Transmission lines

Transmission lines

At the beginning of my experiences on electricity, I came across a piece of coaxial cable with the label “50 ohm” printed along its outer sheath. The coaxial cable is constituted by a single conductor surrounded by a braided wire coating, with a plastic insulating material that separates the two. The outer conductor (braided) completely surrounds the inner conductor (single wire), the two conductors are isolated from each other for the entire length of the cable. This type of wiring is often used to conduct weak voltage signals (of reduced amplitude), due to its excellent ability to shield those signals from external interference.

Construction of coaxial cables.

I was confused by the label “50 ohm” this coaxial cable. In that way the two conductors, isolated from one another by a relatively thick layer of plastic, have 50 ohms of resistance between them? By measuring the resistance between the external conductors and internal I found infinite resistance (open circuit), just as I would have expected two insulated wires. The measurement of the resistances of the two conductors from one end of the other cable indicated almost zero ohms resistance: Once again, exactly what I would expect from continuous lengths and continuous wire. Nowhere I am able to measure 50 Ω of resistance of this cable, no matter what points I connected my ohmmeter.

When asked an expert told me that it is a cable for high frequency signals CA. The continuous current (DC) – as well as that used by my ohmmeter to check the resistance of the cable – It shows that the two conductors are completely isolated from one another, with an almost infinite resistance between the two. However, due to the effects of capacitance and inductance distributed along the length of the cable, the cable response to voltages that rapidly change is such as to act as an impedance, (absorbed current which is proportional to the applied voltage). What we normally think of only a pair of wires becomes an important circuit element in the presence of transient signals and high-frequency AC, with all its characteristic properties. When we express these properties, we refer to the pair of wires as a transmission line .

This chapter explores the behavior of the transmission line. Many effects of the transmission line does not appear significantly in AC circuits of the line frequency, or in continuous circuits, so I was not me occupied them until that time. However, in circuits involving high frequencies and / or long lengths of cable, the effects are very significant. The practical effect of the transmission line applications abound in radio frequency communications circuits, including computer networks, and in the low-frequency subject to voltage transients circuits (“peaks”) like lightning on the power lines.

Circuits and the speed of light

I'll try to explain very complex concepts with simple examples and with the minimum reduced formulas, for mathematical insights in support of the above reference to university treated, it would have been very easy to describe the behavior with mathematical formulas.

The hard part is just that, explain complex things as if you were explaining it to a guy of medium.

Suppose we have a simple circuit consisting of an LED controlled by a switch. When the switch is closed, the LEd will light instantly. When the switch is open, the LED turns off immediately:

The LED seems to respond immediately to the switch.

For all practical purposes, the switch effect is instantaneous compared to the LED status. Although the electrons are moving very slowly through the wires, the overall effect of the electrons that push against one another occurs at the speed of light (circa 300,000 Km al second!).

What would happen, however, if the wires carrying power to the lamp were long 300.000 Km? Since we know that the effects of electricity have a finite speed (although very fast), a set of very long cables should introduce a delay in the circuit, delaying the action switch on the lamp.

At the speed of light, LED respond after 1 second.

Assuming no resistance along the length of 600.000 Km of both threads, the LED will turn on about one second after the switch closes. Although the construction and operation of superconducting cables so long would entail enormous practical problems, it is theoretically possible, and so this “thought experiment” It is valid. When the switch is opened again, the lamp will continue to receive power for one second after the opening of the switch, then deenergises.

One way to imagine the electrons inside a conductor is thinking of them as railway wagons in a train: connected together with a small game in pairs. When a railroad car (electron) starts moving, He pushes the front and pulling the one behind, but not before the mating game is absorbed. In this way, the movement is transferred from wagon to wagon (from electron to electron) at a maximum speed limited by loose coupling, determining a motion transfer much faster from the left end of the train (circuit) the right end than the actual speed of the wagons (electrons):

The movement is transmitted efficiently from one wagon.

Another analogy, perhaps more suitable for the theme of the transmission lines, is that of the waves in the water. Suppose that a flat object is suddenly moved horizontally along the surface of the water, so as to produce a wave in front of him. The wave will travel when the water molecules will collide with each other, transferring the wave motion along the surface of the water much faster than the water molecules themselves are actually traveling:

Swell in water.

Similarly, the “coupling” of movement of the electrons traveling at approximately the speed of light, although the electrons themselves do not move so quickly. In a very long circuit, This speed “coupling” It would become apparent to a human observer in the form of a short time interval between the switch action and the action of the lamp.

characteristic Impedance

Suppose, however, to have a series of parallel wires of length infinity, Without light at the end.

What happens when we close the switch?

As there is no longer a load at the end of the wires, this circuit is open. There would be at current?

An infinite transmission line.

Although we are able to avoid the resistance of the wire through the use of superconductors in this “thought experiment”, we can not eliminate the ability of the wires along the lengths. Any pair of conductors separated by an insulating medium creates capacitance between those conductors:

Equivalent circuit diagram showing parasitic capacitance between the conductors.

