7.2 Electric Circuits: Series, Parallel & Combination Circuit Analysis

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7.2 Electric Circuits: Series, Parallel & Combination Circuit Analysis

Key Takeaways Explain This Topic AI

In a Series circuit, current is identical through all components, while total voltage is shared (V_T = V_1 + V_2 + ...), and total resistance is the sum (R_T = R_1 + R_2 + ...).

In a Parallel circuit, voltage is identical across all branches, while total current is the sum of branch currents (I_T = I_1 + I_2 + ...), and total resistance is less than the smallest branch resistance.

Kirchhoff's Voltage Law (KVL) states the algebraic sum of all voltages around any closed loop equals zero (sum of source voltages = sum of voltage drops).

Kirchhoff's Current Law (KCL) states total current entering any node or junction must equal total current leaving that node.

An open circuit creates an infinite resistance path (zero current, full source voltage across the open), whereas a short circuit creates a zero resistance bypass (excessive current, blown fuses).

Last updated: July 2026

7.2 Electric Circuits: Series, Parallel & Combination Circuit Analysis

Quick Answer: An electric circuit requires a closed, continuous conductive loop for current to flow. In Series circuits , current is constant ( I T = I 1 = I 2 I_T = I_1 = I_2 I T = I 1 = I 2 ) and resistances add directly ( R T = R 1 + R 2 R_T = R_1 + R_2 R T = R 1 + R 2 ). In Parallel circuits , voltage is constant ( V T = V 1 = V 2 V_T = V_1 = V_2 V T = V 1 = V 2 ) and branch currents add ( I T = I 1 + I 2 I_T = I_1 + I_2 I T = I 1 + I 2 ). Master Kirchhoff's Laws (KVL & KCL) and circuit reduction techniques to solve combination-circuit problems.

1. Circuit Fundamentals & Schematic Terminology

An electric circuit is an interconnection of electrical components forming at least one closed path for current to flow. Every functional circuit requires three basic elements:

Voltage Source: Provides the electrical pressure (e.g., Battery, DC Power Supply, Generator).

Load (Resistor/Device): Consumes electrical energy to perform work (e.g., Light bulb, Motor, Heating element).

Conductors & Control Devices: Provide a closed path (Copper wires) and control flow (Switches, Fuses). [ Basic Circuit Conditions ]

CLOSED CIRCUIT OPEN CIRCUIT SHORT CIRCUIT
(Current Flows) (Break in Path) (Bypassed Load)

+-[ Battery ]-+ +-[ Battery ]-+ +-[ Battery ]-+
| | | | | |
(S) [Switch] (S) (S) / [Open] (S) (S) +---------+
| | | | | | (Short) |
[R] [Resistor][R] [R] [R] [R] +---------+
| | | | | |
+-------------+ +-------------+ +-------------+

Circuit States:

Closed Circuit: An unbroken path allowing current to flow continuously from the source, through the load, back to the source.

Open Circuit: A break or interruption in the conductive path (high/infinite resistance), causing current flow to drop to exactly 0 Amperes .

Short Circuit: An unintended zero-resistance path bypassing the load. Current surges to dangerous levels, tripping circuit breakers or blowing fuses.

2. Series Circuit Rules & Analysis

In a Series circuit , electrical components are connected end-to-end, forming a single continuous path for current flow. [ Series Circuit Schematic ]

+--------[ R1 = 10 Ω ]--------[ R2 = 20 Ω ]--------+
| |
( + )| |
[ 60V Source ] [ R3 = 30 Ω ]
( - )| |
+---------------------------------------------------+

The Three Fundamental Rules of Series Circuits:

Current is Constant Throughout: I Total = I 1 = I 2 = I 3 = ⋯ = I n I_{\text{Total}} = I_1 = I_2 = I_3 = \dots = I_n I Total = I 1 = I 2 = I 3 = ⋯ = I n

Because there is only one path, the exact same rate of electron flow passes through every component.

Total Resistance is the Sum of Individual Resistances: R Total = R 1 + R 2 + R 3 + ⋯ + R n R_{\text{Total}} = R_1 + R_2 + R_3 + \dots + R_n R Total = R 1 + R 2 + R 3 + ⋯ + R n

Adding more resistors in series increases the total circuit resistance.

