1.1 Ohm's Law, Power, and DC Circuit Analysis

1.1 Ohm's Law, Power, and DC Circuit Analysis

Quick Answer: Direct-current (DC) aircraft electrical systems operate on the governing relationship $V = I \times R$. Electrical power dissipation is determined by $P = V \times I = I^2 R = \frac{V^2}{R}$. While series circuits maintain a single constant current path with additive resistances ($R_T = R_1 + R_2 + \dots$), aircraft power distribution networks wire loads in parallel so that full bus voltage is supplied across every branch ($V_T = V_1 = V_2$) and the loss of a single component does not interrupt the entire bus. Voltage dividers experience significant output voltage drop ("loading error") whenever an external load resistance $R_L$ is connected in parallel with the output resistor. Per FAA AC 43.13-1B Table 11-6, the continuous-operation voltage drop between the bus and the equipment ground should not exceed 1 V on 28V DC systems or 0.5 V on 14V DC systems.

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Ohm's Law in Aircraft DC Systems

Every avionics system—from basic cockpit annunciators to integrated primary flight computers—relies on direct-current (DC) power derived from aircraft storage batteries, engine-driven DC generators, or transformer-rectifier units (TRUs). The fundamental mathematical framework describing DC circuit behavior is Ohm's Law, formulated by Georg Simon Ohm. It establishes that the current flowing through a conductor between two points is directly proportional to the voltage across the two points and inversely proportional to the resistance between them:

$V = I \times R$

Where:

• $V$ is electrical potential difference (Electromotive Force, EMF), measured in Volts (V).
• $I$ is the intensity of electron flow, measured in Amperes (A), where $1\text{ A} = 1\text{ Coulomb per second} = 6.242 \times 10^{18}\text{ electrons/sec}$.
• $R$ is the net opposition to electrical current flow, measured in Ohms ($\Omega$).

By algebraic manipulation, an aircraft electronics technician calculates current or resistance using the two companion forms:

$I = \frac{V}{R} \quad \text{and} \quad R = \frac{V}{I}$

Practical Aircraft Calculation Example

An avionics technician is installing a 28V DC pitot head heater. The technical specifications specify an internal heating element resistance of $5.6\ \Omega$ at operating temperature. To calculate the continuous current demand on the DC essential bus:

$I = \frac{V}{R} = \frac{28\text{ V}}{5.6\ \Omega} = 5.0\text{ A}$

If ground power supplies a reduced bus voltage of 24V during ramp testing, the current drawn decreases proportionally: $I = 24 / 5.6 = 4.29\text{ A}$.

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Electrical Power and Thermal Derating

Electrical power represents the rate at which electrical energy is converted into work or dissipated as heat. The fundamental power equation states that power ($P$) in Watts (W) equals the product of voltage and current:

$P = V \times I$

Substituting Ohm's law ($V = I \times R$ and $I = V / R$) yields the three foundational power formulas:

$P = V \times I = I^2 R = \frac{V^2}{R}$

Power Derating in Avionics Enclosures

In aircraft electrical installations, heat dissipation is a primary failure mechanism. Resistors dissipate electrical power strictly as thermal energy ($I^2 R$ losses). Operating a resistor at or near its maximum manufacturer-rated wattage causes rapid temperature elevation, resistance value drift, thermal degradation of surrounding printed circuit board (PCB) traces, and premature failure.

A common electronics design practice (not an FAA rule) is to apply a 50% power derating: a resistor should dissipate no more than half of its rated wattage, which matters most in enclosed, poorly ventilated equipment bays. Under that practice, a resistor expected to dissipate 1.0W must be rated for at least 2.0W. When you replace a part, the equipment manufacturer's parts list and derating rules govern.

$\text{Minimum Rating} \ge 2 \times P_{\text{calculated}}$

For example, if a series dropping resistor drops 14V at 0.1A ($P = 14\text{ V} \times 0.1\text{ A} = 1.4\text{ W}$), 50% derating calls for at least $2 \times 1.4 = 2.8\text{ W}$. A standard 2W part is therefore too small, and the next standard size (3W or larger) is selected.

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Series Circuit Analysis

A series circuit provides only a single continuous conducting path for electron flow. As a result, the following rules govern every series circuit:

1. Current is Constant: The current flowing through every component in the series string is identical: $I_{\text{total}} = I_1 = I_2 = I_3 = \dots = I_n$
2. Resistance is Additive: The total equivalent resistance ($R_T$) equals the arithmetic sum of all individual resistances: $R_{\text{total}} = R_1 + R_2 + R_3 + \dots + R_n$
3. Voltage Drops are Additive: The total applied voltage ($V_T$) equals the sum of the individual voltage drops across each resistor: $V_{\text{total}} = V_1 + V_2 + V_3 + \dots + V_n$

If any single component in a series circuit opens, current drops to zero across the entire circuit. For this reason, primary aircraft operational loads are never connected in series.

