Fundamentals Of Electric Circuits

Some basic nomenclature and information on electric circuits.

When we think of an electric circuit, the first thing that comes to mind is a bunch of electric and electronic components put together to make a device that provides a certain functionality.

This is totally right, of course, but it is not a precise definition of an electric circuit. In fact, circuits can be defined in different ways, depending on the particular aspect of them that we want to highlight.

If we want to define a circuit with respect to the wave length of the voltage and current that the circuit handles, we will distinguish between “lumped” and “distributed” circuits.

If instead we want to highlight the kind of electric or electronic components the circuit is made of, then we need to talk about linear and non-linear circuits.

And, finally, depending on the temporal stability of the components, we can talk about “time-invariant” and “time-variant” circuits.

And, of course, we can consider combinations of the properties and, therefore, combinations of different circuit types.

In this article, I will go through the above list of circuit types and provide a proper definition and description of each one of them. Combination of those types will lead to combined definitions and descriptions of the basic types. However, in this context, I will go through only the basic types of circuit, leaving to you the task to provide definitions and descriptions of the combinations.

Lumped and Distributed Circuits

Let’s go now into more details about these kind of circuits. What differentiates these circuits is the size of their components compared to the size of the wavelength of the electric current flowing through them. Hum, well, I guess we need to back up a little bit first. Let’s start with the types of current.

We have two kind of currents: the direct current, or DC, and the alternate current, or AC. DC current always flows in the same direction and never changes. AC current has usually the shape of a sine wave, or some other shape that periodically inverts the direction of the current. The number of times the current inverts its flow depends on the number of times the voltage flips its polarity. When the Voltage goes from positive to negative n number of times in a second, we say that its frequency is n and it is measured in Hertz.

Since the flow of the current in a conductor is not instantaneous, it makes sense to think that a change in the voltage at the ends of a conductor makes the current change progressively through the conductor. And if the voltage keeps changing back and forth, so does the current. At the end, both the instantaneous values of the current and the voltage across the length of the conductor follow in space the same shape of the voltage changes (in time) applied at the ends of the conductor. So, if the voltage changes in time like a sine wave at the ends of the conductor, it will follow a similar shape in space throughout the length of the conductor. The length of such a sine wave in space throughout the conductor is called wavelength of the voltage, or the current. Such length in space depends on the length in time of the corresponding voltage applied at the ends of the conductor.

We can calculate the wavelength using the following formula:

The Greek letter “lambda” represents the wavelength, the f represents the frequency of the voltage, which is how many times the voltage goes from positive to negative and back in one second. And, finally, the letter ‘c’ represents the speed of light. Yes, you got it right, it is the speed of light!

When the wavelength is much longer than the physical size of the components of the circuit, we say that it is a lumped circuit, because the components are just small lumps with respect to the wavelength itself. In such a case voltage and current are practically constant across the whole length of the component.

When the wavelength is comparable with the physical size of the components of the circuit, we say that it is a distributed circuit, because the components are so big compared with the size of the wavelength that the wave itself is distributed across them. In such a case voltage and current will be different in different sections of the component, at any instant in time.

Distributed circuits cannot be analyzed with normal algebra equations. For those, it is necessary to heavily use calculus. Example of such circuits are the microwave circuits, those used to deal with radars and satellite signals.

Linear and Non-Linear Circuits

A component is defined as linear if it can be represented in a I-V (current-voltage) diagram with a straight line.

A component is defined as not linear if its representation on a I-V diagram is not a straight line.

Simply put, a linear circuit is one made with only linear components, while a non-linear circuit is one that has at least one non-linear component.

Note the difference: for a circuit to be linear, ALL the components must be linear; for a circuit to be non-linear, it is enough that ONLY ONE component is non-linear. All the other components can be linear and still the whole circuit is non linear.

Time-Invariant and Time-Variant Circuits

Difference between time-invariant and time-variant circuits is also straightforward.

A time-invariant component is one for which the measurements that define it never change over time.

A time-variant component is one for which the measurements that define it can change over time.

As a result, a time-invariant circuit is one made only with time-invariant components. A time-variant circuit is one made with at least one time-variant component. This is a subtle definition, very similar in its form to the one for linear and non-linear circuits.

Circuits in Series and in Parallel

One last thing I would like to discuss about circuits in general is related on their topology or, in other words, on how the components in a circuit are connected to each other.

There are two main configurations of connected components:

  1. components in series
  2. components in parallel

Components are said to be connected in series when they are traversed by the same current.

