7.2 Electric Circuits: Series, Parallel & Combination Circuit Analysis

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 ($I_T = I_1 = I_2$) and resistances add directly ($R_T = R_1 + R_2$). In Parallel circuits, voltage is constant ($V_T = V_1 = V_2$) and branch currents add ($I_T = I_1 + I_2$). Master Kirchhoff's Laws (KVL & KCL) and circuit reduction techniques to quickly solve complex AFCT combination circuit problems.

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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:

1. Voltage Source: Provides the electrical pressure (e.g., Battery, DC Power Supply, Generator).
2. Load (Resistor/Device): Consumes electrical energy to perform work (e.g., Light bulb, Motor, Heating element).
3. 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.

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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:

1. Current is Constant Throughout: $I_{\text{Total}} = I_1 = I_2 = I_3 = \dots = I_n$ Because there is only one path, the exact same rate of electron flow passes through every component.

2. Total Resistance is the Sum of Individual Resistances: $R_{\text{Total}} = R_1 + R_2 + R_3 + \dots + R_n$ Adding more resistors in series increases the total circuit resistance.

3. Total Voltage Equals the Sum of Voltage Drops: $V_{\text{Total}} = V_1 + V_2 + V_3 + \dots + V_n$ The supply voltage divides proportionally across each resistor based on its resistance value.

Voltage Divider Rule

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

Complete Series Circuit Worked Example

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

1. Find Total Resistance ($R_T$): $R_T = R_1 + R_2 + R_3 = 10\Omega + 20\Omega + 30\Omega = 60\Omega$

2. Find Total Circuit Current ($I_T$): $I_T = \frac{V_T}{R_T} = \frac{60\text{V}}{60\Omega} = 1.0\text{ Ampere}$

3. Find Voltage Drop Across Each Resistor:

   • $V_1 = I_T \times R_1 = 1.0\text{A} \times 10\Omega = 10\text{ Volts}$
   • $V_2 = I_T \times R_2 = 1.0\text{A} \times 20\Omega = 20\text{ Volts}$
   • $V_3 = I_T \times R_3 = 1.0\text{A} \times 30\Omega = 30\text{ Volts}$

4. Verify Kirchhoff's Voltage Law: $V_T = V_1 + V_2 + V_3 = 10\text{V} + 20\text{V} + 30\text{V} = 60\text{ Volts} \quad \checkmark$

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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:

1. Voltage is Constant Across All Parallel Branches: $V_{\text{Total}} = V_1 = V_2 = V_3 = \dots = V_n$ Every branch experiences the full potential difference of the voltage source.

2. Total Current Equals the Sum of Branch Currents: $I_{\text{Total}} = I_1 + I_2 + I_3 + \dots + I_n$ Total current drawn from the power source equals the sum of currents flowing through each parallel branch.

3. Total Resistance Reciprocal Formula: $\frac{1}{R_{\text{Total}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \dots + \frac{1}{R_n}$

Critical AFCT 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): $R_{\text{Total}} = \frac{R_1 \times R_2}{R_1 + R_2}$
• $N$ Equal Resistors in Parallel: $R_{\text{Total}} = \frac{R}{N}$

Current Divider Rule

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

Complete Parallel Circuit Worked Example

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

1. Find Total Equivalent Resistance ($R_T$): $R_T = \frac{R_1 \times R_2}{R_1 + R_2} = \frac{12 \times 24}{12 + 24} = \frac{288}{36} = 8.0\Omega$ (Notice that $8\Omega$ is less than the smallest branch resistor of $12\Omega$!)

2. Find Branch Currents:

   • $I_1 = \frac{V_T}{R_1} = \frac{24\text{V}}{12\Omega} = 2.0\text{ Amperes}$
   • $I_2 = \frac{V_T}{R_2} = \frac{24\text{V}}{24\Omega} = 1.0\text{ Ampere}$

3. Find Total Supply Current ($I_T$): $I_T = I_1 + I_2 = 2.0\text{A} + 1.0\text{A} = 3.0\text{ Amperes}$ (Verification via Ohm's Law: $I_T = \frac{V_T}{R_T} = \frac{24\text{V}}{8\Omega} = 3.0\text{A} \quad \checkmark$)

4. Combination (Series-Parallel) Circuit Analysis

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

Step-by-Step Circuit Reduction Procedure:

1. Identify Parallel Branches: Locate sub-branches connected directly in parallel and calculate their equivalent single resistance ($R_{\text{eq}}$).
2. Redraw the Equivalent Circuit: Replace the parallel group with a single equivalent resistor $R_{\text{eq}}$.
3. Solve Series Components: Add $R_{\text{eq}}$ in series with any remaining series resistors to find total circuit resistance ($R_{\text{Total}}$).
4. Calculate Total Current: Use $I_{\text{Total}} = \frac{V_{\text{Total}}}{R_{\text{Total}}}$.
5. Work Backwards: Expand the circuit step-by-step to calculate individual component voltage drops and branch currents using Ohm's Law.

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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. $\sum V = 0 \quad \implies \quad V_{\text{Source}} = V_{R1} + V_{R2} + V_{R3}$ 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. $\sum I_{\text{entering}} = \sum I_{\text{leaving}} \quad \implies \quad I_{\text{in1}} + I_{\text{in2}} = I_{\text{out1}} + I_{\text{out2}}$ KCL is a statement of the Law of Conservation of Electric Charge.

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6. Circuit Troubleshooting & Fault Analysis

Diagnosing circuit faults is a key component of the AFCT EI test. Understanding how opens and shorts behave in series versus parallel circuits is essential:

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7. Comparative Summary: Series vs. Parallel Properties