Capacitors, Inductors and Resonance

Question File Number 09



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The Inductor

This figure shows a conductor carrying a current. A magnetic field is set up around the conductor as concentric circles.

wirefield

If a coil of wire has a current flowing through it, the magnetic flux due to each turn will link with every other turn and produce the same sort of magnetic field as a permanent magnet. Such a coil is called a solenoid as shown here. It acts as a magnet only when current is flowing through it.

solonoid

The magnetic polarity of the solenoid can be determined from the direction of current flow as seen looking in the ends, as shown in the diagram. From that the direction in which the magnetic field is acting can also be found.

Solenoids or electromagnets are widely used in electronic equipment. Loudspeakers, headphones, moving coil microphones, measuring instruments, transformers and such things, depend on electromagnetism for their operation.

An inductor may be air-cored or have a solid core.

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Magnetic Materials

Magnetic materials in common use for the cores of solenoids are:

Soft iron: easy to magnetise and demagnetise. Used for motor pole-pieces.

Silicon iron: used for transformer laminations and AC motors. Low-loss.

Nickel iron alloy: also known as radio metal and mu-metal, is used for high-class audio transformers and cathode-ray tube screens.

Ferrites: iron oxide based materials used for a wide range of applications in radio and electronics generally. The characteristics depend on the mix of materials in the core and is extremely varied. Also known as ferroxdure and ferroxcube.

Permanent magnets: tungsten steel, and alloys of iron, nickel, aluminium, cobalt, ceramic, and titanium are used. Iron oxides can also be used.

The magnetic field surrounding a coil does not appear immediately the circuit is connected. It takes time to grow from nothing at the moment of switch-on to its maximum. The time taken for this depends on many factors, including the number of turns in the coil, the current, the core material, and the self-resistance of the coil. Similarly, when the current is switched off the field takes time to decay.

It should be noted that the current in the coil takes time to rise to its maximum. This must be compared to the capacitor where at switch-on, the voltage across the capacitor takes time to rise to its maximum (see below).

Inductors in Series and Parallel

lseriespar

A coil has inductance, measured in henries. The values of inductance used in radio range from several henries (H) to parts of a microhenry (H). The inductance of a coil depends on the number of turns and the core details.

When inductors are connected in series, the number of turns is effectively increased. So too is the inductance, and the effective inductance of the circuit is the sum of the individual inductances.

The diagram shows series and parallel inductors. These calculations apply only to inductors which are not coupled magnetically. Where there is coupling between coils, the total inductance is also affected by the amount of coupling.

As with the resistor, for amateur radio examination purposes, you can visualise the resulting value of inductors like this:

Putting two inductors in series in effect increases the number of turns so the inductance value increases.

Putting two inductors in parallel, in effect decreases the effective inductance

Like the resistor, we can use visualised examples to easily work out what happens with two (or more) typical inductors of the same value. Put two in parallel, the value will halve. Put two in series and the effective inductance value will be double.

In a perfect inductor there is no loss of energy. The energy is stored in the magnetic field surrounding the inductor and (in an AC circuit) it changes in magnitude and sign twice in each cycle. The opposition to the flow of current is called the inductive reactance and is denoted by XL. Reactance is an opposition to current resulting from a storage of energy. The relationship is:

XL = 2π f L where f is the frequency in hertz, L is the inductance in henries, and XL is the reactance expressed in ohms. For our purposes, 2π can be taken to be the figure 6.28.

Note that as the frequency rises, the inductive reactance also rises.

In practice there is no such thing as a perfect inductor and it is usual to consider the practical component to be a circuit containing both a resistor R (the inductor's resistance) and L the inductor.

LRcct

Where resistance and inductance (inductive reactance) exist in a circuit together, they are combined into a term, called impedance, representing their combined total opposition to current flow.

Reactances can be added together directly. But resistances and reactances must be added together vectorially (as vectors), to get impedance. More about this follows below.

