7 Passive electronic components
7.1 Passive circuit components
7.1.1 Resistors
The ability of material to transport charge carriers is termed the conductivity s of that material. The reciprocal of conductivity is resistivity r. The latter is defined as the ratio between the electric field strength E (V/m) and the resultant current density J (A/m2), hence: E= ◊r J (7.1) and s r = 1 (7.2)
The conductivity of common materials ranges almost from zero almost up to infinity. It is determined by the concentration of charge carriers and their specific mobility within the material. On the basis of their conductivity, materials can be classified as isolators, semiconductors or conductors (Figure 7.1). Conductivity is a material property that is independent of dimensions (except in special cases as with very thin films).
Figure 7.1 The conductivity of some materials.
The resistance R (unit ohm, W) of a piece of material is defined as the ratio between the voltage V across two points of the material and the resulting current I (Ohm's law). The resistance depends on the resistivity of the material as well as on its dimensions. The reciprocal of the resistance is the conductance G (unit S- 1
, sometimes called siemens, S).
A huge range of resistor types is now commercially available. They are categorized as fixed resistors, adjustable resistors and variable resistors. Important criteria to consider when working with resistors are: the resistance value or range, the inaccuracy (tolerance of the value), the temperature sensitivity (temperature coefficient) and the maximum tolerated temperature, voltage, current and power. Ultimately all these properties will be determined by the material type in use and the construction. With respect to the materials, we may distinguish between carbon film, metal film and metal wire resistors.
Metal film and metal wire resistors have much better stability than the carbon film types of resistors but the latter allow for the realization of much higher resistance values. Metal wire resistors show a self-inductance and capacitance that may not always be ignored. Self-inductance can be minimized by adopting special winding methods like those implemented in expensive, highly accurate types of resistors. Usually the desired resistance value of a film resistor (made of carbon or metal) is achieved by cutting a spiral groove into a film around the cylindrical body. Table 7.1 provides an overview of the main properties of the various resistor types in common use.
Table 7.1. Some properties of different resistor types.
Range tolerance temperature coefficient maximum temperature maximum power carbon film 1 … 107 5 … 10 –500 … +200 155 0.2 … 1 metal film (NiCr) 1 … 10 7 0.1 … 2 50 175 0.2 … 1 wire wound (NiCr) 0.1 … 5·10 4 5 … 10 –80 … +140 350 1 … 20
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Except in very special cases, the resistance values are normalized and fit into various series (International Electro-technical Commission, 1952). The basic norm is the E-12 series where each decade is subdivided logarithmically into 12 parts. The subsequent values between 10 and 100 W are:
10, 12, 15, 18, 22, 27, 33, 39, 47, 56, 68, 82, 100.
Resistors with narrower tolerances belong to other series, for instance the E-24 series (increasing by a factor 2410), E-48, E-96 and E-192. For further information on commercially available resistors the reader is referred to manufacturers’ information books.
7.1.2 Capacitors
A capacitor is a set of two conductors separated from each other by an isolating material called the dielectric. The capacitance C (unit farad, F) of this set is defined as the ratio between the charge Q being displaced from one conductor to the other and the voltage V resulting between them:
C Q
V
= (7.3)
When the charge varies, so too does the voltage. The charge transport to or from the conductor per unit of time is the current I (I = dQ/dt), which is why the relation between the current and the voltage of an ideal capacitor is
I CdV dt
= (7.4)
The capacitance of a capacitor is determined by the conductor and dielectric geometry. The dielectric permittivity or dielectric constant is the ability of a material to be polarized. The dielectric constant of a material that cannot form electric dipoles is, by definition, just 1. In such situations those materials behave as a vacuum.
The capacitance of a capacitor consisting of two parallel flat plates placed distance d apart which have a surface area of A and are situated in a vacuum equals
C A
d
=e0 (7.5)
(this is an approximation which is only valid when the surface dimensions are large compared to d). In this expression, e0 is the absolute or natural permittivity or the permittivity of vacuum : e0 = 8.85·10
-12 F/m. When the space between the conductors
of a capacitor is filled with a material that has the dielectric constant er, then the
C A d
r
=e e0 (7.6)
A large capacitance requires a dielectric material with large relative permittivity. Table 7.2 shows the relative permittivity of several materials.