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Parallel Circuit Analysis

A parallel circuit connects two or more components across the same two common electrical nodes, creating multiple independent branch paths for current flow. The governing rules for parallel circuits are:

1. Voltage is Constant: The voltage drop across every parallel branch is identical and equals the source bus voltage: $V_{\text{total}} = V_1 = V_2 = V_3 = \dots = V_n$
2. Branch Currents Add: The total current supplied by the power source equals the sum of the individual branch currents: $I_{\text{total}} = I_1 + I_2 + I_3 + \dots + I_n$
3. Equivalent Resistance Decreases: Total equivalent resistance is always less than the resistance of the smallest individual branch. The reciprocal formula defines parallel resistance: $\frac{1}{R_{\text{total}}} = \frac{1}{R_1} + \frac{1}{R_2} + \dots + \frac{1}{R_n}$
4. Conductance ($G$): Conductance represents the ease with which current flows, measured in Siemens (S), where $G = 1 / R$. In parallel circuits, total conductance is directly additive: $G_{\text{total}} = G_1 + G_2 + \dots + G_n$

Two-Resistor Shortcut: Product Over Sum

When exactly two resistors are connected in parallel, technicians utilize the simplified product-over-sum formula:

$R_{\text{total}} = \frac{R_1 \times R_2}{R_1 + R_2}$

Numerical Example: An avionics warning panel connects a $60\ \Omega$ lamp in parallel with a $40\ \Omega$ auxiliary relay coil across a 28V DC bus:

$R_{\text{total}} = \frac{60 \times 40}{60 + 40} = \frac{2400}{100} = 24\ \Omega$

Total current supplied by the bus:

$I_{\text{total}} = \frac{V}{R_{\text{total}}} = \frac{28\text{ V}}{24\ \Omega} = 1.167\text{ A}$

Individual branch currents verify Kirchhoff's Current Law:

• $I_1 = 28 / 60 = 0.467\text{ A}$
• $I_2 = 28 / 40 = 0.700\text{ A}$
• $I_{\text{total}} = 0.467 + 0.700 = 1.167\text{ A}$

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Kirchhoff's Laws

Complex avionics networks with interconnected loops and distribution buses are analyzed using Gustav Kirchhoff's two governing conservation laws.

Kirchhoff's Current Law (KCL)

Kirchhoff's Current Law (junction rule) is an expression of the conservation of electrical charge. It states that the algebraic sum of currents entering and leaving any circuit node is zero:

$\sum I_{\text{entering}} = \sum I_{\text{leaving}}$

If an aircraft main DC distribution bus node receives $50\text{ A}$ from Generator 1 and $30\text{ A}$ from Generator 2, the total current exiting that bus node to various sub-buses and equipment circuit breakers must equal precisely $80\text{ A}$.

Kirchhoff's Voltage Law (KVL)

Kirchhoff's Voltage Law (loop rule) is an expression of the conservation of energy. It states that the algebraic sum of all electrical potential differences (source voltages and resistive voltage drops) around any closed loop is zero:

$\sum V_{\text{sources}} - \sum V_{\text{drops}} = 0 \quad \implies \quad V_{\text{source}} = V_1 + V_2 + \dots + V_n$

In a 28V DC circuit supplying a navigation receiver through a switch and circuit breaker, if the wire drops $0.4\text{ V}$, the circuit breaker drops $0.1\text{ V}$, and the airframe ground return drops $0.2\text{ V}$, the remaining voltage available directly at the receiver terminals is $28 - (0.4 + 0.1 + 0.2) = 27.3\text{ V}$.

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Voltage Dividers: Unloaded vs. Loaded

A voltage divider is a series arrangement of two or more resistors used to obtain a specific fractional output voltage from a higher DC supply rail.