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Types of Inductors

Inductors used in radio can range from a straight wire at UHF to large chokes and transformers used for filtering the ripple from the output of power supplies and in audio amplifiers. Values of inductors range from nano-henries to tens of henries. It is convenient to group them into three categories.

Air core: to keep losses to a minimum it is necessary to keep the self-resistance of coils as low as possible.

This means using the thickest possible wire within the space available. Another reason for using thick wire or even tubing is to reduce the skin effect losses at high frequencies. Direct current is spread uniformly over the cross-section of the conductor, but alternating current moves closer to the surface as the frequency increases.

Thus it is necessary to provide a large surface to minimise radio frequency resistance which is known as skin effect. Inductors used can range from a 25 mm diameter tube for a slot antenna on VHF to 50 turns of 22 swg wire on a 7.5 cm diameter former for the tank circuit of a 1.8 MHz transmitter.

The only adjustment available with air core inductors is by tapping all or part of a turn, or by varying the spacing between turns.

Ferrite or iron dust core: by inserting a ferrite or iron dust core in a coil it is possible to double its inductance. This means that it is also possible to halve the size of a coil for a given inductance. If the core is threaded, its position within the coil can be varied to alter the inductance. Some high-grade communications receivers have a system of cam-operated cores which are used for tuning. The type of material used for the cores or slugs is of importance and care must be taken to use the right grade for the right frequency band. This type of coil is used throughout the HF range, and into the VHF, for low-level signal circuits. Losses in the cores make them unsuitable for use in power circuits. Values range from a few microhenries to about a millihenry.

Similar types of coils have been made using brass cores instead of ferrite. The effect of this is to reduce the inductance by about 20%. They also increase the losses.

Iron core: this classification includes chokes and transformers, both of which have laminated iron cores. Transformers are described in the next section. A choke is a single winding and a transformer has two or more windings. Typical values of inductance for chokes range from 0.1 of a henry to 50 henries.

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The Transformer

Any two coils magnetically linked will act as transformers. Transformers come in as many forms as inductors, air or dust cored as well as the more familiar iron-cored type. The iron-core can take several forms.

Transformer

The simple transformer comprises two or more inductors (windings) sharing a common core. See Power Supplies for an explanation.

An alternating current is fed to one of the windings. The operation can explained by considering the magnetic field of the input winding, the primary, sweeping through the secondary winding to induce an AC current in the secondary. These principles are considered in AC

The Turns Ratio

A common task for a transformer is to provide an AC supply at a voltage suitable for rectifying to produce a stated DC output.

The number of turns on each winding determines the output voltage from the transformer. The output voltage from the secondary is proportional to the ratio of the turns on the windings.

For example, if the secondary has half as many turns as there are on the primary, and 100V AC is applied to the primary, the output from the secondary will be 50V.

Transformers can be step-up or step-down (in voltage). With twice as many turns on the secondary as there are on the primary and 100 V applied, the output would be 200V.

The impedance ratio is proportional to the square of the turns ratio. We can use transformers to change impedances. This property is one of the most important properties in the use of transformers.

The power output from the secondary winding cannot exceed the power fed into the primary. Ignoring losses, a step-down in voltage means that an increase in current from that lower-voltage winding is possible. Similarly, a step-up in voltage means a decrease in the current output. So the gauge of wire used for the secondary winding may be different to the wire used for the primary. (The term gauge of wire relates to its cross-sectional area.)

Iron-cored transformers are used for audio frequencies and for power supplies. Audio frequency transformers are designed to give suitable efficiency to frequencies up to 25 kHz. For speech and domestic quality radio reproduction the core material used is stalloy, while the laminations of high-fidelity transformers are made of mu-metal The construction is the same as for chokes and the same considerations of size and power rating apply.

Transformer Losses

There are two main types of loss in a transformer, the iron loss and the copper loss. Copper loss is due to the resistance of the wire used for the windings. Copper loss can be reduced by using large diameter wire for the windings, but there is a limit to the size and weight and some copper loss is unavoidable.

