationship to decide what should be plotted. If the raw relationship is nonlinear, choose a transformation that produces a line whose slope or intercept has a defined physical meaning.
Control and repeat for a reason
Hold variables fixed when they would change the predicted relationship. Repeat trials to estimate scatter; vary the independent quantity over a useful range to test functional dependence. These solve different experimental problems.
Evaluate the model, not only the number
Compare the best-fit result with the theoretical prediction, inspect residual or endpoint behavior when appropriate, and identify whether a discrepancy is random scatter, instrument resolution, or a systematic violation of an ideal-model assumption.
Error-repair matrix
| Failure pattern | Repair move | Self-check |
|---|---|---|
| Calling two resistors series merely because they are drawn next to each other. Series means there is no branching path between them. | Return to the circuit schematic/graph, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Calling two resistors parallel merely because they are drawn side by side. Parallel means both ends connect to the same two nodes. | Return to the Kirchhoff/constitutive or transient equation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Forgetting internal resistance when a battery is specified nonideal, causing terminal voltage to be incorrectly set equal to emf under load. | Return to the circuit schematic/graph, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Putting an ammeter across an element or a voltmeter in series. In ideal models those placements fundamentally alter or destroy the intended circuit behavior. | Return to the Kirchhoff/constitutive or transient equation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Assuming an equivalent resistance tells the individual branch currents without expanding back through the network. | Return to the circuit schematic/graph, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Ignoring the boundary that unequal ideal batteries connected directly in parallel are not assessed. | Return to the Kirchhoff/constitutive or transient equation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
Spiral review: make this topic transferable
A topic is exam-ready only when it can be recognized inside a mixed problem. Use the connections below as transfer tests rather than as optional reading.
Transfer 1
Topic 11.3 provides the resistor model and Ohm’s law used during reduction and back-substitution. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 2
Topic 11.4 turns the solved currents and voltages into element powers and source/load energy-transfer rates. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 3
Topic 11.6 formalizes the loop energy equation used to derive terminal voltage and analyze networks that cannot be reduced by simple series/parallel steps. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Formula-condition retrieval
From memory, write R_eq,series = ΣR_i and one condition that must be true before using it. Then state one situation where a more general derivative, integral, conservation law, or field/force model is required.
Topic 11.5 mastery ladder
Interactive compound DC network
Model a nonideal battery (emf ℰ, internal resistance r) feeding R₁ in series with the parallel pair R₂∥R₃. The tool reduces the network and expands back to terminal voltage and branch currents.
Experimental connection
Build a safe low-voltage compound resistor network. Before measuring, calculate R_eq, source current, branch currents, and branch voltages. Measure with an ammeter in series and voltmeter in parallel. Then intentionally add known series resistance to model a nonideal ammeter or a finite shunt resistance to model a nonideal voltmeter and compare the altered readings with predictions. If available, characterize a battery by measuring terminal voltage at several load currents and fit ΔV_terminal=ℰ−Ir. Include resistor tolerance, meter resistance, battery heating, contact resistance, and uncertainty.
Connections across Physics C
Connection 1
Topic 11.3 provides the resistor model and Ohm’s law used during reduction and back-substitution.
Connection 2
Topic 11.4 turns the solved currents and voltages into element powers and source/load energy-transfer rates.
Connection 3
Topic 11.6 formalizes the loop energy equation used to derive terminal voltage and analyze networks that cannot be reduced by simple series/parallel steps.
Connection 4
Topic 11.7 formalizes current conservation at junctions, which underlies branch-current checks.
Connection 5
Topic 11.8 adds capacitors so currents and voltages become time dependent.
Related lessons and resources
Use these links when the current model depends on a prerequisite or becomes the foundation for the next model.
Common mistakes
Avoid this
Calling two resistors series merely because they are drawn next to each other. Series means there is no branching path between them.
Avoid this
Calling two resistors parallel merely because they are drawn side by side. Parallel means both ends connect to the same two nodes.
Avoid this
Forgetting internal resistance when a battery is specified nonideal, causing terminal voltage to be incorrectly set equal to emf under load.
Avoid this
Putting an ammeter across an element or a voltmeter in series. In ideal models those placements fundamentally alter or destroy the intended circuit behavior.
Avoid this
Assuming an equivalent resistance tells the individual branch currents without expanding back through the network.
Avoid this
Ignoring the boundary that unequal ideal batteries connected directly in parallel are not assessed.
Mastery check
- Define the main quantity without relying on an equation.
- Choose the correct representation and sign convention.
- Explain why the calculus relationship matches the physics.
- Design or interpret an experiment that could test the model.
Frequently asked questions
How do I know whether resistors are truly in series?
There must be one unbranched path through them so the same current must pass through each.
How do I know whether resistors are truly in parallel?
Both ends of each resistor must connect to the same two circuit nodes, so they share the same potential difference.
Why is terminal voltage lower than emf for a discharging real battery?
Current through the battery’s internal resistance produces an internal drop Ir, leaving ΔV_terminal=ℰ−Ir for the external circuit.
Why does an ideal ammeter have zero resistance?
It must carry the measured current in series without changing that current. Zero resistance makes its voltage drop zero in the ideal model.
Why does an ideal voltmeter have infinite resistance?
It is connected in parallel across two points but should draw no current, so the ideal model uses infinite resistance.
When should I stop reducing series/parallel groups?
Stop when the remaining topology is not reducible by true series/parallel definitions; then use Kirchhoff’s loop and junction rules.