Unloaded Voltage Divider

In an open-circuit (unloaded) condition, no current is drawn from the output terminal. The current flowing through both resistors is identical ($I = V_{\text{in}} / (R_1 + R_2)$). The output voltage taken across $R_2$ is governed by the voltage divider formula:

$V_{\text{out}} = V_{\text{in}} \times \left(\frac{R_2}{R_1 + R_2}\right)$

Example: An avionics sensor bias network connects $R_1 = 18\text{ k}\Omega$ and $R_2 = 10\text{ k}\Omega$ in series across a 28V DC bus:

$V_{\text{out}} = 28\text{ V} \times \left(\frac{10\text{ k}\Omega}{18\text{ k}\Omega + 10\text{ k}\Omega}\right) = 28 \times \left(\frac{10}{28}\right) = 10.0\text{ V}$

The Loading Effect

When an external device with load resistance $R_L$ is connected across $R_2$, $R_L$ is in parallel with $R_2$. The equivalent lower resistance ($R_{\text{eq}}$) becomes:

$R_{\text{eq}} = \frac{R_2 \times R_L}{R_2 + R_L}$

Because $R_{\text{eq}}$ is strictly less than $R_2$, the loaded output voltage drops below the design voltage:

$V_{\text{out(loaded)}} = V_{\text{in}} \times \left(\frac{R_{\text{eq}}}{R_1 + R_{\text{eq}}}\right)$

Worked Loading Calculation: If a $10\text{ k}\Omega$ load ($R_L = 10\text{ k}\Omega$) is connected to the divider above:

$R_{\text{eq}} = \frac{10\text{ k}\Omega \times 10\text{ k}\Omega}{10\text{ k}\Omega + 10\text{ k}\Omega} = 5.0\text{ k}\Omega$

$V_{\text{out(loaded)}} = 28\text{ V} \times \left(\frac{5\text{ k}\Omega}{18\text{ k}\Omega + 5\text{ k}\Omega}\right) = 28 \times \left(\frac{5}{23}\right) = 6.09\text{ V}$

The output voltage sags from $10.0\text{ V}$ down to $6.09\text{ V}$—a massive $39.1\%$ loading error! To minimize loading error in precision avionics circuits, the bleeder current flowing through the divider resistors must be at least 10 times greater than the anticipated load current ($I_{\text{bleeder}} \ge 10 \times I_{\text{load}}$), or an active operational-amplifier buffer must isolate the divider.

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Circuit Troubleshooting via Voltage Drop Analysis

Voltage drop troubleshooting is the most powerful in-circuit diagnostic method available to an aircraft electronics technician. Rather than disconnecting wires or desoldering components to measure resistance, the technician measures dynamic potential drops while the circuit is energized under normal load.

FAA AC 43.13-1B Allowable Voltage Drop Limits

FAA Advisory Circular AC 43.13-1B, Chapter 11, Section 5 (paragraph 11-66 and Table 11-6) lists the maximum acceptable voltage drop in load circuits between the bus and the utilization equipment ground:

A separate, tighter limit applies upstream of the bus: paragraph 11-66b says the drop in the main power wires from the generator or battery to the bus should not exceed 2 percent of the regulated voltage with the generator carrying rated current. Do not confuse that 2% source-to-bus figure with the Table 11-6 bus-to-load values.

Any conductor, relay contact, or ground bond that pushes a circuit past these limits wastes power as heat and starves the avionics of voltage. AC 43.13-1B adds that if a circuit's measured drop does not exceed the limit set by the aircraft or product manufacturer, its resistance may be considered satisfactory.

Diagnosing Circuit Faults

1. High-Resistance Fault (Corrosion / Loose Terminals):
   • Symptom: The load operates sluggishly, intermittently, or displays low bus warnings.
   • Method: Connect the digital multimeter (DMM) in DC Volts mode across individual conductors, switches, and ground connections while current is flowing.
   • Interpretation: A good conductor or closed switch displays $< 0.1\text{ V}$. A corroded terminal block lug, pitted relay contact, or oxidized airframe ground stud will drop significant voltage (e.g., $1.5\text{ V}$ to $5.0\text{ V}$), starving the downstream load.
2. Open-Circuit Fault (Broken Wire / Blown Fuse / Open Switch):
   • Symptom: Load is completely inoperative; circuit current is $0.0\text{ A}$.
   • Interpretation: Across any functional, closed switch or intact fuse, voltage drop is $0.0\text{ V}$. When measuring directly across an open switch, open contact, or blown fuse in an energized circuit, the meter displays full source voltage ($28\text{ V DC}$). This occurs because the meter's internal $10\text{ M}\Omega$ input impedance completes the circuit loop to ground, dropping the entire supply voltage.
3. Short-to-Ground Fault:
   • Symptom: The circuit breaker immediately trips or fuse violently ruptures upon activation.
   • Method: De-energize the circuit, pull the breaker, isolate the load, and measure resistance from the conductor to airframe ground with a DMM in Ohms mode.
   • Interpretation: An ungrounded avionics power lead should measure infinite resistance ($> 10\text{ M}\Omega$) to ground. A reading of $0.0\ \Omega$ to $0.2\ \Omega$ confirms an insulation chafing short against airframe structure.