Components are said to be connected in parallel when the voltage on each one of them is the same.

When we talk about connections in series and in parallel, of course, we refer to components directly attached to one another, at least on one terminal, or lead. Components that are far away in the circuit diagram, or that are not directly connected together cannot be defined as in series or in parallel.

So, now, can you tell me what kind of circuit is the one at the very beginning of this article? Put your answer in the comments and let’s see who gets it right.

How To Choose A Resistor

How do we choose the right resistor when designing and building an electronic circuit? Here are the major parameters that should be kept into account.

bunch_of_resistors

A resistor is a component made out of a poor conducting material, so that it can offer a resistance to the flow of the current.

You can think to resistance in terms of the obstacles that charges encounter when moving from one end to the other of a conductor. The more obstacles, the higher the resistance. In a metallic wire, for example, the charges are the electrons of the conduction band (see this post and this other one for further details).

In today’s post I would like to address an issue that sometimes is underestimated when designing an electronic circuit: how to choose the right resistor for the job.

Resistors are not all the same. Besides the resistance value that distinguishes one from the other, there are other factors that are important as well.

Here is a list of all the important factors, why they are important, and what are the consequences of not choosing a resistor based on each specific factor.

  • The first thing that comes to mind is the tolerance, which is usually provided on the body of the resistor itself, along with its resistance value.

resistor_color_bands

In color coded resistors, the tolerance is defined by the band that is far away from all the others. In the above picture, for example, it is the gold band, which means that the tolerance is of 5%. In other resistors, where the resistance is explicitly written on the body of the resistor, the tolerance is usually written in clear along with the resistance. More in general, you’ll have to refer to the data sheet provided by the constructor to figure out its tolerance.
Tolerance is an important factor for those circuits that require very precise resistors, like measuring instruments and the like. It is also important when the resistor is used for the polarization of a critical component. If the resistors used in the project have a tolerance that is too high, the whole circuit may not function properly because the actual value of the resistor is too different from the one that was required.

  • Operating Temperature. This depends both from the ambient conditions and by the temperature raise produced by the power dissipation. There are two reasons to keep the temperature range into account. First, resistors slightly change their resistance with the change of the temperature. Using the resistor outside its temperature range would cause a variation greater than the one considered by the tolerance. Second, but not last, when the resistor is traversed by current it heats up. As long as the current stays within a range for which the power dissipation is not exceeded, everything is fine. Otherwise, the resistor can easily overheat and burn.

scorched_resistor

  • Maximum Voltage. Operating a resistor above its maximum voltage rating may cause sparks that would destroy the resistor.

burned_resistor

Resistors used in low power circuits usually have a maximum voltage in the order of at least 100V, and that’s why people usually don’t care or it doesn’t even know that there is such a parameter. In fact, low voltage circuits will normally never exceed the maximum voltage of any resistor. However, there are specific applications where voltages in the circuits can be above the 100V threshold. In such cases, it is important to verify that the resistors used in the circuit can withstand those voltages.

  • Temperature coefficient. This is the parameter that tells us how much the resistance changes per degree Celsius. It depends on the material the resistor is made of, but also on the heat dissipation capability of the component. Some resistors are built with an embedded heat sink to reduce the value of this factor.

power_resistor

This information becomes important in those cases where it is known that the resistor is going to dissipate a considerable amount of power. Based on that, it is possible to figure out if the resistor needs an external heat sink and, eventually, the heat sink thermal resistance.

  • Parasitic Capacitance and Inductance. A real resistor does not have only a resistance but also a very low value of capacity and inductance that may affect its functionality at high frequencies.

equivalent_resistor

These parasitic capacitance and inductance are caused by the physical dimensions and shape of the component and cannot be avoided. When working at high frequencies, these values need to be taken into account, since they will generate both capacitive and inductive reactance that will affect the value of the resistor at the particular frequency it is going to be used.

  • Packaging. This keeps into account where and how the resistor is going to be mounted. It can be a through holes resistor, which is provided with two leads to make the connections. The leads are usually inserted in the holes of a perforated board or of a Printed Circuit Board (PCB). Or, the resistor can be a Surface Mounted one. This has no wires, just two pads that can be directly soldered on a Surface Mounted technology (SMT) PCB. Other factors affecting the packaging include the possibility of attaching it to an external heat sink, and/or the necessity to properly ventilate it, to guarantee enough heat dissipation.

 

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.