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newSchoolWorks.com › VoTechWorks › Technology & Design › Electronics / Telecom Technician › DC and AC Circuit Theory and Measurement › Lesson 6
TEC-ELEC48-02 · Lesson 6 · 14.5 clock hours · lab

Series-Parallel DC Circuits and Systematic Troubleshooting

Objective. Students will analyze a series-parallel DC circuit by reducing it to equivalent resistances, calculate all node voltages and branch currents, and locate an instructor-inserted fault using a half-split troubleshooting method.

⚠ Pending professional review

This lesson, its worked examples and its practice questions has not yet been verified by an industry-certified professional or by a certification or approval body. It is a study draft and may contain errors. Do not use it as instructions for work on a job site or on live equipment. Always follow your instructor, your supervising professional, and current official references and regulations.

VoTechWorks program approval and accreditation are still in progress. Credentials are issued only by the certifying bodies named in each program, after their own exams.

Series-Parallel DC Circuits

Most real-world circuits are neither purely series nor purely parallel. A series-parallel circuit contains branches that share a single current path (series) combined with branches that provide alternate current paths (parallel). Analyzing these circuits requires you to reduce them systematically to a single equivalent resistance, then work back out to find every voltage and current.

Step-by-Step Reduction Method

1. Identify groups. Trace the circuit and mark resistors that carry the same current — they are in series. Mark resistors whose ends connect to the same two nodes — they are in parallel.

2. Reduce parallel groups first. Calculate the equivalent resistance of each parallel group using the reciprocal formula: 1/R_eq = 1/R1 + 1/R2 + … For two resistors this simplifies to R_eq = (R1 × R2)/(R1 + R2).

3. Combine series segments. Add equivalent resistances that now appear in series: R_total = Ra + Rb + …

4. Repeat until a single total resistance remains.

5. Find total current. Apply Ohm's Law: I_total = V_source / R_total.

6. Work back outward. Use voltage divider and current divider relationships to recover voltages and currents in each branch.

Always redraw simplified versions of the circuit as you go — a clear diagram prevents errors and is required on your lab worksheet.

Node Voltages and Branch Currents

A node is any point where two or more components connect. Choose one node as the reference (ground, 0 V). Every other node voltage is measured with respect to that reference. Once you know the voltage across a parallel group, each branch current in that group is found by I_branch = V_node / R_branch. Kirchhoff's Voltage Law (KVL) and Kirchhoff's Current Law (KCL) are your verification tools: voltage drops around any closed loop must sum to the source voltage, and currents entering a node must equal currents leaving it.

Using a DMM Safely in This Lab

Systematic Troubleshooting: Half-Split Method

When a circuit does not behave as calculated, use the half-split method (also called binary search or divide-and-conquer):

1. Define the known-good and known-bad boundaries. The source voltage is present (good) and some downstream measurement is wrong (bad).

2. Measure at the midpoint of the circuit between those boundaries.

3. If the midpoint measurement is correct, the fault is in the second half; if incorrect, it is in the first half.

4. Bisect the faulty half and repeat until the fault is isolated to a single component or connection.

5. Document every step in the order you performed it — your step log is graded, not just the final answer.

Four Common Faults and Their Signatures

| Fault | Effect on circuit | DMM clue |

|---|---|---|

| Open resistor | Current path broken; downstream voltage drops to 0 V (or node floats) | Full source voltage appears across the open component |

| Shorted resistor | Resistance drops to near 0 Ω; current in that branch rises sharply | Near 0 V across the shorted component; other voltages shift |

| Wrong resistor value | All voltages and currents shift predictably but not to designed values | Readings off but proportional; resistance check reveals wrong value |

| Loose wire | Intermittent or complete open at connection point | Voltage present on one side, absent on the other |

Passing This Lab

You must submit a complete data table with every measured value within ±5% of your calculated value and a troubleshooting step log showing a logical half-split sequence that correctly names the fault. Both elements are required. Neither partial submission alone earns a pass.

Key terms8

Series-parallel circuitA circuit that contains both series segments (single current path) and parallel branches (multiple current paths) combined in the same network.
Equivalent resistanceA single resistance value that represents the combined effect of two or more resistors; calculated by series addition or the parallel reciprocal formula.
Node voltageThe electrical potential at a circuit node measured with respect to a chosen reference node (ground).
Branch currentThe current flowing through one specific path or component within a parallel group.
Half-split methodA troubleshooting strategy that bisects the circuit repeatedly to locate a fault, each measurement eliminating half the remaining suspect circuit.
Open faultA break in a conductive path causing current to stop flowing; the full source voltage appears across the open component.
Short faultAn unintended low-resistance connection that bypasses a component, causing near-zero voltage across it and increased current in that branch.
Kirchhoff's Current Law (KCL)States that the total current entering a node equals the total current leaving that node; used to verify branch current calculations.