One of the principal iron losses is caused by eddy currents flowing in the core. The magnetic circuit (core) can be considered to be a one-turn coil and heavy currents could flow causing very high losses. To reduce this eddy current loss the core is made up from many thin slices of iron called laminations which are insulated from each another.

Toroidal Core Transformers

If the core of a transformer is of specially-selected material and is formed into a complete loop as shown in this diagram, nearly all the flux lies within the core and there is very little leakage, or flux outside the core.

toroid

This results in very little unwanted coupling to adjacent magnetic circuits, and is a very desirable feature in some circuits. An application is in the common SWR Bridge. Measurements

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The Capacitor

Cap1

The capacitor has a wide range of uses in radio. Its fundamental construction comprises a pair of metal plates. The plates are separated by a dielectric which may be air or some insulating material. The diagram shows a diagrammatic capacitor with its circuit symbol.

A capacitor exhibits capacitance, a value measured in farads (F). In practice, capacitors range from a few picofarads (pF) to many microfarads (F) in value.

The value of a capacitor is determined by the dimensions of its plates, the distance between the plates, (see the arrows on the above diagram) and the characteristics of the dielectric.

For amateur radio examination purposes, you can visualise the value of a capacitor like this:

The capacity is proportional to the area of the plates. So putting two capacitors in parallel in effect increases the size of the plates so the capacity value increases.

The capacity is inversely proportional to the distance between the plates - increase the spacing (the thickness of the dielectric) and the capacitance will decrease.

Putting two capacitors in series, in effect does the same thing, it increases the effective distance between plates so the value decreases.

Like the resistor, we can use this visualised example to easily work out what happens with two typical capacitors of the same value. Put two in parallel, the value will double. Put two in series and the effective value will be half.

Voltage Ratings

The dielectric space between the capacitor plates may be made very small, to achieve a high capacitance in a component of small physical size. A high voltage applied between the plates may cause a break-down in the dielectric causing a short-circuit or other damage. So each capacitor must be given a voltage rating in addition to its capacitance rating, by its manufacturer.

In practice, capacitors may sometimes be wired in series to achieve higher effective voltage ratings for special applications. Two capacitors of the same voltage rating wired in series will produce a resulting capacitor of double the effective voltage rating but of half the capacity. (In practice, there are other things that should be taken into consideration too, but these are not of concern for this amateur radio examination.)

Most types of capacitors have low leakage. This means that they can hold high levels of charge for long periods after voltage has been removed, and for this reason should be treated with caution when servicing equipment in which high voltages are used.

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Capacitor Types and Characteristics

Ctypes

Air-spaced: are used mainly as variable capacitors for tuning. Air-spaced capacitors consist of a set of fixed plates, with a set of moving plates, mounted on a spindle, that exactly mesh with the fixed plates. The moving plates are controlled by a knob or by a screwdriver adjustment. Values from 5 to 500 pF and voltage rating up to 500 V for receivers, and several thousand volt for transmitters are available. There are special types for some applications.

Electrolytic: in this type of capacitor the dielectric is a very thin layer of aluminium oxide formed on the plates by a conducting chemical compound when a DC potential is applied. The surface of the plates, which are made of aluminium foil, may be etched to increase the surface area and hence the capacitance value. The large surface area and the very thin dielectric means that a very large capacitance value can be obtained. Another similar type uses tantalum oxide as the dielectric. Tantalums, range from 0.01 to 3000 F with voltages up to 100 volts, while aluminium types range from 0.1 F to nearly 1 F with a voltage range of from 3 to 700 volts. The higher the value of capacitance, the lower is the voltage range. In both types a leakage current is essential to maintain the dielectric, and they are generally used on a DC voltage for smoothing and de-coupling. The polarising voltage must also be high enough to keep the leakage current flowing, otherwise the capacitance value will be reduced.

Capacitive Reactance

In a capacitor, energy is not dissipated but is continually being stored and released. Opposition is to current flow which results from energy storage rather than energy loss is called reactance. It is dependent on frequency and is denoted by Xc.