Table 7.2. The dielectric constants of various materials.
material er material er
Vacuum 1 ceramic, Al2O3 10
air (0°C, 1 atm) 1.000576 porcelain 6 - 8
water, at 0°C 87.74 titanate 15 - 12,000
at 20°C 80.10 plastic, PVC 3 - 5
glass, quartz 3.75 teflon 2.1
Pyrex 7740 5.00 nylon 3 - 4
Corning 8870 9.5 rubber, Hevea 2.9
mica 5 - 8 silicone 3.12 - 3.30
There is an enormous range of capacitor types. Like with resistors, capacitors are distinguished in three ways as: fixed, adjustable (trimming capacitors) and variable. Common dielectric materials are: air (for trimming capacitors), mica, ceramic materials, paper, plastic and electrolytic materials.
A capacitor never behaves as a pure capacitance; it shows essential anomalies. The leading deviations are linked to the dielectric loss, the temperature coefficient and the breakthrough voltage. We will discuss these features separately below.
When an AC voltage is applied to the terminals of a capacitor, the dipoles in the dielectric must continuously change their direction. The resulting dissipation (heat loss) is known as dielectric loss. It is modeled by a resistance in parallel to the capacitance. A quality measure for dielectric loss is the ratio between the loss resistance current and the capacitance current. This ratio is called the loss angle d, defined as tan d = |IR|/|IC| = 1/wRC (Figure 7.2).
Figure 7.2. A model of a capacitor with dielectric losses.
The temperature dependence of the capacitance is caused by the expansion coefficient of the materials and the temperature sensitivity of the dielectric constant. There are capacitors with built-in temperature compensation. The non-linearity of the capacitance is generally negligible.
A very large capacitance value is achieved by having a very thin dielectric. However, electrical field strength increases as thickness decreases, thus limiting the breakthrough voltage of the capacitor.
Very high capacitance values are achieved with electrolytic capacitors. Their dielectric consists of a very thin oxide layer, formed from the material of one of the conductors
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(usually aluminum). The surface of the conductor is first stained which creates pores in the metal surface, thus increasing the active area. Then the surface material is anodically oxidized. The resulting aluminum oxide can withstand a high electrical field strength (of up to 700 V/µm), allowing a very thin layer to form without breakthrough occurring. In the wet aluminum capacitor the counter electrode is connected to the oxide by a layer of paper impregnated with boric acid and a second layer of (non-anodized) aluminum. In the dry type, the paper is replaced by a fibrous material (glass) impregnated with manganine. Instead of aluminum, tantalum can also be used with anodic tantalum oxide as the dielectric. Electrolytic capacitors are unipolar, in other words, they can only function correctly at the proper polarity of the voltage (as indicated on the encapsulation layer).
The highest values of the capacitance that can be obtained are 1 F (electrolytic types); the lowest values are around 0.5 pF and the inaccuracy is about 0.3 pF (ceramic types). Accurate and stable capacitances are realized when plastic film (for instance polystyrene) or mica are used as the dielectric. Mica capacitors have a loss angle that corresponds to tan d ª 0.0002 and are very stable: 10-6
/°C. They can withstand high electric field strength (60 kV/mm) and therefore also high voltage (up to 5 kV). 7.1.3 Inductors and transformers
Inductors and transformers are components based on the phenomenon of induction: a varying magnetic field produces an electric voltage in a wire surrounding that magnetic field (Figure 7.3a). Quantities that describe this effect are the magnetic induction B (unit tesla, T) and the magnetic flux F (unit weber, Wb). With a uniform magnetic induction field (the field has the same magnitude and direction within the space being considered), the flux equals
F =B A◊ (7.7)
where A is the surface area of the loop formed by the conductor (Figure 7.3a). In vector notation, the magnetic flux is defined as
F =
ÚÚ
B dAr r◊A
(7.8)
This definition also holds for non-uniform fields. Both B and F are defined in such a way that the induced voltage satisfies the expression
V d
dt
ind= -
F
(7.9)
(i.e. Faraday's law of induction). The induced voltage equals the rate of change of the magnetic flux. In a uniform magnetic field the induced voltage is vind = –AdB/dt.