The voltage applied between two conductors creates an electric field between these conductors. Energy is stored in this electric field, and this accumulation of energy translates into an opposition to voltage change. The reaction of a capacity against voltage changes is described by the equation I = C (from / dt), which tells us that the current will be proportional to the voltage variation speed in time. Therefore, when the switch is closed, the capacitance between the conductors will react against the sudden increase in voltage charging and drawing current from the source. According to equation, an instantaneous increase of the applied voltage (as a product from the perfect closure of the switch) It gives rise to a current of infinite charge.

However, the current absorbed by a pair of parallel wires will not be infinite, since there is an impedance of long wires due to the inductance series.

Equivalent circuit diagram showing parasitic capacitance and inductance.

The current through any conductor develops a magnetic field of magnitude proportional to the current itself. Energy is stored in this magnetic field, This accumulation of energy translates into an opposition to the current change.

Each wire develops a magnetic field as it carries charging current for the capacitance between the wires, and thereby it drops the voltage according to the equation of inductance

and L = (of / dt).

This voltage drop limits the voltage variation velocity through the distributed capacitance, preventing current to always attain infinite intensity:

Because the electrons in the two wires transfer the movement towards each other and almost at the speed of light, the “wave front” the voltage and current will propagate along the length of the wires to the same speed, resulting in the distributed capacitance and inductance which is charged progressively to full voltage and current, respectively, in this way:

loaded transmission line.

Start wave propagation.

Continue wave propagation.

The end result of these interactions is a constant current of limited amplitude supplied by the battery. In other words, this pair of cables draws current from the source until the switch is closed, acting as a constant load. The wires are not only more conductors of electrical current and voltage bearers, but now they constitute a circuit component itself, with unique features. The two wires are no longer simply a pair of conductors, but rather a transmission line.

As a constant load, the response of the transmission line to the applied voltage is resistive rather than reactive, despite being constituted solely by inductance and capacitance (assuming superconducting cables with zero resistance). We can say this because there is no difference from the perspective of the battery between a resistor that dissipates energy and eternally an infinite transmission line that absorbs energy eternally. The impedance (resistance) This line is called in ohms characteristic impedance, and it is fixed by the geometry of the two conductors. For a parallel-wire line with air insulation, the characteristic impedance can be calculated as such:

If the transmission line is coaxial construction, the characteristic impedance follows a different equation:

In both equations, They must be used measurement units identical in both terms of the fraction. If the insulating material is different from the air (or by vacuum), will be influenced by both the characteristic impedance that the speed of propagation. The relationship between the real propagation speed of a transmission line and the speed of light in vacuum is called factor of speed of this line.

The speed factor is purely a factor of the relative permittivity of the insulating material (otherwise known as dielectric constant), defined as the ratio between the permittivity of the electric field of a material and that of a pure vacuum. The velocity factor of any type of cable, coaxial or other, It can simply be calculated by the following formula:

The characteristic impedance is also known as natural impedance It refers to the equivalent resistance of a transmission line if it were infinitely long, due to the distributed capacitance and inductance when “at where” Voltage and current are propagated along its length

It can be seen in one of the first two equations that the characteristic impedance of a transmission line (FROM) increases as the spacing of the conductor. If the conductors are moved away from each other, the distributed capacitance will decrease (greater distance between the “plates” condenser) and distributed inductance will increase (less cancellation of the two opposing magnetic fields). A minor parallel capacity and a greater inductance in series result in a lower current drawn from the line for any given amount of applied voltage, which by definition it is a higher impedance. The other way around, approaching the two conductors increases the parallel capacitance and decreases the inductance in series. Both changes produce a higher current consumption for a given applied voltage, equivalent to a lower impedance.

Barring any dissipative effect as “dispersion” dielectric and conductor resistance, the characteristic of a transmission line impedance is equal to the square root of the ratio of the line inductance per unit length divided by the capacity of the line per unit length:

Lines for length transmission over

A transmission line of infinite length is an abstraction interesting, but physically impossible. All transmission lines have a finite length, and as such does not behave exactly like an endless line. If that piece of cable from 50 Oh “RG-58 / The” I measured with a ohmmeter years ago it was infinitely long, in reality I would have been able to measure 50 Ω of resistance between the inner and outer conductors. But it was not infinite and therefore measured as “open” (infinite resistance).

However, the characteristic impedance of a transmission line is also important when it comes to limited lengths. An older term for the characteristic impedance, I like it for its descriptive value, is peak impedance. If a transient voltage (a “impulse”) It is applied to the end of a transmission line, the line absorbs a current proportional to the magnitude of the peak voltage divided by the surge impedance of the line (I = E / FROM).

If the end of a transmission line is open, ie not connected, the current “then” which propagates along the length of the line must stop at the end, because the electrons can not flow where there is not a continuous path. This sudden cessation of current at the end of the line causes a “accumulation” along the entire length of the transmission line, since electrons can no longer find any place to go.

