Chapter 6 Inductance, Capacitance, and Mutual Inductance 6.1 The inductor 6.2 The capacitor 6.3 Series-parallel combinations of inductance and capacitance 6.4 Mutual inductance 6.5 Closer look at mutual inductance 1 Overview In addition to voltage sources, current sources, resistors, here we will discuss the remaining 2 types of basic elements: inductors, capacitors. Inductors and capacitors cannot generate nor dissipate but store energy. Their current-voltage (i-v) relations involve with integral and derivative of time, thus more complicated than resistors. 2 Key points di Why the i-v relation of an inductor isv L ? dt dv Why the i-v relation of a capacitor isi C ? dt Why the energies stored in an inductor and a capacitor are: 1 1 w Li, 2 , 2 Cv respectively? 2 2 3 Section 6.1 The Inductor 1. Physics 2. i-v relation and behaviors 3. Power and energy 4 Fundamentals An inductor of inductance L is symbolized by a solenoidal coil. Typical inductance L ranges from 10 H to 10 mH. The i-v relation of an inductor (under the passive sign convention) is: di v L , dt 5 Physics of self-inductance (1) Consider an N1 -turn coil C1 carrying current I1 . The resulting magnetic fieldB 1() r 1 N 1(Biot- I will pass through Savart law) C1 itself, causing a flux linkage 1 , where B1() r 1 N 1 , 1 B1 r1() d s1 P 1, 1 N I S 1 P1 is the permeance. 2 1 PNI 1 11. 6 Physics of self-inductance (2) The ratio of flux linkage to the driving current is defined as the self inductance of the loop: 1 2 L1NP 1 1, I1 which describes how easy a coil current can introduce magnetic flux over the coil itself. 7 Examples Solenoidal & toroidal coils: ~2 cm L = 270 H L = 36 H RG59/U coaxial cable: L = 351 nH/m. 8 The i-v relation Faraday’s law states that the electromotive force v (emf, in units of volt) induced on a loop equals the time derivative of the magnetic flux linkage : d d d v , v Li L i. dt dt dt Note: The emf of a loop is a non-conservative force that can drive current flowing along the loop. In contrast, the current-driving force due to electric charges is conservative. 9 Behaviors of inductors di v L dt DC-current: inductor behaves as a short circuit. Current cannot change instantaneously in an inductor, otherwise, infinite voltage will arise. Change of inductor current is the integral of voltage during the same time interval: 1 t i()() t i0 t v(). d L t0 10 Inductive effect is everywhere! Nearly all electric circuits have currents flowing through conducting wires. Since it’s difficult to shield magnetic fields, inductive effect occurs even we do not purposely add an inductor into the circuit. 11 Example 6.1: Inductor driven by a current pulse 0 , t 0 i() t 5t 10 te , t 0 The inductor voltage is: Inductor voltage can Memory-less di 0 , t 0 v() t L in steady 5t jump! dt (e 1 5 t ), t 0 state. 12 Power & energy (1) Consider an inductor of inductance L. The instantaneous power in the inductor is: di p vi Li. dt Assume there is no initial current (i.e. no initial energy), i(t =0)=0, w(t =0)=0. We are interested in the energy W when the current increases from zero to I with arbitrary i(t). 13 Power & energy (2) dw di W I p dwLi , Li di, dw, L idi dt dt 0 0 I i2 1 1 WL LI 2 , i.e.w Li2 2 2 0 2 How the current changes with time doesn’t matter. It’s the final current I determining the final energy. Inductor stores magnetic energy when there is nonzero current. 14 Example 6.3: Inductor driven by a current pulse t <0.2, p>0, w, charging. t >0.2, p<0, w, discharging. In steady state (t ), i 0, v0, p0, w0 (no energy). 15 Example 6.3: Inductor driven by a voltage pulse 1 t i() t (v ) d ( i 0 ) L 0 p>0, w, always charging. In steady state (t), i 2 A, v0, p0, w200 mJ (sustained current and With memory in steady state. constant energy). 