Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 13.1 Capacitance and Electric Fields  Introduction  Capacitors and Capacitance  Alternating Voltages and Currents  The Effect of a Capacitor’s Dimensions  Electric Fields  Capacitors in Series and Parallel  Voltage and Current  Sinusoidal Voltages and Currents  Energy Stored in a Charged Capacitor  Circuit Symbols Chapter 13

Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 13.2 Introduction  We noted earlier that an electric current represents a flow of charge  A capacitor can store electric charge and can therefore store electrical energy  Capacitors are often used in association with alternating currents and voltages  They are a key component in almost all electronic circuits 13.1

Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 13.3 Capacitors and Capacitance  Capacitors consist of two conducting surfaces separated by an insulating layer called a dielectric 13.2

Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 13.4  A simple capacitor circuit –when switch is closed electrons flow from top plate into battery and from battery onto bottom plate –charge produces an electric field across the capacitor and a voltage across it

Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 13.5  For a given capacitor the stored charge q is directly proportional to the voltage across it V  The constant of proportionality is the capacitance C and thus  If the charge is measured in coulombs and the voltage in volts, then the capacitance is in farads

Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 13.6  Example – see Example 13.1 in course text A 10  F capacitor has 10 V across it. What quantity of charge is stored in it? From above

Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 13.7 Alternating Voltages and Currents  A constant current cannot flow through a capacitor –however, since the voltage across a capacitor is proportional to the charge on it, an alternating voltage must correspond to an alternating charge, and hence to current flowing into and out of the capacitor –this can give the impression that an alternating current flows through the capacitor 13.3

Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 13.8  A mechanical analogy may help to explain this –consider a window - air cannot pass through it, but sound (which is a fluctuation in air pressure) can

Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 13.9 The Effect of a Capacitor’s Dimensions  The capacitance of a capacitor is directly proportional to its area A, and inversely proportional to the distance between its plates d. Hence C  A/d –the constant of proportionality is the permittivity  of the dielectric –the permittivity is normally expressed as the product of the absolute permittivity  0 and the relative permittivity  r of the dielectric used 13.4

Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT Electric Fields  The charge on the capacitor produces an electric field with an electric field strength E given by the units of E are volts/metre (V/m) 13.5

Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT  All insulating materials have a maximum value for the field strength that they can withstand –the dielectric strength E m  To produce maximum capacitance for a given size of capacitor we want d to be as small as possible –however, as d is decreased the electric field E is increased –if E exceed E m the dielectric will break down –there is therefore a compromise between physical size and breakdown voltage

Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT  The force between positive and negative charges is described in terms of the electric flux linking them –measured in coulombs (as for electric charge) –a charge Q will produce a total flux of Q coulombs  We also define the electric flux density D as the flux per unit area  In a capacitor we can almost always ignore edge effects, and

Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT  Combining the earlier equations it is relatively easy to show that  Thus the permittivity of the dielectric within a capacitor is equal to the ratio of the electric flux density to the electric field strength.

Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT Capacitors in Series and Parallel  Capacitors in parallel –consider a voltage V applied across two capacitors –then the charge on each is –if the two capacitors are replaced with a single capacitor C which has a similar effect as the pair, then 13.6

Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT  Capacitors in series –consider a voltage V applied across two capacitors in series –the only charge that can be applied to the lower plate of C 1 is that supplied by the upper plate of C 2. Therefore the charge on each capacitor must be identical. Let this be Q, and therefore if a single capacitor C has the same effect as the pair, then

Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT Voltage and Current  The voltage across a capacitor is directly related to the charge on the capacitor  Alternatively, since Q = CV we can see that and since dQ/dt is equal to current, it follows that 13.7

Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT  Consider the circuit shown here –capacitor is initially discharged  voltage across it will be zero –switch is closed at t = 0 –V C is initially zero  hence V R is initially V  hence I is initially V/R –as the capacitor charges:  V C increases  V R decreases  hence I decreases  we have exponential behaviour

Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT  Time constant –charging current is determined by R and the voltage across it –increasing R will increase the time taken to charge C –increasing C will also increase time taken to charge C –time required to charge to a particular voltage is determined by the product CR –this product is the time constant  (greek tau)  See Computer Simulation Exercises 13.1 and 13.2 in the course text

Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT Sinusoidal Voltages and Currents  Consider the application of a sinusoidal voltage to a capacitor –from above I = C dV/dt –current is directly proportional to the differential of the voltage –the differential of a sine wave is a cosine wave –the current is phase-shifted by 90  with respect to the voltage 13.8

Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT  Since I = C dV/dt the magnitude of the current is related to the rate of change of the voltage –in sinusoidal voltages the rate of change is determined by the frequency –hence capacitors are frequency dependent in their characteristics  We will return to look at frequency dependence in later lectures.

Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT Energy Stored in a Charged Capacitor  To move a charge Q through a potential difference V requires an amount of energy QV  As we charge up a capacitor we repeatedly add small amounts of charge  Q by moving them through a voltage equal to the voltage on the capacitor  Since Q = CV, it follows that  Q = C  V, so the energy needed E is given by 13.9

Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT  Alternatively, since V = Q/C  Example – see Example 13.7 in the course text Calculate the energy stored in a 10  F capacitor when it is charged to 100 V. From above:

Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT Circuit Symbols 13.10

Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT Key Points  A capacitor consists of two plates separated by a dielectric  The charge stored on a capacitor is proportional to V  A capacitor blocks DC but appears to pass AC  The capacitance of several capacitors in parallel is equal to the sum of their individual capacitances  The capacitance of several capacitors in series is equal to the reciprocal of the sum of the reciprocals of the individual capacitances  In AC circuits current leads voltage by 90  in a capacitor  The energy stored in a capacitor is ½CV 2 or ½Q 2 /C