# Inductance

Summary: Excessive power bus inductance results in increased impedance and ringing in the Digital Command Control signal.

## Inductance and Impedance Reduction

Managing the impedance of your track bus is important. While this is an advanced topic, it is important to understand the basic principles to avoid creating problems when wiring.

*Main article: Wire_Sizes_and_Spacing/Twisting_the_Bus*

*A conductor in free space will have inductance when current flows through it.*

*Nature abhors a change in flux . (D. J. Griffiths) *

A length of wire has **impedance** caused by the resistive and reactive properties of the wire. A larger gauge wire will have less impedance, as its resistance will be lower, and self-induction (reactive) properties are also reduced. The mathematics required to demonstrate this is complex and beyond the scope of The DCCWiki.

### Three Steps to Managing Inductance

- Use heavy gauge wire
- Avoid long runs
- Keep the pair of bus wires close together.

It is important to understand that the reactive components of your wiring's impedance have an effect on phase relationships. Since a DCC signal is a pulse made from a fundamental frequency and its multiples (harmonics), as those relationships are altered by the reactive properties of the wiring distortion of the DCC signal results. This results in some frequencies travelling faster than others, distorting the pulse shape. Excessive inductance increases any ringing which may be present. By reducing the inductance, ringing is reduced.

#### DCC Issues

- Excessive track bus impedance can result in a multifunction decoder's PWM pulses being superimposed onto the DCC signal. This distortion can create problems such as runaway locomotives.
- Ringing: Excessive inductance increases ringing in the DCC signal.
^{[1]} - The amount of inductance in the bus directly relates to the degree of DCC waveform distortion and other problems such as large voltage spikes.
- Large voltage spikes created during intermittent short circuits caused by derailments or other track electrical issues may result in a decoder reset occurring.
- Many boosters rely on
*rate of change*to detect a short circuit. Excessive power bus impedance impairs the ability of the booster to detect a short by increasing the RL time constant of the bus. This impedes the rate of change, resulting in excessive currents flowing and damaging rolling stock or track work. - Excessive voltage drop

### Terms

- Inductance (Unit: Henry (H)) is the property of a wire to store current in the magnetic field around it, and oppose any change in the current.
- Impedance is a complex number consisting of the resistance and the reactance. This quantity is a vector, having both amplitude and a direction.
- Reactance is a property of inductors and capacitors, where their resistance varies with frequency, denoted by the symbol X. It is expressed as an imaginary quantity in calculations.

### Inductance

Two properties of wire which impact inductance are *gauge* and *length*. Longer wires have **increased** inductance. Increasing the gauge (reducing the diameter) **increases** the inductance. To counteract the inductance of a long power bus, a heavy guage wire should be used.

The impedance of the wire is dominated by the inductance. When the inductance is lowered, so is the impedance. A 1m length of 10AWG has an inductance of 1.32µH, which at 8kHz results in a inductive reactance component of 66mΩ. The same length of wire has a resistive component of 3.3mΩ. The impedance (Z) computes to 66mΩ. This is why heavier wire is used than that of analog applications. A 14AWG wire has an impedance of 71mΩ per metre. If the wiring loop is 5m long (10m total) 10AWG is 660mΩ compared to the 710mΩ of 14AWG.

An effective method to decrease the inductance/impedance is keeping the wires close. The closer the better. The most effective solution is to twist the wires, with about three to five twists per metre. Doing so alters the phase relationships between the two wires, reducing any induced currents in the wire. Tying the bus wires together is also a very effective approach. Not only does it reduce the inductance, the increased capacitance reduces ringing in the bus. This is sometimes referred to as *Closing the Loop*. ^{[2]}

Attaching feeder wires to twisted bus wires gets complicated if you are not consistent with your colour coding, however, if the wires are not twisted a great deal, it shouldn't be too difficult. Tying the two wires together with cable ties is just as effective, plus it is easier to connect feeders.

#### Self Induction

Inductors store *current* in a magnetic field, when current flow increases, it resists that change. Lenz's Law states that a voltage, which will be opposite in polarity, opposes any increases in flux and current. This voltage is called *Counter–Electromotive Force*, or C–EMF, which is similar to Back EMF.

When the current decreases, such as when the booster output switches phase, the magnetic field collapses inducing an Electromotive Force into the inductor. When this happens the polarity of the self-induced voltage is *reversed* as the inductor attempts to maintain current flow using the polarity of the induced voltage such that it *adds* to the source voltage, raising the total voltage in an attempt to maintain current flow in the circuit.

Relays and solenoids do the same thing, which is why they have circuits added to prevent/reduce the damage the kick back can cause. As shown in the picture, a diode provides a path for the discharge current to flow when the voltage source is disconnected. Otherwise, the current would flow into the transistor and the microcontroller as it seeks a path. At the same time a large voltage spike would be imposed across the transistor. The diode clamps that voltage to about 1V, protecting the transistor and microcontroller.

#### Mutual Inductance

**Mutual Inductance** is the property where flux around one conductor induces a current in another. This is the principle by which transformers work. Mutual inductance can be additive or subtractive in nature. By keeping the power bus wires close together, their mutual inductance reduces the total inductance. As *k*, the coefficient of coupling increases, the resulting inductance decreases, as per the formula:

- Total Inductance = L
_{1}+ L_{2}– 2M^{[3]}

- Where M = k × √(L
_{1}× L_{2})

- Where M = k × √(L

- k is a value from 0 to 0.9. Perfect coupling is considered to be 0.9.

For example, if each bus wire were to possess an inductance of 50µHenries:

- k= 0.8 for a tightly coupled pair, L
_{Total}= 20µH - k = 0.4, for a loose coupling such as wires several inches apart, L
_{Total}= 60µH - With no coupling, (k = 0 the total inductance of the loop would be 100µH.
- The k values are just for illustration of the effect of mutual inductance and coupling effects.

#### Impedance

In DCC, the *impedance* of the wire is important. The *Reactive Component* can be much larger than the resistive properties of the wire, controlling the Reactive Inductive component is important for good electrical performance of your wiring.

Impedance (**Z**) is calculated using the following formula: Z = √(R^{2} + X^{2})

- Where X is the reactive component, calculated as (X
_{L}– X_{C})

- X
_{L}is Reactive Inductance - X
_{C}is Reactive Capacitance - Frequency determines the Reactive component.