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ELE 1001: Basic Electrical Technology Lecture 6 Capacitors Overview of Topics What is a Capacitor? Transient behavior of a Capacitive circuit. Capacitor as an energy storage element. Department of Electrical & Electronics Engineering Capacitors A capacitor is a passive electric device that can store energy in the electric field between a pair of closely spaced conductors Capacitance (C): Property which opposes the rate of change of voltage; Unit – Farad (F) The capacitive current is proportional to the rate of change of voltage across it ; 𝒊𝒄 = 𝑪. 𝒅𝑽𝒄 𝒅𝒕 Circuit representation is Department of Electrical & Electronics Engineering Terminologies Electric field strength, E = V / d volts / m Electric flux density, D = Q / A coulombs / m2 Permittivity of free space, 0 = 8.85 x 10-12 F /m Relative permittivity, r Capacitance of parallel capacitor C = 0 r A / d plate Department of Electrical & Electronics Engineering Equivalent Capacitance In Series: 𝟏 𝟏 𝟏 𝟏 = + + ……………+ 𝑪𝒆𝒒 𝑪𝟏 𝑪𝟐 𝑪𝒏 In Parallel: 𝑪𝒆𝒒 = 𝑪𝟏 + 𝑪𝟐 + … . . + 𝑪𝒏 Department of Electrical & Electronics Engineering Charging of a Capacitor through a Resistor 𝐖𝐫𝐢𝐭𝐢𝐧𝐠 𝐊𝐂𝐋 𝐞𝐪𝐮𝐚𝐭𝐢𝐨𝐧; 𝒗𝒄 − 𝑽 𝒅𝒗𝒄 −𝑪 =𝟎 𝑹 𝒅𝒕 Initial conditionsat t=0 seconds, Vc= 0 Solving: 𝟏 −𝑹𝑪 𝒕 𝒗𝒄 = 𝑽 𝟏 − 𝒆 𝒊𝒄 = 𝑽 𝑹 Department of Electrical & Electronics Engineering 𝒆 − 𝟏 𝑹𝑪 𝒕 Charging of a Capacitor through a Resistor Time Constant, = RC Time taken by the voltage of the capacitor to reach its final steady state value, had the initial rate of rise been maintained constant vc ic Department of Electrical & Electronics Engineering Discharging of Capacitor through a Resistor Capacitor is initially charged to a voltage V. At t= 0 sec, switch is moved from position a to b t=0 𝑣𝑐 𝑑𝑣𝑐 +𝐶 =0 𝑅 𝑑𝑡 𝒗𝒄 = 𝑽 𝒆 a V Solving, 𝟏 −(𝑹𝑪 )𝒕 𝒊𝒄 = −𝑰𝒆 R 𝟏 −(𝑹𝑪 )𝒕 Department of Electrical & Electronics Engineering b i C + vc Discharging of Capacitor through a Resistor Department of Electrical & Electronics Engineering Energy stored in a Capacitor Instantaneous power 𝒅𝒗𝒄 𝒑 = 𝒗𝒄 ∗ 𝒊 = 𝑪 𝒗𝒄 𝒅𝒕 Energy supplied during ‘dt’ time is ; 𝒅𝒘 = 𝑪 𝒗𝒄 𝒅𝒗𝒄 Energy stored when potential rises from 0 to ‘V’ volts is, 𝑽 𝑾= 𝟎 𝟏 𝑪 𝒗𝒄 𝒅𝒗𝒄 = 𝑪. 𝑽𝟐 𝐉𝐨𝐮𝐥𝐞𝐬 𝟐 Department of Electrical & Electronics Engineering Summary Capacitor stores energy in its electrical field. When a series RC circuit is connected to a d.c voltage source, there is an exponential growth of voltage across the capacitor and it decays exponentially when the voltage source is removed. Time constant of a series RC circuit is 𝑅𝐶 𝑠𝑒𝑐𝑜𝑛𝑑𝑠. Department of Electrical & Electronics Engineering Illustration 1 A Capacitor consists of 2 metal plates each 400 by 400 mm spaced 6 mm apart. The space between the metal plate is filled with a glass plate 5 mm thick and a layer of paper 1 mm thick. The relative permittivity of glass and paper are 8 and 2 respectively. Calculate the equivalent capacitance. Also find the electric field strength in each dielectric due to a potential difference of 10 kV between the metal plate. Ans: Cg = 2.265 x 10-9 F; Cp = 2.832 x 10-9 F; Ceq = 1.258 x 10-9 F Q = 1.258 x 10-5 C Vg = 5.554 kV; Vp = 4.446 kV Eg = 1.11 kV / mm ; Ep = 4.446 kV / mm Department of Electrical & Electronics Engineering Illustration 2 A 10F capacitor is initially charged to 100 V dc. It is then discharged through a resistance R Ω and the p.d. across the capacitor after 20 seconds is 50 V. Calculate the value of R. Ans: 2.886 MΩ Department of Electrical & Electronics Engineering Illustration 3 An 8uF capacitor is connected in series with a 0.5MΩ resistor, across a 200 V d.c supply through a switch.At t=0 seconds, the switched is turned on, Calculate Time constant of the circuit Initial charging current. Time taken for the potential difference across the capacitor to grow to 160V. Current & potential difference across the capacitor 4.0 seconds after the switch is turned on. Derive the expressions used. Ans: (i) 4 seconds, (ii) 400 μA , (iii) 6.44 seconds (iv) 126.424 V & 147.15 μA Department of Electrical & Electronics Engineering