Inductors Basics

Describing basic functionality of the inductors and how they are treated when connected in series or in parallel.

What is an inductor? How does it work? And how we handle inductors when they are connected in series or in parallel? Here are the answers.

An inductor is an electric device capable of storing energy in the form of a magnetic or electromagnetic field.

inductor

In its basic form, an inductor can be made of a single loop of wire, or several loops (solenoid). These loops can be arranged in air or on a ferromagnetic core.

When an inductor is connected to a battery, a current starts flowing in the circuit. The current that flows inside the inductor generates a magnetic field, like the one that would be generated by an actual magnet. This field stores an amount of energy, the same way an electric field does.

inductor_circuit

If the battery is suddenly disconnected, the energy that was accumulated in the inductor must be somehow released. but the energy cannot be released instantaneously, it needs to be released a little bit at a time. And since the energy depends on the current flowing in the inductor, the inductor tries to keep the it running, even if the battery is no more connected. To do so, it uses the energy stored into the magnetic field to generate a voltage at its terminals to keep the current going.

inductor_open_circuit

However, since the inductor is now connected nowhere, current cannot flow, unless the voltage is so high that the current can flow in the thin air. And that is exactly what happens: the voltage increases so much that there is a sudden discharge of current through the air, in the form of a spark, that dissipates all the energy that was stored in the inductor. This spark is the one you may sometimes notice when opening a switch that is powering a lamp or a motor, or when you pull the plug from a device that was working using a considerable amount of current.

Similarly to the case where the current is suddenly removed, an inductor generates a voltage also when the current is just changed in intensity. In this case, the voltage is created to react to the change in current, trying to keep it to the same value, so the energy can be conserved.

In both cases, the amount of voltage is proportional to the change in current (ΔI) and inversely proportional to the amount of time in which the current changes (Δt). In other words, the faster the current change, the higher is the voltage.

For a specific inductor, the ratio between the change of current and the interval in which that happens equals the voltage generated by the inductor divided by a constant that depends on the physics dimensions of the inductor. Such constant is called inductance, represented with the letter L, and can be calculated with the following experimental formula:

inductance_formula

where:

μ = permeability of the material inside the coil

N = number of turns making the coil

A = area of the cross section of the coil

l = length of the coil

L is measured in Henry.

μ is the product of the permeability of the void (or air) and the relative permeability of the material:mu

The voltage at the terminals of the inductor is therefore calculated as:

vdit

We can now calculate the energy stored in the magnetic field of an inductor as the integral of the power, which is obtained multiplying the voltage at the inductor and the current that flows through it:

inductor_energy

which, considering the value of the voltage previously calculated, can be solved as follows:

inductor_energy_value

where I is the current flowing through the inductor at the time the energy is calculated.

When choosing an inductor for a circuit, the following parameters must be considered:

  • the value of the inductance in Henry

  • the max current the inductor can sustain; failure to specify that could cause the inductor to overheat, since the wire could be too thin to deal with the required current;

  • the max voltage that can be applied to the inductor; an excessive voltage on the inductor could cause sparks due to insufficient insulation of the wire.

Inductors In Series

Let’s consider a series of inductors of different inductance values and let’s calculate the equivalent inductance.

inductors_in_series

All the inductors, being in series, are traversed by the same current. And since each inductor has its own inductance value, each one will store a different amount of energy:

inductors_series_energies.png

The total energy stored in the inductors is therefore:

inductors_series_total_energy.png

So, the equivalent inductance is clearly:

series_inductance.png

which we can generalize as:

series_inductance_gen.png

Inductors In Parallel

In the case of inductors in parallel, they are all subject to the same voltage and are traversed by a different current:

inductors_in_parallel.png

parallel_inductors_voltages.png

From these equations we can find the currents by integration:

parallel_inductors_currents.png

The total amount of current is therefore:

parallel_inductors_total_current.png

So we can say that the equivalent inductance of a parallel of inductors can be determined through the formula:

parallel_inductors_formula_1.png

or, more in general:

parallel_inductors_formula_2.png

All the formulas presented here are very general and can be applied to both DC and AC circuits. Note, however, that since AC circuits have a variable voltage and current, the application of the formulas in AC is a little more challenging then in DC. But this is a story for another time.

Deep Dive into the Theremin v.2 Audio Amplifier

See YouTube video https://www.youtube.com/watch?v=90T0ZkN-oos&t=3s for further details.

audio_amp_schematic

The Theremin v2 audio amplifier and preview audio amplifier share the same schematic.

The input of the amplifier is on pins P1 and P2, where P2 is the ground connection. The input is supposed to be DC decoupled with a small electrolytic capacitor, as suggested by the data sheet of the integrated circuit TDA2003, which is the main component of this amplifier. However, the capacitor was not put in this schematic because it will find its place directly on the output of the audio generator stage. No reason to have two capacitors in series: one at the output of the audio generator and one at the input of the audio amplifier.

The signal coming from P1 goes to a voltage divider consisting of the resistor R1 and the potentiometer R2. When the potentiometer cursor is all the way toward ground, there will be no signal going to pin 1 of the IC and, therefore, the amplifier will be silent. When the potentiometer cursor is all the way toward resistor R1, the signal coming from P1 is divided exactly in half, since the value of the potentiometer R2 and of resistor R1 are exactly the same. This is the position where the volume will be the maximum possible.