16 Section 6.2 The Capacitor 1. Physics 2. i-v relation and behaviors 3. Power and energy 17 Fundamentals A capacitor of capacitance C is symbolized by a parallel-plate . Typical capacitance C ranges from 10 pF to 470 F. The i-v relation of an capacitor (under the passive sign convention) is: dv i C . dt 18 Physics of capacitance (1) If we apply a voltage V12 between two isolated conductors, charge Q will be properly distributed over the conducting surfaces such that the resulting electric field E() r satisfies: 2 E() r 1 1 B1 E(), r d l 12 V 2 which is valid for any integral path linking the two conducting E surfaces. 19 Physics of capacitance (2) IfVV 12 12 , QQ while the spatial distribution of charge remains such that 1 E r( ) E ( r ), V 12 E ( r ) d12 l . V 2 The ratio of the deposited charge to the bias voltage is defined as the capacitance of the conducting pair: Q C , V describing how easy a bias voltage can deposit charge on the conducting pair. 20 Examples Ceramic disc & electrolytic: RG59/U coaxial cable: C = 53 pF/m. 21 The i-v relation From the definition of capacitance: Q d d d C, q ( t ) Cv ( ),q t Cv, i C v . V dt dt dt Note: Charge cannot flow through the dielectric between the conductors. However, a time-varying voltage causes a time-varying electric field that can slightly displace the dielectric bound charge. It is the time-varying bound charge contributing to the “displacement current”. 22 Polarization charge 23 Behaviors of capacitors dv i C dt DC-voltage: capacitor behaves as an open circuit. Voltage cannot change instantaneously in an capacitor, otherwise, infinite current will arise. Change of capacitor voltage is the integral of current during the same time interval: 1 t v()() t v0 t i(). d C t0 24 Capacitive effect is everywhere! A Metal-Oxide-Semiconductor (MOS) transistor has three conducting terminals (Gate, Source, Drain) separated by a dielectric layer with one another. Capacitive effect occurs even we do not purposely add a capacitor into the circuit. (info.tuwien.ac.at) charges 25 Power & energy (1) Consider a capacitor of capacitance C. The instantaneous power in the capacitor is: dv p vi Cv . dt Assume there is no initial voltage (i.e. no initial energy), v(t =0)=0, w(t =0)=0. We are interested in the energy W when the voltage increases from zero to V with arbitrary v(t). 26 Power & energy (2) dw dv W V p dwCv , Cv dv, dw C , vdv dt dt 0 0 2 V v 1 2 1 WC CV , i.e.w Cv2 2 0 2 2 How the voltage increases with time doesn’t matter. It’s the final voltage V determining the final energy. Capacitor stores electric energy when there is nonzero voltage. 27 Example 6.4: Capacitor driven by a voltage pulse Capacitor current can jump! t <1, p>0, w, charging. t >1, p<0, w, discharging. In steady state (t), i 0, v Memory-less in 0, p0, w0 (no energy). steady state. 28 Section 6.3 Series-Parallel Combinations 1. Inductors in series-parallel 2. Capacitors in series-parallel 29 Inductors in series i1 i 2i , i3 v v1 v2 , 3 v di v L j , j j dt di di di v L L L 1 dt 2 dt 3 dt di L , eq dt n LLeq j j1 30 Inductors in parallel 1 t v v 1 v 2 v, 3 i1 i 2 , i 3 j iv i( ) d j i (0 t ), t 0 L j 3 t 3 t 3 1 1 i v()() d j i0 t v() d ()j i0 t t0 t0 j1 L j j1 L j j1 n 1 t 1 1 (v ) d ( i0 ),t t 0 Leq LLeq j1 j 31 Capacitors in series = v v1 v 2 3 v, i 1 i 2 i, 3 i n 1 1 1 t 1 t v j id( vj 0 ), t v id ( v0 ), t t0 t0 CCeq j1 j C j Ceq 32 Capacitors in parallel = v v v v, i i i, i 1 2 3 1 2 3 n dv dv CCeq j i C ,i C , j1 j j dt eq dt 33 Section 6.4, 6.5 Mutual Inductance 1. Physics 2. i-v relation and dot convention 3. Energy 34 Fundamentals Mutual inductance M is a circuit parameter between two magnetically coupled coils. The value of M satisfiesM k 1 L 2 , where L 0 k 1is the magnetic coupling coefficient. The emf induced in Coil 2 due to time-varying current in Coil 1 is proportionalM to di1 . dt 35 The i-v relation (1) Coil 1 of N1 turns is driven by a time-varying current i1 , while Coil 2 of N2 turns is open. The flux components linking (1) only Coil 1, (2) both coils, and (3) total flux linking Coil 1 are: B11 r1() d s11 P 1, 1 N B 21 I r1() d s21 P 1.