lr I

Figure PH.23 Simple electrical circuit consisting of a source of electrical potential (V), current (l) and a resistance (R)

When a current of 1 ampere flows for 1 s is an electrical circuit, a charge of 1 C will pass any point in that circuit.

An electrical resistance (R) provides a restriction to the motion of electrons. If the positive and negative poles of an electrical cell were connected with a wire of very low resistance, then the electrons would flow very rapidly and there would be a high current. However, connecting them by a wire that does not conduct so well reduces the current. In fact, the current (I) is inversely proportional to the resistance, but proportional to the driving potential, difference (V).

Resistance is measured in ohms and those encountered in the human body are normally thousands or millions of ohms.

When electrical charge flows through a resistor, energy is dissipated and the resistor heats up. In mechanics, the work done is the product of the applied force and the distance moved. The electrical equivalent of force is the electromotive force (voltage), and the equivalent of distance is the amount of charge which is moved. Thus, the total work done, E, is expressed by:

or the electrical power (the rate of work done), W, is given by:

Work is measured in joules (J) and power is measured in watts (W). Power can be written in two other ways by eliminating either current or voltage from the equation above, thus:

The heating effect in electrical equipment with fixed resistance increases in proportion to the square of the supply voltage, or the square of the current drawn.

Capacitance and Inductance

Capacitance

A circuit containing only a supply voltage (Va), a resistance and a switch is shown in Figure PH.24.

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Figure PH.24 Purely resistive electrical circuit where the current rises instantly when the switch is closed

When the switch is closed the voltage across the resistor (Vm) and the current through it (I) instantly rise in unison as in the schematic graphs. However, the introduction of capacitance into this simple circuit changes the relationship between the applied voltage and the current.

Consider the capacitive circuit shown in Figure PH.25. A capacitor consists of two conducting plates placed close together, but separated by an insulating layer. When the switch is first closed, electrons that are crowded together at the cathode flow through the resistor and onto the left-hand plate of the capacitor. However, as the current flows, the electrons on the capacitor plate become as crowded together as those at the cathode, the voltage on the capacitor has reached the supply voltage. At the same time, the current steadily diminishes to zero as the capacitor becomes charged'. Both the rise in the capacitor voltage and the reduction in current follow exponential forms.

Initially, only the resistance in the circuit determines the current, since at first the capacitor offers no impedance to the flow of electrons. The greater the resistance, the lower will be the initial current, and the longer the capacitor will take to charge. However, as the capacitor charges, it begins to determine the current, until fully charged and then allows no more current to flow. The larger the capacitor, the longer it will take to charge. The product of the capacitance and resistance, CR, is the time constant for the circuit. This also determines the rate at which the current falls towards zero.

A similar circuit is shown in Figure PH.26, but this includes an inductor. Inductors consist of coils of wire wrapped around a magnetizable material (the inductor core), which is generally made from an iron compound. As current begins to flow, the core becomes magnetized, the degree of magnetization being proportional to the current. As the magnetic flux changes, it induces a voltage in the coil of wire in the opposite sense to the applied voltage, thus reducing the current flow. It is the change in current that causes this opposing voltage, so that when the current is steady, the inductor has no effect. The time constant for this circuit is L/R, since the larger the inductance the greater will be the opposition to changing current, but the larger the resistance the lower will be the changes in current and opposition to those changes.

The capacitor and inductor can be thought of as having opposite effects in an electrical circuit. The capacitor presents a high impedance to a steady applied voltage, but a low impedance when there are changes in the voltage. In contrast the impedance of an inductor is lowest when there is a steady voltage, and high when there are rapid changes in the applied voltage. In a capacitive circuit, the current changes more quickly than the voltage, while in an inductive circuit, current changes lag behind changes in voltage.

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Figure PH.26 Resistive/inductive circuit where the inductor prevents rapid changes in current, but conducts freely when a steady current is reached

Consideration of the effect of oscillating voltages on such circuits requires the concept of impedance. This term has been used loosely interchangeably with the term 'resistance'. However, resistance should only be used in describing electrical components where the electrical energy is lost as heat or some other form of energy. For components where the energy is stored rather than dissipated (such as capacitors and inductors), the correct term for their opposition to the flow of electrical current is impedance.