Total Voltage Equals the Sum of Voltage Drops: V Total = V 1 + V 2 + V 3 + ⋯ + V n V_{\text{Total}} = V_1 + V_2 + V_3 + \dots + V_n V Total = V 1 + V 2 + V 3 + ⋯ + V n

The supply voltage divides proportionally across each resistor based on its resistance value.

Voltage Divider Rule

The voltage drop ( V x V_x V x ) across any specific resistor ( R x R_x R x ) in a series circuit can be calculated directly using the Voltage Divider Formula : V x = V Total × ( R x R Total ) V_x = V_{\text{Total}} \times \left( \frac{R_x}{R_{\text{Total}}} \right) V x = V Total × ( R Total R x )

Complete Series Circuit Worked Example

Refer to the schematic above: A 60 V 60\text{V} 60 V DC power supply is connected in series with three resistors: R 1 = 10 Ω R_1 = 10\Omega R 1 = 10Ω , R 2 = 20 Ω R_2 = 20\Omega R 2 = 20Ω , and R 3 = 30 Ω R_3 = 30\Omega R 3 = 30Ω .

Find Total Resistance ( R T R_T R T ): R T = R 1 + R 2 + R 3 = 10 Ω + 20 Ω + 30 Ω = 60 Ω R_T = R_1 + R_2 + R_3 = 10\Omega + 20\Omega + 30\Omega = 60\Omega R T = R 1 + R 2 + R 3 = 10Ω + 20Ω + 30Ω = 60Ω

Find Total Circuit Current ( I T I_T I T ): I T = V T R T = 60 V 60 Ω = 1.0 Ampere I_T = \frac{V_T}{R_T} = \frac{60\text{V}}{60\Omega} = 1.0\text{ Ampere} I T = R T V T = 60Ω 60 V = 1.0 Ampere

Find Voltage Drop Across Each Resistor:

V 1 = I T × R 1 = 1.0 A × 10 Ω = 10 Volts V_1 = I_T \times R_1 = 1.0\text{A} \times 10\Omega = 10\text{ Volts} V 1 = I T × R 1 = 1.0 A × 10Ω = 10 Volts

V 2 = I T × R 2 = 1.0 A × 20 Ω = 20 Volts V_2 = I_T \times R_2 = 1.0\text{A} \times 20\Omega = 20\text{ Volts} V 2 = I T × R 2 = 1.0 A × 20Ω = 20 Volts

V 3 = I T × R 3 = 1.0 A × 30 Ω = 30 Volts V_3 = I_T \times R_3 = 1.0\text{A} \times 30\Omega = 30\text{ Volts} V 3 = I T × R 3 = 1.0 A × 30Ω = 30 Volts

Verify Kirchhoff's Voltage Law: V T = V 1 + V 2 + V 3 = 10 V + 20 V + 30 V = 60 Volts ✓ V_T = V_1 + V_2 + V_3 = 10\text{V} + 20\text{V} + 30\text{V} = 60\text{ Volts} \quad \checkmark V T = V 1 + V 2 + V 3 = 10 V + 20 V + 30 V = 60 Volts ✓

3. Parallel Circuit Rules & Analysis

In a Parallel circuit , components are connected across common electrical nodes, creating multiple independent parallel branches for current flow. [ Parallel Circuit Schematic ]

+-------------------+-------------------+
| | |
( + )| | |
[ 24V Source ] [ R1 = 12 Ω ] [ R2 = 24 Ω ]
( - )| | |
| | |
+-------------------+-------------------+

The Three Fundamental Rules of Parallel Circuits:

Voltage is Constant Across All Parallel Branches: V Total = V 1 = V 2 = V 3 = ⋯ = V n V_{\text{Total}} = V_1 = V_2 = V_3 = \dots = V_n V Total = V 1 = V 2 = V 3 = ⋯ = V n

Every branch experiences the full potential difference of the voltage source.

Total Current Equals the Sum of Branch Currents: I Total = I 1 + I 2 + I 3 + ⋯ + I n I_{\text{Total}} = I_1 + I_2 + I_3 + \dots + I_n I Total = I 1 + I 2 + I 3 + ⋯ + I n

Total current drawn from the power source equals the sum of currents flowing through each parallel branch.