Imagine the previous example train walking on the rails: if the main car crashes suddenly on an immense barricade, it will stop, causing the arrest of a wagon behind the first as soon as the loose coupling is resumed, which causes the next railcar stops as soon as the next slack is loosened, and so on until the last car does not stop.

reflected Wave

When this “accumulation” of electrons propagates, the current to the battery ceases and the line acts as a simple open circuit. Everything happens very quickly for a reasonable length of transmission lines, and therefore a measure of the line ohmmeter never reveals the short period in which the line is actually behaves like a resistor. For a long hollow one km with a velocity factor of 0.66 (the signal propagation speed equal to 66% the speed of light, O 198,000 Km per second), you need only 1 / 198.000 second (5.050 microseconds) a signal to travel from one extreme to another. In order for the current signal reaches the end of the line and is “reflects” back to the source, the round-trip time is twice that figure, or 10,101 μs.

The high-speed measuring instruments are able to detect this new transit time from the source to the end of the line and vice versa, and they can be used in order to determine the length of cable. This technique can also be used to determine the presence and the localization of a break in one or both of the conductors of the cable, since a current “It will be reflected” on the breakage of its wire as if it were an open circuit. The instruments designed for such purposes are called reflectors of the time domain (TDR). The basic principle is identical to that of the sonar detection field: generating a sound pulse and measuring the time required for the echo return.

A similar phenomenon occurs if the end of a transmission line is shorted: when the opposite voltage waveform reaches the end of the line, It is reflected back to the source, because the voltage can not exist between two electrically common points. When this reflected wave reaches the source, the source sees the entire transmission line as a short circuit.

A simple experiment illustrates the phenomenon of reflection of waves in transmission lines. Take a piece of string on one side and “whip” with a quick up and down motion of the wrist. You can see a wave traveling along the rope until it completely dissipates because of friction (forgive me but it is intuitive that are not a professional designer):

Transmission line loss

This is analogous to a long transmission line with internal loss: the signal becomes steadily weaker as it propagates along the length of the line, never go back to the source. However, if the far end of the rope is attached to a solid object in a previous point to the total incident wave dissipation, a second wave will be reflected on your hand:

reflected Wave

Usually, the purpose of a transmission line is to convey electrical energy from one point to another. Even if the signals are intended only for information and not to supply a significant load device, the ideal situation would be that all the energy of the original signal to pass from the source to the load, and therefore it is completely absorbed or dissipated by the load for maximum signal-to-noise ratio. Therefore, “loss” along the length of a transmission line is undesirable, as well as the reflected waves, since the reflected energy is energy not delivered to the terminal device.

The reflections can be eliminated from the transmission line if the load impedance is exactly equal to the characteristic impedance (“surge”) the line. Eg, coaxial cable 50 Ω which is open or short-circuited will reflect all the energy incident at the source. However, if at the end of a resistor is connected by cable 50 Oh, there will be no reflected energy, all the signal energy will be dissipated by the resistor.

This makes perfect sense if we go back to our hypothetical example of an infinitely long transmission line. A transmission line with characteristic impedance of 50 Ω and infinite length behaves exactly like a resistance of 50 Ω measured from end.

If we cut this line to a finite length, It will behave as a resistor 50 Ω to a constant source of DC voltage for a short period, but then they behave like an open circuit or short, depending on the conditions in which the line is finished.

However, HE we finish the line with a resistor 50 Oh, the line will still result in a time as a resistor 50 Oh, as if it were of infinite length .

Basically, a terminal resistor that matches the natural impedance of the transmission line causes the line “appear” infinitely long from the perspective of the source, since a resistor eternally has the ability to dissipate the energy in the same way in which an infinitely long transmission line is capable of absorbing energy eternally.

The reflected waves occur even if the terminating resistor is not exactly equal to the characteristic impedance of the transmission line, not only. Even if the reflected energy will not be total with an impedance of slight misalignment termination, It will be partial. This happens regardless of whether the terminating resistor is greater O lower the characteristic impedance of the line.

Reflections of a reflected wave may also occur at the end of origin of a transmission line, if the internal impedance of the source (Thevenin equivalent impedance) It is not exactly equal to the characteristic impedance of the line. A reflected wave which returns to the source will be completely dissipated if the impedance of the source corresponds to the line, but it will be reflected towards the end of the line as another incident wave, at least partially, if the source impedance does not match the line. This type of reflection can be particularly annoying, since it seems that the source has passed another boost.

For now I'll stop here, I want to give a chance to assimilate the concepts accurately, as soon as I integrate the topic of transmission lines, my intent is to arrive at last describe the waveguides and resonant cavities.

Give me the time to organize the best way to explain it in a simple manner without an endless series of formulas that would scare most readers.

Amilcare Greetings

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2 replies
  1. Avatar
    romanelli says:

    Good examples clarifiers.
    Lines, antennas and everything that revolves around the electromagnetic waves is very difficult to represent in concrete, without abstractions and mathematics.
    I encourage the author to continue because I feel very interesting content.
    Also I think I have a lot to learn for how to present the arguments.

    Approvals

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