The induced voltage is not influenced when short-circuiting the wire loop. The current that will flow there will be equal to vind/R and R will be the resistance of the loop. The
The induced voltage also increases as the loop area and number of turns increases (Figure 7.3b). Each loop undergoes the same flux change and is connected in series, so vind = –ndF/dt, with n as the number of turns.
A varying magnetic field produces an induction voltage in a conductor. The reverse effect also exists: when current I passes through a conductor this produces a magnetic field, its strength H (unit A/m) satisfies the Biot and Savart law:
Figure 7.3. (a) A varying magnetic field produces an induction voltage in a conductor, (b) the induction voltage increases with the number of turns, (c) a current through a
conducting wire generates a magnetic field.
H I
r =
2p (7.10)
This empiric law, which only applies to a straight wire, shows that the magnetic field strength is inversely proportional to the distance r from the wire (Figure 7.3c). Another form of this law is Ampère's law in which the integral of the magnetic field strength over a closed contour equals the total current through the contour:
r r H ds i C n n N
Ú
◊ =Â
=1 (7.11)This law shows that the magnetic field produced by a group of conducting wires is the sum of the fields from each individual wire.
When two conductors are mutually coupled, an electric current in one conductor produces a magnetic field which, as it varies over the course of time, induces a voltage in the other conductor. This is called mutual inductance. Inevitably, this effect also occurs in a single conductor because when a varying current goes through the conductor it will produce a varying magnetic field that induces a voltage in the same conductor. This phenomenon is called self-inductance. The direction of the induced voltage is such that it tends to reduce its origin (Lenz's law). The degree of self- inductance depends very much on the geometry of the conductor. Self-inductance increases when the conductor is coupled more closely to itself, such as inside a coil.
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We have introduced two magnetic quantities: magnetic induction B and magnetic field strength H. Because of the different definitions, these two quantities do not have the same dimensions. They are joined by the equation
B=m0H (7.12)
where µ0 = 4p·10 -7
Vs/Am, the permeability of vacuum. Obviously, the inductive coupling between two conductors (or within one conductor) depends on the properties of the medium between the conductors. The property that accounts for the ability of the material to be magnetized (which is magnetically polarized) can be described on the basis of relative permeability, µr. For a vacuum, the relative permeability is 1. The
relation between B and H in a magnetizable medium is
B=m m0 rH (7.13)
A high value of µr means a higher flux for the same magnetic field strength. For most
materials µr is about 1. Some electronic applications require materials with a high
permeability (for instance supermalloy: µr = 250,000). The relative permeability of
iron is about 7500.
Magnetic materials are much less ideal than dielectric materials because the magnetic losses are high and the material shows magnetic saturation, thus resulting in strong non-linearity and hysteresis.
The self-inductance L of a conductor is the ratio between magnetic flux F and current i through the conductor:
L i
=F (7.14)
The relation between the (AC) current through the inductor and the voltage across it is
v Ldi dt
= (7.15)
We saw that the capacitance is increased by applying a high permittivity dielectric. Likewise, the self-inductance of an inductor is increased if materials that have high relative permeability are used. Inductors are constructed as coils (with or without a magnetic core) and come in various shapes (for instance solenoidal, toroidal).
Whenever possible inductors are avoided in electronic circuits as they are expensive and bulky (especially when designed for low frequencies). They display non-linearity and hysteresis (due to the magnetic properties of the core material), resistance in the wires and capacitance between the turns. Inductors are introduced to filters for signals in the medium and higher frequency ranges.
A transformer is based on signal transfer by mutual inductance. In its most basic form a transformer has two windings (primary and secondary windings) around a core of a material with high permeability. The ratio between the number of turns in the primary and secondary windings is the turns ratio n. For an ideal transformer with ratio 1:n turns, the voltage transfer (from primary winding to secondary winding) is just n, and the current transfer is 1/n. A proper coupling between the two windings is achieved by overlaying them and by applying a toroidal core.
Transformers are mainly used for system power supply, for down converting mains voltage or for up converting voltages in battery-operated instruments. They are also found in special types of (de)modulators in the middle and high frequency ranges.