Worked examples4

1. R1 = 100 Ω is in series with a parallel combination of R2 = 220 Ω and R3 = 330 Ω. The supply is 12 V. Find total resistance, total current, the voltage across the parallel group, and each branch current.
  1. Step 1 – Reduce the parallel group: R_parallel = (R2 × R3)/(R2 + R3) = (220 × 330)/(220 + 330) = 72600/550 = 132 Ω
  2. Step 2 – Add series resistor: R_total = R1 + R_parallel = 100 + 132 = 232 Ω
  3. Step 3 – Total current: I_total = V/R_total = 12/232 ≈ 0.0517 A (51.7 mA)
  4. Step 4 – Voltage across the parallel group (same current flows through R1 first): V_parallel = I_total × R_parallel = 0.0517 × 132 ≈ 6.82 V
  5. Step 5 – Verify with KVL: V_R1 = I_total × R1 = 0.0517 × 100 ≈ 5.17 V; 5.17 + 6.82 ≈ 12.0 V ✓
  6. Step 6 – Branch currents: I_R2 = V_parallel/R2 = 6.82/220 ≈ 31.0 mA; I_R3 = 6.82/330 ≈ 20.7 mA
  7. Step 7 – Verify KCL: 31.0 + 20.7 = 51.7 mA = I_total ✓
Answer: R_total = 232 Ω; I_total ≈ 51.7 mA; V_parallel ≈ 6.82 V; I_R2 ≈ 31.0 mA; I_R3 ≈ 20.7 mA
2. A five-resistor series-parallel circuit: R1 = 47 Ω in series with the network. That network has two parallel paths: Path A is R2 = 100 Ω alone; Path B has R3 = 150 Ω and R4 = 68 Ω in series. A fifth resistor R5 = 33 Ω is in series after the parallel network. Supply = 15 V. Find R_total.
  1. Step 1 – Find Path B series combination: R_B = R3 + R4 = 150 + 68 = 218 Ω
  2. Step 2 – Find parallel network (Path A ∥ Path B): R_parallel = (100 × 218)/(100 + 218) = 21800/318 ≈ 68.55 Ω
  3. Step 3 – Add all series elements: R_total = R1 + R_parallel + R5 = 47 + 68.55 + 33 = 148.55 Ω
  4. Step 4 – Total current: I_total = 15/148.55 ≈ 0.1010 A (101.0 mA)
Answer: R_total ≈ 148.6 Ω; I_total ≈ 101 mA
3. During the fault-finding practical, a student measures node voltages on a 12 V circuit and finds 12 V across one resistor that should read about 4 V. What fault does this indicate and why?
  1. Step 1 – Recall the open-fault rule: when a resistor opens, no current flows through it, so no voltage drops across any series resistors on that branch, and the full source voltage appears across the open component.
  2. Step 2 – The measurement shows the full 12 V across one resistor — exactly matching an open (broken) resistor.
  3. Step 3 – Confirm by setting DMM to resistance (power off, component isolated if possible) — an open resistor reads OL (overrange/infinite) instead of its rated value.
Answer: The resistor is open. Full source voltage across a component with zero current through the rest of the branch is the classic open-fault signature.
4. A student calculates I_R2 = 45 mA in a branch. The DMM reads 42.3 mA. Is this within the ±5% lab tolerance?
  1. Step 1 – Calculate 5% of the calculated value: 0.05 × 45 mA = 2.25 mA
  2. Step 2 – Find the difference between calculated and measured: |45 − 42.3| = 2.7 mA
  3. Step 3 – Compare: 2.7 mA > 2.25 mA, so the reading is outside ±5% tolerance.
  4. Step 4 – Student should check connections, verify the actual resistor value with the DMM (power off), and confirm the DMM is zeroed and in the correct range before re-measuring.
Answer: 2.7 mA error exceeds the 2.25 mA (5%) limit — the measurement does NOT pass tolerance. The student must investigate and re-measure.

The math in this lesson2

Learners reduce (220 x 330)/(220 + 330) = 132 ohms, add 100 ohms, and find 51.7 mA total current.
A-SSE.1A-CED.4
Brush up in the math halls at newSchoolWorks.com:
In class: Analysis drill: three series-parallel circuit schematics of increasing complexity; students reduce step by step and solve for all unknowns on a structured worksheet; instructor reviews with class. · Breadboard build and full measurement: construct a five-component series-parallel circuit; use DMM to measure every node voltage referenced to ground and every branch current; complete a data table comparing measured to calculated. · Fault-finding practical: instructor inserts one of four faults (open resistor, shorted resistor, wrong resistor value, loose wire) into a pre-built series-parallel circuit; student documents half-split approach in a step log and identifies the fault within 15 minutes.
Materials: Breadboards, resistors, DC power supply, DMM, fault-inserted reference board prepared by instructor, troubleshooting step-log worksheet, lab report template
Check: Data table complete with all measured values within ±5%. Fault correctly identified and documented; troubleshooting steps logical and written in order. Both elements required to pass.

Practice check8

Answer each question, then check it. Nothing is recorded; this is your own practice.

ETA CET — DC circuit analysis
1. R1 = 200 Ω is in series with a parallel combination of R2 = 400 Ω and R3 = 400 Ω. What is the total equivalent resistance?
ETA CET — DC measurement technique
2. When measuring branch current with a DMM, the meter must be connected:
ETA CET — systematic troubleshooting
3. A node voltage measures 0 V when several volts are expected downstream of a resistor. The resistor's upstream node reads normal voltage. Using the half-split method, the technician's NEXT action should be:
ETA CET — fault analysis
4. A shorted resistor in a parallel branch will cause:
ETA CET — circuit theory
5. Kirchhoff's Current Law (KCL) states that at any node:
ETA CET — safe measurement practice
6. Before measuring resistance in-circuit with a DMM, the technician MUST:
ETA CET — DC circuit analysis
7. In a series-parallel reduction, which group should be simplified FIRST?
ETA CET — measurement accuracy and lab practice
8. A student's calculated branch current is 80 mA. The DMM reads 75 mA. The lab pass criterion is ±5% of the calculated value. Does this reading pass?
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