I chose the value of 100K for both R1 and R2 so to have a relatively high impedance toward the audio generator, so its output won’t be affected by the movement of the potentiometer. At the same time, I wanted a value of R2 low enough to keep the input of the amplifier solidly on the ground when there is no input signal.

R2 is a logarithmic potentiometer, which means that its value changes according to a logarithmic scale. This is done to compensate for how we perceive the sound volume: the potentiometer increases the power of the sound wave in the opposite way as our ear perceives it. This way, we feel that the volume increases and decreases linearly when we turn the potentiometer.

The output from pin 4 of the IC goes to the loudspeaker through capacitor C3, which is used to prevent that a DC current goes all the way through. In addition, capacitor C5 and R6 provide a high pass filter that shorts to ground all the high frequencies that are not supposed to go to the loudspeaker. Values of capacitors C3 and C5 and the resistor R6 reflect the suggestion on the data sheet.

data_sheet_schematic

The power supply comes in through pins P5 and P6 and is filtered by capacitors C1 and C2.

C2 takes care of shunting toward ground any unwanted high frequency signal coming from/to the power supply. Capacitor C1, which is a high value electrolytic capacitor, is mostly used to boost the input current when the amplifier requires sudden increases due to peaks of the volume. These capacitors have also the values suggested by the data sheet.

The last part of the amplifier is the negative feedback loop, connected between pins 4 and 2 of the TDA2003. The loop is composed of capacitors C4 and C6, and the resistors R3, R4, and R5.

The negative feedback is used to limit the output power of the amplifier outside the region of frequencies where it is requested, thus preventing auto oscillations of the amplifier, which would otherwise behave like an oscillator.

The values of these components are again those suggested by the data sheet. However, C4 and R5, which correspond to Cx and Rx in the data sheet, are calculated based on the data sheet formulas visible in the above picture, where B is the maximum frequency of the signal that we want to amplify, which I set to 10kHz. I don’t believe we want to hear from the Theremin sounds that are above that frequency.

For further information on this amplifier and for a demonstration on how it works, please refer to my corresponding YouTube video.

A Tester For Zener Diodes

Zener diodes are used for several purposes, from providing a reference voltage, to protecting sensitive circuits from being destroyed by the wrong input.

Today, I will show you how these diodes work and how to build a simple circuit to measure their most important characteristic, the reverse breakdown voltage. To know more on this topic, please watch the companion video posted on YouTube.

A zener diode looks like a regular diode and actually behaves as such when directly biased (positive voltage on the anode).

However, when inversely biased (negative voltage on the anode), a zener diode behaves in a completely different way.

Let’s take a look at its characteristic I-V diagram:

zener_characteristic

You can see that in the region of direct (or forward) bias, the zener behaves just like any diode. It also seems like in the inverse bias it behaves like a regular diode.  However, there is a big difference between the two.

For a regular diode, the reverse breakdown voltage is very high, in the order of 100V or more, sometimes much more. Such high that you never think at it when you use regular diodes, and you assume that with inverse bias the diode just does not conduct electric current.

For a zener diode, instead, the reverse breakdown voltage is low, in the order of one or two digit volts. Therefore, it is very easy in an electronic circuit to bring this kind of diode to reach the condition when it will start conducing electric current even if inversely polarized.

We actually exploit this behaviour to create reference voltages, or to provide a protection against unwanted voltages at the input of certain circuits, or a ton of other things.

The behaviour of a diode depends in fact upon the way it was fabricated, and in particular upon how it was doped. Regular diodes are lightly doped, while zener diodes are heavily doped. Depending on the amount of doping on both the P and the N side of the junction, the reverse breakdown voltage changes. That way, manufacturers can create zener diodes within a large range of breakdown voltages.

Problem is, manufacturers often don’t put the value of the breakdown voltage on the body of the components. Instead, they put some internal code or, sometimes, nothing at all.

So, if you had a number of such diodes on your workbench, how to distinguish them from one another?

Meet the zener tester.

It is a device that allows you to measure the reverse breakdown voltage, so you know if the diode works and what that voltage is.

How such a tester works? From the I-V diagram above, you can see that the characteristic of the zener diode is an almost vertical line when polarized in the reverse bias region. For any current value in that vertical line, the voltage is always the same and corresponds to the breakdown voltage. So, if we circulate a current at any point of that vertical line, we can measure at the terminals of that diode its breakdown voltage.

The zener tester I’m showing you today does just that: forces a current into the zener diode so we can measure the value of the breakdown voltage. We choose this current in such a way that it is high enough to stay away from the point where the characteristic is not linear, but low enough to avoid dissipating inside the diode a power that the diode itself cannot handle.

The following link allows you to download an archive containing the schematic of such device, along with the OpenSCAD code to 3D print the box for the device.

zener_diodes_tester_files

In the schematic you’ll see that I used a ready-made boost converter and a digital voltmeter. Here are the links to the store where I bought them. Of course you are free to use any other equivalent component. It will work as well.

BOOST CONVERTER

DIGITAL VOLTMETER

Please make sure to watch the YouTube video that completes the information I provided in this post. Between the two, you should have a complete view of the design of the device and should be able to build it.

Happy experiments!