6.2 Inductance, Laws and Time Constants

Inductance, Laws, and Time Constants in Aviation Systems

Inductance is the electrical property that opposes any change in current flowing through a circuit. In aviation systems, inductors (often called coils or chokes) are vital for filtering electrical noise, smoothing generator outputs, and stepping up voltages in ignition systems. However, inductance also introduces transient behaviors that present significant challenges to aircraft electrical systems. The rapid collapse of a magnetic field in an inductive component, such as a relay coil or an actuator motor, can induce an extremely high voltage spike (inductive kickback). In modern aircraft equipped with sensitive digital avionic systems, these spikes can cause severe damage or corrupt data networks. Technicians must understand inductive principles, electromagnetic laws, and transient timing to properly maintain aircraft circuits and troubleshooting suppressors, such as flyback diodes and RC snubbers.

The Core Laws of Electromagnetic Induction

Electromagnetic induction is the process of generating an electromotive force (EMF) by moving a conductor through a magnetic field or by changing the magnetic flux passing through a stationary conductor.

Faraday's Law of Induction

Michael Faraday formulated the fundamental relationship between a changing magnetic field and the induced electrical potential. Faraday's Law states that the magnitude of the induced EMF in a circuit is directly proportional to the time rate of change of the magnetic flux linkage. Mathematically, it is expressed as:

$e = -N \cdot \frac{d\Phi}{dt}$

Where:

• $e$ is the induced instantaneous electromotive force (Volts).
• $N$ is the number of turns in the coil.
• $\frac{d\Phi}{dt}$ is the rate of change of magnetic flux (Webers per second).

The formula reveals that the induced voltage depends on three primary factors: the number of turns in the winding, the strength of the magnetic field, and the speed of relative motion between the conductor and the magnetic field.

Lenz's Law

Hinrich Lenz added a critical refinement to Faraday's law, which explains the negative sign in the equation. Lenz's Law states that the direction of the induced EMF (and any resulting current) is always such that its own magnetic field opposes the change in magnetic flux that created it.

Lenz's law is a direct consequence of the conservation of energy. For example, if a permanent magnet's North pole is pushed into a hollow coil of wire, the induced current in the coil will flow in a direction that establishes a North pole at the entry end of the coil, repelling the incoming magnet. Work must be done to push the magnet against this repulsive force, and this mechanical work is converted into the electrical energy of the induced current. If the coil's field attracted the magnet, the magnet would accelerate on its own, violating the law of conservation of energy by creating electrical energy from nothing.

Self and Mutual Induction

Self-Induction

When a changing current flows through a conductor, it creates an expanding or contracting magnetic field around that conductor. This changing magnetic field cuts across the conductor itself, inducing a voltage in it. This process is called self-induction. The induced voltage is known as the back EMF (or counter EMF) because it opposes the applied voltage and limits the rate of change of the current.

The constant of proportionality is Inductance ($L$), measured in Henries ($H$):

$e_L = -L \cdot \frac{di}{dt}$

A circuit component has an inductance of $1\text{ Henry}$ if a rate of change of current of $1\text{ Ampere per second}$ induces a back EMF of $1\text{ Volt}$. The inductance of a coil is determined by its physical construction:

$L = \frac{N^2 \cdot \mu \cdot A}{l}$

Where $N$ is the number of turns, $\mu$ is the core's absolute permeability, $A$ is the cross-sectional area, and $l$ is the physical length of the coil. Note that doubling the number of turns increases the inductance fourfold because $L \propto N^2$.

Energy is stored in the magnetic field of an active inductor. The formula for the stored energy ($E$, in Joules) is:

$E = \frac{1}{2} \cdot L \cdot I^2$

Mutual Induction

When two coils are placed in close proximity, a changing current in the first coil (primary) creates a changing magnetic field that cuts across the turns of the second coil (secondary). This induces an EMF in the secondary coil. This phenomenon is mutual induction. The unit of mutual inductance ($M$) is also the Henry:

$e_2 = -M \cdot \frac{di_1}{dt}$

Mutual induction is the operating principle of power transformers, step-up ignition coils, and linear variable differential transformers (LVDTs) used in flight control surface position sensing.

The LR Time Constant and Transient Behavior

When a DC voltage is applied to an RL circuit (resistor and inductor in series), the current does not instantly reach its maximum value ($I_{max} = V/R$). Instead, the back EMF opposes the rise of current, resulting in an exponential growth curve. Conversely, when the voltage source is removed and the circuit is closed, the collapsing magnetic field induces a voltage that maintains current flow, resulting in an exponential decay curve.

The Time Constant ($\tau$)

The rate at which current changes in an LR circuit is determined by the time constant ($\tau$), measured in seconds:

$\tau = \frac{L}{R}$

• Growth Phase: The current at any time $t$ after applying voltage is given by: $i(t) = I_{max}(1 - e^{-t/\tau})$.
• Decay Phase: The current at any time $t$ after removing the source is given by: $i(t) = I_{initial} e^{-t/\tau}$.

The 5-$\tau$ Rule

For practical engineering and maintenance, an RL circuit is considered to have completed its transient phase and reached a steady state after five time constants ($5\tau$):

Worked Exam Calculation Scenario

Problem: An aircraft landing gear selector solenoid has a coil inductance of $1.5\text{ Henries}$ and an internal resistance of $6\ \Omega$. To limit current, it is connected in series with a $24\ \Omega$ resistor. The circuit is powered by a $30\text{ VDC}$ battery. Calculate:

1. The time constant of the circuit.
2. The current flowing through the circuit exactly $1$ time constant after the switch is closed.
3. The time required for the circuit to reach steady-state.

Step 1: Calculate the total resistance ($R_{total}$). $R_{total} = R_{internal} + R_{series} = 6\ \Omega + 24\ \Omega = 30\ \Omega$

Step 2: Calculate the time constant ($\tau$). $\tau = \frac{L}{R_{total}} = \frac{1.5\text{ H}}{30\ \Omega} = 0.05\text{ seconds} \quad (\text{or } 50\text{ ms})$

Step 3: Calculate the maximum steady-state current ($I_{max}$). $I_{max} = \frac{V}{R_{total}} = \frac{30\text{ V}}{30\ \Omega} = 1.0\text{ Ampere}$

Step 4: Calculate the current at $1\tau$ ($t = 0.05\text{ s}$). After $1\tau$, the current reaches $63.2%$ of its maximum value: $i(1\tau) = 0.632 \cdot I_{max} = 0.632 \cdot 1.0\text{ A} = 0.632\text{ Amperes} \quad (\text{or } 632\text{ mA})$

Step 5: Calculate the time to reach steady-state ($5\tau$). $t_{steady} = 5 \cdot \tau = 5 \cdot 0.05\text{ s} = 0.25\text{ seconds} \quad (\text{or } 250\text{ ms})$

Exam Traps and Tips

• Neglecting Internal Coil Resistance: EASA exams frequently state that a coil has a specific resistance. Do not forget to add this internal resistance to any external resistors in series before calculating $\tau = L/R$ or $I_{max}$.
• Inductor Voltage vs. Current: Remember that current through an inductor cannot change instantaneously (it takes time to build the magnetic field). However, the voltage across an inductor can change instantly. At the instant a switch is closed ($t=0$), the inductor acts as an open circuit ($i=0$), and the entire source voltage drops across it. At steady-state, it acts as a short circuit (only its winding resistance limits current).
• Energy Scaling: If the current through a coil is doubled, the stored energy is multiplied by four ($2^2$) because $E = \frac{1}{2} L I^2$.