1.3 Series and Parallel DC Circuits

1.3 Series and Parallel DC Circuits

Quick Takeaway: Kirchhoff's Voltage Law (KVL) states that the sum of voltage drops around any closed loop equals the source voltage ($\sum E = 0$). Kirchhoff's Current Law (KCL) dictates that total current entering a junction equals total current exiting ($\sum I_{\text{in}} = \sum I_{\text{out}}$). In series circuits, current is uniform and resistances add directly; in parallel circuits, voltage is uniform across branches and equivalent resistance is less than the smallest branch resistance.

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Series DC Circuit Laws and Rules

A series circuit is defined as an electrical configuration providing only a single continuous path for electron flow. Because electrons cannot accumulate at any point along a single unbranched loop, fundamental rules govern all series DC circuits:

        +---[ R1 ]---+---[ R2 ]---+---[ R3 ]---+
        |                                       |
    +-(E_source)-                           (Ground)
        |                                       |
        +---------------------------------------+

1. Uniform Current Rule

The current through every series component is identical: $I_T = I_1 = I_2 = I_3 = \dots = I_n$

2. Resistance Summation Rule

The total equivalent resistance ($R_T$) of series-connected resistors equals the sum of their individual resistances: $R_T = R_1 + R_2 + R_3 + \dots + R_n$

3. Voltage Division Rule

The total applied voltage ($E_T$) equals the sum of all individual voltage drops across series components: $E_T = V_1 + V_2 + V_3 + \dots + V_n$

The voltage drop across any individual resistor ($R_x$) in a series network is directly proportional to its resistance value relative to total resistance: $V_x = I \cdot R_x = E_T \cdot \left( \frac{R_x}{R_T} \right)$

4. Total Power Summation

Total power consumed in a series circuit equals the sum of powers dissipated by each component: $P_T = P_1 + P_2 + P_3 + \dots + P_n$

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Parallel DC Circuit Laws and Rules

A parallel circuit provides two or more independent branches connected across a common voltage source. Parallel topology is the standard architecture for aircraft electrical distribution systems because it ensures that each flight instrument or load receives full bus voltage independently of other equipment status.

                  +---+-------------+-------------+
                  |   |             |             |
              (E_source) [Branch 1]  [Branch 2]  [Branch 3]
                  |   |             |             |
                  +---+-------------+-------------+

1. Uniform Voltage Rule

The voltage across every parallel branch is equal to the source voltage: $E_T = E_1 = E_2 = E_3 = \dots = E_n$

2. Reciprocal Resistance Rule

The reciprocal of the total equivalent resistance ($R_T$) equals the sum of the reciprocals of individual branch resistances: $\frac{1}{R_T} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \dots + \frac{1}{R_n}$

Key Characteristic: Total equivalent resistance of a parallel network is always less than the resistance of the smallest branch.

Special Shortcut Formulas:

• Two Resistors in Parallel (Product-Over-Sum): $R_T = \frac{R_1 \cdot R_2}{R_1 + R_2}$
• $n$ Equal Resistors in Parallel: $R_T = \frac{R}{n}$

3. Current Summation Rule

Total current ($I_T$) drawn from the voltage source equals the sum of individual branch currents: $I_T = I_1 + I_2 + I_3 + \dots + I_n$

Branch current varies inversely with branch resistance ($I_x = E_T / R_x$).

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Kirchhoff's Circuit Laws

Formulated by Gustav Kirchhoff in 1845, these two fundamental laws allow technicians to solve complex, multi-loop aircraft DC networks.

1. Kirchhoff's Voltage Law (KVL) - The Loop Rule

Statement: The algebraic sum of all potential differences (voltages) around any closed loop in a circuit must equal zero: $\sum E_{\text{sources}} - \sum V_{\text{drops}} = 0 \quad \text{or} \quad \sum E = 0$

KVL is a direct statement of the Law of Conservation of Energy. In walking around any closed electrical loop, the electrical energy gained from voltage sources must equal the energy dissipated across resistive loads.

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

Statement: The total electrical current entering any circuit junction or node must equal the total electrical current exiting that junction: $\sum I_{\text{entering}} = \sum I_{\text{exiting}} \quad \text{or} \quad \sum I_{\text{node}} = 0$

KCL is a direct statement of the Law of Conservation of Electric Charge. Electric charge cannot accumulate or vanish at a wiring node.

       I1 (Entering: 6A) ----->+-----> I2 (Exiting: 4A)
                               |
                               +-----> I3 (Exiting: 2A)
                     Node: 6A in = 4A + 2A out

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Series-Parallel Combination Circuits & Network Reduction

Most practical aircraft circuits combine series and parallel elements—such as a series dimming rheostat controlling parallel panel backlights, or wire resistance ($R_{\text{wire}}$) in series with parallel bus loads.

Step-by-Step Network Reduction Method:

1. Identify Parallel Branches: Replace parallel resistor combinations with single equivalent resistances ($R_{\text{eq}}$) using product-over-sum or reciprocal rules.
2. Combine Series Resistors: Add remaining series elements to solve for total circuit resistance ($R_T$).
3. Apply Ohm's Law: Calculate total source current ($I_T = E_T / R_T$).
4. Expand Back: Use KVL and KCL to calculate node voltages and branch currents throughout the network.

Worked Example: Aircraft Instrument Dimming Circuit

A 28.0 VDC bus powers two identical panel light bulbs connected in parallel ($R_1 = 20\ \Omega$, $R_2 = 20\ \Omega$). A series dimming resistor ($R_{\text{dim}} = 10\ \Omega$) is placed in series with the combination to adjust intensity.

1. Calculate parallel lighting bank resistance ($R_{\text{bank}}$): $R_{\text{bank}} = \frac{20 \times 20}{20 + 20} = 10.0\ \Omega$
2. Calculate total circuit resistance ($R_T$): $R_T = R_{\text{dim}} + R_{\text{bank}} = 10.0\ \Omega + 10.0\ \Omega = 20.0\ \Omega$
3. Calculate total circuit current ($I_T$): $I_T = \frac{E_T}{R_T} = \frac{28.0\text{ V}}{20.0\ \Omega} = 1.40\text{ A}$
4. Calculate voltage drop across dimming resistor ($V_{\text{dim}}$): $V_{\text{dim}} = I_T \times R_{\text{dim}} = 1.40\text{ A} \times 10.0\ \Omega = 14.0\text{ V}$
5. Calculate voltage across parallel lamps ($V_{\text{lamps}}$): By KVL: $V_{\text{lamps}} = E_T - V_{\text{dim}} = 28.0\text{ V} - 14.0\text{ V} = 14.0\text{ V}$.
6. Calculate current per bulb ($I_{\text{bulb}}$): $I_{\text{bulb}} = \frac{14.0\text{ V}}{20.0\ \Omega} = 0.70\text{ A} \quad (\text{Note: } 0.70\text{ A} + 0.70\text{ A} = 1.40\text{ A by KCL})$

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Avionics Trap: Open Neutral / Ground in Parallel Networks

In an aircraft parallel distribution bus, an open fault in one branch stops current in that specific branch without affecting remaining branches (except for a minor rise in bus voltage due to reduced generator load).

However, if an open fault occurs in a shared common series return ground wire ($R_{\text{ground}}$) servicing multiple avionics LRUs, current from high-power loads will attempt to return through the internal circuitry of lower-power digital sensors connected to the same ground stud. This creates a "floating ground" condition, back-feeding 28 VDC potential into low-voltage sensor return paths, destroying delicate analog-to-digital converters (ADCs) and generating false cockpit sensor warnings across unrelated systems.