Total Resistance Reciprocal Formula: 1 R Total = 1 R 1 + 1 R 2 + 1 R 3 + ⋯ + 1 R n \frac{1}{R_{\text{Total}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \dots + \frac{1}{R_n} R Total 1 = R 1 1 + R 2 1 + R 3 1 + ⋯ + R n 1

Critical classification practice Parallel Rule: The total equivalent resistance of a parallel circuit is ALWAYS less than the resistance of the smallest individual branch resistor .

Shortcut Formulas for Parallel Resistance:

Two Resistors in Parallel (Product-Over-Sum): R Total = R 1 × R 2 R 1 + R 2 R_{\text{Total}} = \frac{R_1 \times R_2}{R_1 + R_2} R Total = R 1 + R 2 R 1 × R 2

N N N Equal Resistors in Parallel: R Total = R N R_{\text{Total}} = \frac{R}{N} R Total = N R
Current Divider Rule

In a two-resistor parallel circuit, current divides inversely proportional to resistance: I 1 = I Total × ( R 2 R 1 + R 2 ) and I 2 = I Total × ( R 1 R 1 + R 2 ) I_1 = I_{\text{Total}} \times \left( \frac{R_2}{R_1 + R_2} \right) \quad \text{and} \quad I_2 = I_{\text{Total}} \times \left( \frac{R_1}{R_1 + R_2} \right) I 1 = I Total × ( R 1 + R 2 R 2 ) and I 2 = I Total × ( R 1 + R 2 R 1 )

Complete Parallel Circuit Worked Example

Refer to the schematic above: A 24 V 24\text{V} 24 V battery powers two parallel branches containing R 1 = 12 Ω R_1 = 12\Omega R 1 = 12Ω and R 2 = 24 Ω R_2 = 24\Omega R 2 = 24Ω .

Find Total Equivalent Resistance ( R T R_T R T ): R T = R 1 × R 2 R 1 + R 2 = 12 × 24 12 + 24 = 288 36 = 8.0 Ω R_T = \frac{R_1 \times R_2}{R_1 + R_2} = \frac{12 \times 24}{12 + 24} = \frac{288}{36} = 8.0\Omega R T = R 1 + R 2 R 1 × R 2 = 12 + 24 12 × 24 = 36 288 = 8.0Ω

(Notice that 8 Ω 8\Omega 8Ω is less than the smallest branch resistor of 12 Ω 12\Omega 12Ω !)

Find Branch Currents:

I 1 = V T R 1 = 24 V 12 Ω = 2.0 Amperes I_1 = \frac{V_T}{R_1} = \frac{24\text{V}}{12\Omega} = 2.0\text{ Amperes} I 1 = R 1 V T = 12Ω 24 V = 2.0 Amperes

I 2 = V T R 2 = 24 V 24 Ω = 1.0 Ampere I_2 = \frac{V_T}{R_2} = \frac{24\text{V}}{24\Omega} = 1.0\text{ Ampere} I 2 = R 2 V T = 24Ω 24 V = 1.0 Ampere

Find Total Supply Current ( I T I_T I T ): I T = I 1 + I 2 = 2.0 A + 1.0 A = 3.0 Amperes I_T = I_1 + I_2 = 2.0\text{A} + 1.0\text{A} = 3.0\text{ Amperes} I T = I 1 + I 2 = 2.0 A + 1.0 A = 3.0 Amperes

(Verification via Ohm's Law: I T = V T R T = 24 V 8 Ω = 3.0 A ✓ I_T = \frac{V_T}{R_T} = \frac{24\text{V}}{8\Omega} = 3.0\text{A} \quad \checkmark I T = R T V T = 8Ω 24 V = 3.0 A ✓ )

Loading diagram... Series-Parallel Combination Circuit Reduction Process

4. Combination (Series-Parallel) Circuit Analysis

Most real-world electronics combine series and parallel subnetworks into combination circuits . To solve combination circuits in practice, apply the Step-by-Step Circuit Reduction Method :

Step-by-Step Circuit Reduction Procedure:

Identify Parallel Branches: Locate sub-branches connected directly in parallel and calculate their equivalent single resistance ( R eq R_{\text{eq}} R eq ).

Redraw the Equivalent Circuit: Replace the parallel group with a single equivalent resistor R eq R_{\text{eq}} R eq .

