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Chapter 4 Inductance and Capacitance Mdm Shahadah Ahmad Inductance and Capacitance • Inductor • Relationship between voltage, current, power and energy • Capacitor • Relationship between voltage, current, power and energy • Series-parallel combinations for inductance and capacitance Inductor Inductor concept • An inductor consist of a coil of conducting wire. • Inductance, L is the property whereby an inductor exhibits opposition to the change of current flowing through it, measured in henrys (H). Inductance • Inductance, L N A L 2 L = inductance in henrys (H). N = number of turns µ = core permeability A = cross-sectional area (m2) ℓ = length (m) Inductance and Capacitance • Inductor • Relationship between voltage, current, power and energy • Capacitor • Relationship between voltage, current, power and energy • Series-parallel combinations for inductance and capacitance Relationship between voltage, current, power and energy Inductor symbol Inductor Voltage di(t ) v(t ) L dt Inductor current 1 t i (t ) v( ) d i (t0 ) L t0 di p vi Li dt 1 t v v dt it 0 L t0 Power t w(t ) w(t0 ) p( ) d t0 1 2 1 2 Li (t ) Li (t0 ) 2 2 • Assuming that energy is zero at time t=t0, then inductor energy is: 1 w(t ) Li (t ) 2 2 Inductance and Capacitance • Inductor • Relationship between voltage, current, power and energy • Capacitor • Relationship between voltage, current, power and energy • Series-parallel combinations for inductance and capacitance CAPACITOR Capacitor physical concept: • A capacitor consists of two conducting plates separated by an insulator (or dielectric). • Capacitance, C is the ratio of the charge on one plate of a capacitor to the voltage difference between the two plates, measured in farads (F). • The amount of charge stored, represented by q, is directly proportional to the applied voltage v, q Cv q = cas dalam coulomb (C) C = kapasitans dalam farad (F) v = voltan dalam volt (V) • Capacitance, C: eA C d C = Capacitance in farads (F) e = permittivity of dielectric material between the plates (C2/N∙m2) A = surface area of each plates (m2) d = distance between the plates (m) Inductance and Capacitance • Inductor • Relationship between voltage, current, power and energy • Capacitor • Relationship between voltage, current, power and energy • Series-parallel combinations for inductance and capacitance Relationship between voltage, current, power and energy • Capacitor symbol Capacitor current dv( t ) i( t ) C dt 1 t v(t ) i ( )d v(t0 ) C t0 Capacitor voltage Power: p(t ) v(t ) i (t ) dv(t ) v(t ) C dt • Energy stored in a capacitor from time t to t0: t w(t ) w(t0 ) p( )d t0 dv( ) v( ) C d t0 d t C v (t ) v (t0 ) v( )dv( ) v (t ) 1 2 C v( ) 2 v (t 0 ) 1 1 2 2 C v(t ) C v(t0 ) 2 2 • Capacitor is not discharge at t=-∞, therefore the voltage is zero. 1 2 w(t ) C v(t ) 2 Energy capacitor Inductance and Capacitance • Inductor • Relationship between voltage, current, power and energy • Capacitor • Relationship between voltage, current, power and energy • Series-parallel combinations for inductance and capacitance Series and parallel capacitors • The equivalent capacitance, Ceq of N parallel-connected capacitors is the sum of the individual capacitances. v i1 i2 iN C1 C2 CN • Using KCL, dv I n Cn dt I I1 I 2 ......... I N dv dv dv I C1 C2 ........... C N dt dt dt dv C1 C2 ....... C N dt N dv dv Cn Ceq dt n 1 dt • Equivalent circuit for the parallel capacitor, N Ceq Cn n 1 is v Ceq • The equivalent capacitance, Ceq of N series-connected capacitors is the reciprocal of the sum of the reciprocals of the individual capacitances. i Vs C1 C2 V1 V2 VN CN • Using KCL, 1 Vn Cn t i ( ) d V V1 V2 ........... VN 1 1 1 V ........ CN C1 C2 N 1 t i ( )d C n 1 n 1 Ceq t i ( )d t i ( )d • Equivalent circuit for the series capacitor, i N 1 1 Ceq n 1 Cn Vs Ceq Series and parallel inductors • The equivalent inductance, Leq of N series-connected inductors is the sum of the individual inductances. i L1 V1 Vs L2 V2 VN LN • Using KVL, di Vn Ln dt V V1 V2 ......... VN di di di V V1 L2 ........... LN dt dt dt di L1 L2 ....... LN dt N di di Ln Leq dt n 1 dt • Equivalent circuit for the series inductor, i N Leq Ln n 1 Vs Leq • The equivalent inductance, Leq of N parallel-connected inductors is the reciprocal of the sum of the reciprocals of the individual capacitances. is i1 i2 iN V L1 L2 LN • Using KVL, I I1 I 2 ........... I N 1 1 1 I ........ LN L1 L2 N 1 t v( )d L n 1 n 1 Leq t v( )d t v( )d 1 t In v ( ) d Ln • Equivalent circuit for the parallel inductor, N 1 1 i Leq n 1 Ln s V Leq Question 1 Obtain the total of capacitance. 4F CT 2F 3F 12F 1F Answer • Short circuit, then: CT 3F Question 2 • Voltage stored in a 10µF capacitor is shown in figure below. Obtain the graph for current of the capacitor. Answer • Capacitor voltage: 10 v(t ) t 6 5 10 0 0 t 5 saat t 5 saat dv(t ) • current: i (t ) C dt 10 10 10 6 5 10 20 A 0 t 5 saat 6 • Thus: Question 3 • Determine the voltage across a 2 µF capacitor if the current through it is i(t ) 6e 3000t mA Assume that initial capacitor voltage is zero Answer • Capacitor voltage: 1 t v(t ) i (t )dt v(0) C 0 v ( 0) 0 1 t v(t ) i (t )dt v(0) C 0 t 1 3000t 3 6 e 10 dt 6 0 2 10 3 10 3000t e 3000 3 (1 e 3000t )V t 0