Mixed Circuits

Part of Electrical Theory

Analyzing circuits with both series and parallel sections — how to simplify, calculate, and troubleshoot real-world electrical networks.

Why This Matters

Almost no real circuit is purely series or purely parallel. A battery feeds multiple parallel loads, but the battery’s internal resistance is in series with everything. A building circuit has one main feed wire (series) supplying multiple outlets (parallel). Motors have series resistance windings plus parallel field coils. Practical circuit analysis means handling these mixed topologies.

Mixed circuit analysis is the step between “I can solve simple circuits” and “I can analyze any circuit I encounter.” Once you can reduce a complex circuit to simpler equivalent components, you can calculate currents, voltages, and power anywhere in the network.

The Strategy: Simplify by Substitution

The key technique for mixed circuits is progressive simplification: find groups of purely series or purely parallel components, replace each group with its equivalent single component, repeat until only one element remains. Then work backward to find individual voltages and currents.

Step-by-step method:

1. Identify portions that are purely series (same current through each)
2. Identify portions that are purely parallel (same voltage across each)
3. Combine series portions: R_series = R1 + R2 + … + Rn
4. Combine parallel portions: 1/R_parallel = 1/R1 + 1/R2 + … + 1/Rn
5. Repeat until circuit reduces to one equivalent resistance
6. Apply Ohm’s law to find total current
7. Work backward to find voltage across each group
8. Then find current through each element within each group

Worked Example 1: Simple Mixed Circuit

Circuit: 24V source with R1=4Ω in series, then R2=12Ω parallel with R3=6Ω.

Step 1: Simplify R2 parallel R3: R_parallel = (12 × 6) / (12 + 6) = 72/18 = 4Ω

Step 2: Now circuit is R1=4Ω in series with R_parallel=4Ω: R_total = 4 + 4 = 8Ω

Step 3: Total current from source: I_total = V/R_total = 24/8 = 3A

Step 4: Voltage across R1 (series): V_R1 = I_total × R1 = 3 × 4 = 12V

Step 5: Voltage across parallel combination: V_parallel = 24 - 12 = 12V (or I_total × R_parallel = 3 × 4 = 12V)

Step 6: Current through each parallel resistor: I_R2 = V_parallel / R2 = 12/12 = 1A I_R3 = V_parallel / R3 = 12/6 = 2A

Verify KCL: I_R2 + I_R3 = 1 + 2 = 3A = I_total ✓

Worked Example 2: Three-Level Mixed Circuit

Circuit: 12V source, R1=2Ω in series, then two parallel branches:

• Branch A: R2=3Ω in series with R3=3Ω
• Branch B: R4=12Ω alone

Step 1: Simplify Branch A (R2 + R3 in series): R_A = 3 + 3 = 6Ω

Step 2: Now parallel combination is R_A=6Ω parallel R_B=12Ω: R_parallel = (6 × 12)/(6 + 12) = 72/18 = 4Ω

Step 3: Total circuit: R1=2Ω + R_parallel=4Ω: R_total = 6Ω

Step 4: Total current: I = 12/6 = 2A

Step 5: Voltage drops: V_R1 = 2 × 2 = 4V V_parallel = 12 - 4 = 8V

Step 6: Branch currents: I_A = 8/6 = 1.333A I_B = 8/12 = 0.667A Check: 1.333 + 0.667 = 2A ✓

Step 7: Voltage across R2 and R3 in Branch A: V_R2 = 1.333 × 3 = 4V V_R3 = 1.333 × 3 = 4V Check: V_R2 + V_R3 = 8V = V_parallel ✓

Internal Resistance: A Practical Mixed Circuit

Every battery and generator has internal resistance — resistance inside the source itself. This creates an always-present series element.

Circuit: Battery EMF=12V, internal resistance r=0.5Ω, external load R=5.5Ω.

This is a series circuit: r and R in series. I = EMF / (r + R) = 12 / (0.5 + 5.5) = 12/6 = 2A

Terminal voltage (voltage actually available at battery terminals): V_terminal = EMF - I×r = 12 - 2×0.5 = 12 - 1 = 11V

Or equivalently: V_terminal = I × R = 2 × 5.5 = 11V ✓

Why internal resistance matters:

• Battery terminal voltage drops under load — measured unloaded, battery looks healthy; under load, voltage sags
• Determines maximum current: I_max = EMF/r (short circuit current)
• For lead-acid batteries, internal resistance increases as battery ages
• A “weak” battery: high internal resistance causes large voltage sag under load

Testing internal resistance:

1. Measure open-circuit voltage (V_oc)
2. Connect known load resistor R, measure terminal voltage V_L and current I
3. r = (V_oc - V_L) / I

Power Calculations in Mixed Circuits

Power dissipated in each element: P = I² × R (or V²/R or V×I)

For the first example (R1, R2, R3):

• P_R1 = 3² × 4 = 36W
• P_R2 = 1² × 12 = 12W
• P_R3 = 2² × 6 = 24W
• Total = 36 + 12 + 24 = 72W

Verify from source: P_source = V × I_total = 24 × 3 = 72W ✓

Conservation of energy holds: all power from source equals sum of power in all resistors.

This calculation tells you which elements get hottest (highest power), which ones you need to make more robust, and whether the circuit’s total power is within the source’s capacity.

Building and Wiring Circuits with Mixed Topology

Identifying the topology in a real circuit:

• Elements in the same physical wire path (one wire, no branches): in series
• Elements connected between the same two nodes (parallel paths between same two junction points): in parallel
• Complex: use KCL to identify current sharing and KVL to identify voltage sharing

Common mixed circuit in practice: Battery + parallel loads + wire resistance

Battery → [wire resistance: 0.3Ω] → junction → [Load 1: 10Ω] parallel [Load 2: 15Ω] → back to battery

Step 1: R_loads = (10×15)/(10+15) = 6Ω Step 2: R_total = 0.3 + 6 = 6.3Ω (assuming 12V battery with negligible internal resistance) Step 3: I_total = 12/6.3 = 1.9A Step 4: V_wire = 1.9 × 0.3 = 0.57V Step 5: V_loads = 12 - 0.57 = 11.43V Step 6: I_L1 = 11.43/10 = 1.14A; I_L2 = 11.43/15 = 0.76A

This is exactly the calculation for sizing wire in a practical installation — if V_loads is too low (too much wire resistance), use thicker wire.

Troubleshooting Mixed Circuits

Symptom: One branch dark, others normal

• That branch has an open circuit (broken wire or blown component)
• Voltage across the open: full supply voltage (all of the source voltage drops across the open)
• Current in that branch: zero

Symptom: All loads dim

• Series element has increased resistance (corroded connection, partial break)
• Check series elements first: battery, fuse, main switch, wire connections

Symptom: One branch much brighter, others dimmer

• Short circuit in one parallel branch reduces total parallel resistance
• More current flows, more voltage drops across series elements, less voltage left for loads
• The shorted branch is very bright (current-limited mainly by series resistance) or blows a fuse

Systematic analysis — simplify, calculate, measure, compare — turns mixed circuit troubleshooting from art into science.