The unit for capacitive reactance is ohms. The formula, where f is the frequency in hertz, C is the capacity in farads, and Xc is the capacitive reactance in ohms, is:

CapX

Note that capacitive reactance is expressed in ohms, and for purposes of easier explanation, we here give it a negative sign. This will be considered again in the explanations following below. This is a to convenient way for us to make a distinction between capacitive reactance and inductive reactance.

Note that as the frequency rises, the capacitive reactance decreases. (Compare this with inductive reactance - see above.)

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Current, Voltage and Phase

As mentioned earlier, resistance and reactance must be added vectorially. For a first and elementary understanding for the purposes of this amateur radio examination, a visual approach is possible.

When a capacitor is first connected to a supply, a large current flows while the capacitor builds up its charge. The current leads on the voltage. At full-charge, the voltage across the capacitor will be high but the current will be zero.

This is in contrast with an inductor. When an inductor is first connected, there is a large voltage drop across it which decreases as the current rises as the magnetic field builds up. The voltage leads on the current. After a period, the voltage across the inductor will be low but the current through it will be high.

The behaviour of inductors, capacitors, and resistors (L, C, and R), in AC circuits is more complex than it is in DC circuits. Fortunately we can envisage the basic principles.

Resistance is an opposition to current that results in power loss, while reactance is an opposition to current resulting from a storing of energy. For our purposes (at the moment), keep resistance and reactance separate.

Resonance

The resistor-inductor-capacitor (LCR) or series circuit

SeriesCct

The diagram shows a series circuit of R, L, and C. The total reactance is the resulting difference between XL and XC. The voltage across each component adds up to the total input voltage, VT.

If we start off with VT at a low frequency, the voltage across the capacitor is much greater than that across the inductor, and the resultant reactance is capacitive. As the frequency increases, the capacitive reactance decreases and the inductive reactance increases, and at very high frequencies the resultant reactance is inductive.

At some intermediate frequency the two reactances are equal (but of opposite sign). At this frequency, the impedance of the circuit is purely resistive due to R, and there is no resultant reactance. The sum of the two reactances is zero. This is the resonant frequency.

The resonant frequency f (in Hz) is given by this formula, in which L is in henries and C is in farads.

FreqForm

(This formula can be easily derived from the relationship XL + XC.= 0 (at resonance) by substituting for XL and for XC and then solving for f.)

There is only one frequency at which resonance occurs. The reason for resonance is that L and C have exact opposite numerical values at resonance. In an inductor the voltage leads the current by 90 degrees while in a capacitor the voltage lags the current by 90 degrees. We have observed this earlier by adopting a different sign for capacitive reactance.

It is important to note that in a series circuit at resonance the reactances (being equal but of different sign), in effect disappear (i.e.= 0), leaving only the resistance R.

Note the effect of the square root in this reciprocal relationship. If the inductance (or, separately, the capacitance) is quadrupled in value, the frequency is reduced to half.

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The Q of a Circuit

ResCurve

As the input frequency signal moves away from resonance, the impedance of the circuit rises and the current through it falls. This diagram shows the variation of current with frequency for a series LCR circuit for two different values of R.

Note that the current at high and low frequencies is fixed by inductive and capacitive reactances respectively, while the current at resonance is determined by the resistance. Having a low value of resistance in a tuned circuit is very desirable to be sure of maximum selectivity. The effectiveness of a resonant circuit is stated by its Q.

The ratio of the voltage across the inductor to the input voltage VT is the magnification factor or quality factor of the circuit. Expressed another way, Q is the reactance divided by the resistance. The smaller the resistance, the higher the Q.

Q is a ratio of two numerical values and hence is a figure alone and has no unit.

Note that the term Q can also apply to capacitors as well as inductors, they are both storers of energy.

The Q or magnification factor, or quality factor, of the circuit, depends on the coil construction, and can range from 5 to 500 in transmitting tuned circuits, to 150 in transformers and to 3000 or so in helical and VHF cavity resonators.

