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Transcript

Capacitance and Electric Fields Chapter 13 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 Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 13.‹#› Introduction 13.1 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 Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 13.‹#› Capacitors and Capacitance 13.2 Capacitors consist of two conducting surfaces separated by an insulating layer called a dielectric Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 13.‹#› 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.‹#› 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 C Q V 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.‹#› 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 Q C V Q CV 10 5 10 100 μC Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 13.‹#› Alternating Voltages and Currents 13.3 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 Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 13.‹#› 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.‹#› The Effect of a Capacitor’s Dimensions 13.4 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 C A d 0 r A d Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 13.‹#› Electric Fields 13.5 The charge on the capacitor produces an electric field with an electric field strength E given by E V d the units of E are volts/metre (V/m) Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 13.‹#› All insulating materials have a maximum value for the field strength that they can withstand – the dielectric strength Em 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 Em 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 13.‹#› 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 DQ A Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 13.‹#› Combining the earlier equations it is relatively easy to show that D E 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 13.‹#› Capacitors in Series and Parallel 13.6 Capacitors in parallel – consider a voltage V applied across two capacitors – then the charge on each is Q VC 1 Q VC 1 2 2 – if the two capacitors are replaced with a single capacitor C which has a similar effect as the pair, then Charge stored on C Q Q 1 2 VC VC VC 1 C C C 1 2 2 Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 13.‹#› 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 C1 is that supplied by the upper plate of C2. 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 V V V 1 2 Q QQ C C C 1 2 1 1 1 C C C Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 1 2 OHT 13.‹#› Voltage and Current 13.7 The voltage across a capacitor is directly related to the charge on the capacitor 1 Q V Idt C C Alternatively, since Q = CV we can see that dQ dV C dt dt and since dQ/dt is equal to current, it follows that dV I C dt Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 13.‹#› Consider the circuit shown here – capacitor is initially discharged voltage across it will be zero – switch is closed at t = 0 – VC is initially zero hence VR is initially V hence I is initially V/R – as the capacitor charges: VC increases VR decreases hence I decreases we have exponential behaviour Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 13.‹#› 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 13.‹#› Sinusoidal Voltages and Currents 13.8 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 Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 13.‹#› 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 13.‹#› Energy Stored in a Charged Capacitor 13.9 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 = CV, so the energy needed E is given by 1 E CV dV CV 2 0 2 V Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 13.‹#› Alternatively, since V = Q/C 2 2 1 1 Q 1 Q E CV 2 C 2 2 C 2C 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: 1 1 2 E CV 10 5 1002 50 mJ 2 2 Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 13.‹#› Circuit Symbols Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 13.10 OHT 13.‹#› 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 ½CV2 or ½Q2/C Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 13.‹#›