A sinusoidally varying voltage (V = Vmax sin rat) is applied to a circuit with a resistor and capacitor in series in Figure PH.27. As the voltage passes through zero, the current reaches a maximum since the capacitor is being charged at the highest rate. However, when the voltage reaches its maximum, no current flows in the circuit since the capacitor is fully charged. Under conditions like this, where the voltage reaches a maximum at a different time to the maximum current, it is said that the phases of the voltage and current differ.

The impedance of a circuit like this, where the voltage and current are out of phase, is defined as the maximum voltage divided by the maximum current:

The impedance of this resistive-capacitive circuit changes with the oscillation frequency. At low frequencies the impedance is high, since the capacitor behaves as an open switch when the applied voltage is steady. As the frequency increases, the impedance of the capacitor decreases until, at very high frequencies, the circuit behaves as if the capacitor were replaced by a conducting piece of wire. A resistive-inductive circuit behaves in the opposite matter. The impedance of the inductor increases with frequency, while for steady voltages the inductor behaves like a short circuit. When resistors, inductors and capacitors are combined in series in a single circuit, the impedance reaches a minimum (equal simply to the resistance) when the effects of the capacitor and inductor cancel each other out. The frequency at which this happens is known as the resonance frequency and this circuit forms the basis of the oscillators found in radio transmitters and receivers.

Filters

The properties of capacitive and inductive circuits are employed in filters. Filters are designed to allow signals

Figure PH. 27

Resistive/capacitive circuit with an oscillating applied voltage. A sinusoidal applied voltage waveform leads to sinusoidal current waveform, but the current lags behind the voltage

Resistive/capacitive circuit with an oscillating applied voltage. A sinusoidal applied voltage waveform leads to sinusoidal current waveform, but the current lags behind the voltage only within a certain frequency range to pass through a system. The resistive/capacitive circuit acts as filter to cut out all but high frequencies (a 'high-pass' filter). At high frequencies the impedance is very low, while for low frequencies the filter behaves like an open circuit (very high impedance). Similarly, an inductive circuit removes high frequencies (a 'low-pass' filter) from an oscillating voltage, while an inductive/capacitive circuit allows only frequencies near the resonance frequency to pass through (a 'band-pass' filter).

Transformers are found widely in electrical equipment and are used either to increase or decrease an alternating voltage. Transformers use electromagnetic principles. A transformer is a pair of inductors that share a common magnetic core. A schematic representation of a transformer is shown in Figure PH.28. An oscillating voltage is applied to the left-hand conducting circuit which is wrapped around an iron core. The conducting wire is electrically insulated from the magnetic iron. However, the oscillating current in the conductor induces an oscillating magnetic field in the core, which is present throughout the core; therefore, the coil of wire on the right-hand side experiences the same oscillating magnetic field. This oscillating field induces a voltage in the second coil, the relative magnitude of which depends only on the relative numbers of turns in the two coils on the left and right and sides. In Figure PH.28, an oscillating voltage Vi is applied to the left-hand circuit, which is electrically isolated from, but linked through magnetic linkage to, the right-hand circuit. The ration of the two voltages depends on the ratio of the number of turns on the coils.

Thus:

If the number of turns on the right is larger than on the left the voltage is increased (a step-up transformer), while if it is lower, then the voltage is decreased (a step-down transformer). For a step-up transformer, this may at first seem paradoxical; an increase in voltage at no cost. However, if the increased voltage were used to do some work, for example to drive an electric motor, it would be found that the current in the left-hand circuit would be higher than in the right hand side. Since power is the product of the current and the voltage, the power provided in the left-hand circuit would be just equal to the power dissipated in the right-hand circuit.

Circuit Diagram Symbols

The more common electrical components are identified by individual symbols. These are illustrated in Figure PH.29.

Electrical Safety

Electrical hazards result from either mishandling or faulty electrical equipment. Electrical currents cause muscular contraction, either directly through muscle stimulation or indirectly through stimulation of nerves. This can cause injury and death either by asphyxiation as the muscles in the thorax and abdomen cannot relax to allow normal breathing, or through disruption of the normal rhythmic myocardial muscle contraction (fibrillation). In extreme cases, the heart muscles contracts tonically. The risk from muscle contraction decreases as the frequency of the alternating current increases, and is worst for direct currents passing through the body.

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