The Parallel LCR Circuit

Parallel cct

The other common LCR circuit is that with L and C in parallel. This diagram shows the parallel circuit. In practice there is always some resistance in the inductor, so it is usual to consider a resistor in series with the inductor.

At frequencies below resonance, the reactance of the inductor is much less than that of the capacitor and the circuit is mainly inductive reactive. At frequencies higher than resonance, the capacitor has the lower reactance and the circuit is mainly capacitive reactive. In between the two frequencies, there is a frequency at which the two reactances are equal. Only a very small current flows in the circuit at this frequency.

The parallel resonant circuit obeys the same formula for resonant frequency as the series resonant one, but at resonance the parallel resonant circuit has a very high impedance. The resistance at resonance offered by the parallel resonant circuit is very high if the resistance of the inductance is very small, and is known as the dynamic resistance.

The parallel tuned circuit is used to select one particular signal frequency from among others. It does this by rejecting the resonant frequency because of its high impedance. For this reason, it is sometimes called the rejector circuit.

Both series and parallel resonant circuits may be found in radio receivers and transmitters. In oscillators and transmitters they are sometimes known as tank circuits. This comes from their ability to store energy in the electrostatic field of a capacitor or in the magnetic field of an inductor.

The selectivity of a tuned circuit is the ability of the tuned circuit to select a signal at the resonant frequency and reject other signals that are close to that frequency. A measure of the selectivity is the Q, or quality factor.

Q can be the quality factor of either the capacitor or the inductor in the circuit, but it is generally taken as the quality factor of the inductor. Inductors have Q at any frequency, not just at resonance. The Q of an inductor falls with frequency. Q is dependent on the resistance of the coil up to about 30 MHz, and above this losses in the capacitor may be significant.

Measuring Q

One way to determine the the Q of a tuned circuit is to measure the band-width (BW) between the two points referred to as the half-power points.

In a series-tuned circuit. these are the two frequency points at which the current has fallen to 0.707 of its value at resonance. There are two of these half-power points in a resonant circuit, one above and one below resonance.

If the frequency difference between the half-power points is given as the bandwidth BW, (the difference in frequency is a number), and if the resonant frequency is f, then:

Qform

Because Q is a ratio, f and BW can be in the same units, Hz, kHz, or MHz.

Bandwidth

BWidth

As we have seen, Q is related to the bandwidth of the response curve. Different bandwidths are optimum for different modes. For example: 1. TV signals require 5.5 MHz or more, 2. AM broadcast stations require 9 kHz or more, 3. Single sideband (SSB) signals require 2.4 kHz, 4. Continuous wave (CW ) signals require 100 Hz.

This diagram shows the ideal shape of a response curve, a square response, with the width exactly the bandwidth required. Receiver Operation

This ideal shape is difficult and expensive to achieve. The bandwidth is measured at the half-power or - 3 dB points which are where the voltage has fallen to 0.707 of its value at resonance (See Decibels). This is also called the nose of the curve. The skirt bandwidth is usually measured at the - 60 dB point and the ratio between the two is a measure of the quality of tuned circuits, particularly filters. The ideal is 1 to 1, but is never achieved.

The Quartz Crystal

xtal

Crystals are thin slices cut from various planes of a quartz crystal. In its simplest construction form, a quartz crystal can be considered as a very thin slice of quartz sandwiched between two metal plates.

This diagram shows the equivalent circuit of a crystal. Quartz crystals are a much-used form of resonant circuit. The piezoelectric properties of quartz are used to produce the equivalent of a highly-stable resonant circuit with a very high Q. (See Oscillators)

C2 is the capacitance of the crystal holder. The Q of a crystal is of the order of 20 000. The crystal has two resonances, one series and one parallel. C1 and L resonate as a series circuit. C2 together with L and C1 resonant as a parallel circuit.

Crystals are not high power devices, at low frequencies their power dissipation is limited to about 10 mW and for the higher frequencies it is limited to 2 mW.



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Compiled Mon Nov 29 2010 at 8:11:49pm


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