Solve Series Components: Add R eq R_{\text{eq}} R eq in series with any remaining series resistors to find total circuit resistance ( R Total R_{\text{Total}} R Total ).

Calculate Total Current: Use I Total = V Total R Total I_{\text{Total}} = \frac{V_{\text{Total}}}{R_{\text{Total}}} I Total = R Total V Total .

Work Backwards: Expand the circuit step-by-step to calculate individual component voltage drops and branch currents using Ohm's Law.

5. Kirchhoff's Circuit Laws

Kirchhoff's Laws allow analysis of complex circuits that cannot be simplified by basic series/parallel rules alone: Kirchhoff's Voltage Law (KVL) Kirchhoff's Current Law (KCL)

Sum of Loop Voltages = 0 Node Current In = Node Current Out

+---[ R1 ]---+ I1 \ / I3
| | \ /
[Source] [R2] *(Node)
| | / \
+---[ R3 ]---+ I2 / \ I4

V_source - V1 - V2 - V3 = 0 I1 + I2 = I3 + I4
1. Kirchhoff's Voltage Law (KVL) - Loop Rule

Statement: The algebraic sum of all electrical potential differences (voltages) around any closed loop in a circuit must equal zero. ∑ V = 0 ⟹ V Source = V R 1 + V R 2 + V R 3 \sum V = 0 \quad \implies \quad V_{\text{Source}} = V_{R1} + V_{R2} + V_{R3} ∑ V = 0 ⟹ V Source = V R 1 + V R 2 + V R 3

KVL is a statement of the Law of Conservation of Energy .

2. Kirchhoff's Current Law (KCL) - Junction/Node Rule

Statement: The total electrical current entering any junction or node in a circuit must equal the total electrical current leaving that node. ∑ I entering = ∑ I leaving ⟹ I in1 + I in2 = I out1 + I out2 \sum I_{\text{entering}} = \sum I_{\text{leaving}} \quad \implies \quad I_{\text{in1}} + I_{\text{in2}} = I_{\text{out1}} + I_{\text{out2}} ∑ I entering = ∑ I leaving ⟹ I in1 + I in2 = I out1 + I out2

KCL is a statement of the Law of Conservation of Electric Charge .

6. Circuit Troubleshooting & Fault Analysis

Diagnosing circuit faults is a key component of the Electronics Information practice. Understanding how opens and shorts behave in series versus parallel circuits is essential:

Fault Type Circuit Type Effect on Total Resistance Effect on Total Current Behavior of Faulted Component

Open Component Series Increases to Infinite ( ∞ \infty ∞ ) Drops to 0 A 0\text{ A} 0 A Full supply voltage appears across the open break

Open Component Parallel Increases slightly Decreases (loses that branch current) Other parallel branches continue operating normally

Shorted Component Series Decreases Increases Voltage drop across shorted component falls to 0 V 0\text{V} 0 V

Shorted Component Parallel Drops to 0 Ω 0\Omega 0Ω Surges to maximum (Fuse blows) Bypasses all parallel branches; total circuit voltage drops to near 0 V 0\text{V} 0 V

7. Comparative Summary: Series vs. Parallel Properties

Parameter Series Circuit Parallel Circuit

Current ( I I I ) Same through all components ( I T = I 1 = I 2 I_T = I_1 = I_2 I T = I 1 = I 2 ) Sum of branch currents ( I T = I 1 + I 2 I_T = I_1 + I_2 I T = I 1 + I 2 )

Voltage ( V V V ) Divides across components ( V T = V 1 + V 2 V_T = V_1 + V_2 V T = V 1 + V 2 ) Same across all branches ( V T = V 1 = V 2 V_T = V_1 = V_2 V T = V 1 = V 2 )

Total Resistance ( R T R_T R T ) R T = R 1 + R 2 + R 3 R_T = R_1 + R_2 + R_3 R T = R 1 + R 2 + R 3 (Increases with added R) 1 R T = 1 R 1 + 1 R 2 \frac{1}{R_T} = \frac{1}{R_1} + \frac{1}{R_2} R T 1 = R 1 1 + R 2 1 (Decreases with added R)

Effect of Open Component Entire circuit stops functioning Only the opened branch stops functioning

Primary Application Switches, fuses, string lights, voltage dividers Home electrical wiring, car electrical systems