Series and Parallel Circuits

Most robotic circuits combine series and parallel elements. Understanding how components behave in each configuration is essential for circuit design, troubleshooting, and optimization.

Series Circuits

Definition and Characteristics

Series circuit: Components connected in a single path where current flows through each component sequentially.

Battery → Component 1 → Component 2 → Component 3 → Back to Battery

Key characteristics:

Figure: Series connection using male-female jumper wires

Series Resistance

Total resistance is sum of individual resistances:

R_total = R1 + R2 + R3 + ... + Rn

Example: Three resistors (10Ω, 20Ω, 30Ω) in series

R_total = 10 + 20 + 30 = 60Ω

Series Current

Since current is the same everywhere:

I = V_total / R_total

Example: 12V battery with 60Ω total resistance

I = 12V / 60Ω = 0.2A = 200mA

Current through each resistor: 200mA

Series Voltage Division

Voltage divides proportionally to resistance:

V1 = V_total × (R1 / R_total)
V2 = V_total × (R2 / R_total)
V3 = V_total × (R3 / R_total)

Example from above:

V1 = 12V × (10/60) = 2V
V2 = 12V × (20/60) = 4V
V3 = 12V × (30/60) = 6V
Check: 2 + 4 + 6 = 12V ✓

Series Battery Connection

Batteries in series add voltages:

3 × 1.5V alkaline cells in series = 4.5V
2 × 3.7V Li-ion cells in series = 7.4V
4 × 3.7V Li-ion cells in series = 14.8V

Problem: All batteries must be removed if one dies!

Parallel Circuits

Definition and Characteristics

Parallel circuit: Components connected across the same voltage, each with its own path.

        ┌─ Component 1 ─┐
Battery ┤ Component 2 ├ Back to Battery
        └─ Component 3 ─┘

Key characteristics:

Parallel Resistance

Reciprocal formula:

1/R_total = 1/R1 + 1/R2 + 1/R3 + ...

For two resistors:

R_total = (R1 × R2) / (R1 + R2)

Example: 30Ω and 60Ω in parallel

R_total = (30 × 60) / (30 + 60)
        = 1800 / 90
        = 20Ω

Check using reciprocal:
1/R = 1/30 + 1/60 = 2/60 + 1/60 = 3/60 = 1/20
So R = 20Ω ✓

Parallel Current Division

Current divides inversely proportional to resistance:

I1 = I_total × (R_total / R1)

Example: 12V source with 30Ω and 60Ω in parallel

R_total = 20Ω (from above)
I_total = 12V / 20Ω = 0.6A

I_30Ω = 0.6A × (20/30) = 0.6A × 0.667 = 0.4A
I_60Ω = 0.6A × (20/60) = 0.6A × 0.333 = 0.2A

Check: 0.4A + 0.2A = 0.6A ✓

Parallel Battery Connection

Batteries in parallel increase current capacity but maintain voltage:

2 × 12V 5Ah batteries in parallel = 12V 10Ah
(double the amp-hours, same voltage)

Advantage: Better reliability (one can fail, other carries load) Requirement: All batteries must have same voltage!

Mixed Series-Parallel Circuits

Common Configuration

Most real circuits combine series and parallel:

Analysis Method

Step-by-step approach:

1. Simplify parallel sections (find equivalent resistance)
2. Add series resistances
3. Calculate total current
4. Work backwards to find individual currents and voltages

Example: Power Distribution Network

Robot power system:

        ┌─ Motor 1 (6Ω) ─┐
12V ─ Wire(0.2Ω) ┤ Motor 2 (6Ω) ├─ Back
        └─ Motor 3 (6Ω) ─┘

Step 1: Parallel resistance (3 × 6Ω)

1/R_parallel = 1/6 + 1/6 + 1/6 = 3/6 = 1/2
R_parallel = 2Ω

Step 2: Total resistance

R_total = R_wire + R_parallel = 0.2Ω + 2Ω = 2.2Ω

Step 3: Total current

I_total = 12V / 2.2Ω = 5.45A

Step 4: Voltage at parallel motors

V_motors = 12V - (5.45A × 0.2Ω) = 12V - 1.09V = 10.91V

Step 5: Current in each motor

I_each = 10.91V / 6Ω = 1.82A per motor
Check: 1.82 + 1.82 + 1.82 = 5.45A ✓

Practical Robotic Examples

Example 1: LED Indicator Array

Requirement: 4 indicator LEDs (red, green, yellow, blue) all powered from 5V supply

Approach: Parallel with individual current-limiting resistors

        ┌─ R_red ─ LED_red ─┐
5V ─────┤ R_green ─ LED_green ├─ GND
        ├ R_yellow ─ LED_yellow ├
        └─ R_blue ─ LED_blue ─┘

Design:

• Each LED: 2V forward voltage, 20mA current
• Resistor needed: (5V - 2V) / 20mA = 150Ω
• Use 150Ω or 180Ω resistor for each

Total current: 4 × 20mA = 80mA (acceptable)

Figure: Parallel LED circuit with individual resistors for current limiting

Example 2: Sensor Network

Requirement: Multiple sensors (each 50Ω equivalent) need 3.3V from microcontroller GPIO

Problem: GPIO can only source 20mA max

Solution: Don't put sensors directly in parallel; use buffer

MCU (3.3V) → OpAmp Buffer → Parallel Sensors
            (can drive 100mA)

Example 3: Motor with Brake and Capacitor

Circuit:

12V ─ MOSFET ─ Motor ─┬─ GND
                      │
                   Capacitor
                    (parallel)

Purpose: Capacitor provides short-term current during switching

Component Combinations in Robotics

Series Applications

When to use series:

• LEDs in long chains (voltage multiplication)
• Batteries to increase voltage
• Resistors for high precision (stack tolerances)
• Inductors for high-frequency filtering

Parallel Applications

When to use parallel:

• Motors for redundancy
• Batteries for capacity
• Capacitors for energy storage
• Resistors for power handling

Series-Parallel Design Checklist

Summary

Series Circuits:

• ✓ Single path for current
• ✓ Current same everywhere, voltage divides
• ✓ Total R = sum of individual resistances
• ✓ One failure breaks circuit
• ✓ Good for voltage multiplication (batteries)

Parallel Circuits:

• ✓ Multiple paths for current
• ✓ Voltage same everywhere, current divides
• ✓ Total R = 1/(sum of reciprocals)
• ✓ One failure doesn't affect others
• ✓ Good for capacity and redundancy

Mixed Circuits:

• ✓ Simplify parallel sections first
• ✓ Add series resistances
• ✓ Calculate total current
• ✓ Work backwards for individual values
• ✓ Verify with measurements