电气课件电路理论-and inductors

1、introductionResistors: dissipate energy;Capacitors: store electric energy;Inductors: store magnetic energy.Chapter 6 Capacitors and Inductors6.1 Capacitors6.2 Series Capacitors6.3 Parallel Capacitors6.4 Inductors6.5 Series Inductors6.6 Parallel Inductors引线引线绝缘介质金属片Circuit symbolA capacitor consists

2、of two conducting plates separated by an insulator(or dielectric).6.1、电容器 CAPACITORSA capacitor is a passive element designed to store energy in its electric field.Capacitors are widely used in the turning circuits of radio receivers and as dynamic memory elements in computer systems.Ciu+1. Circuit

3、theory definition: a two_terminal element will be called a capacitor if at any time t, its charge q(t) and its voltage v(t) satisfy a relation defined by a curve in the q_v Plane.qu0Capacitance: is the ratio of the charge on one plate of a capacitor to the voltage difference between the two plates,

4、measured in farads(F).F= C/V = As/V = s/ Nonlinear capacitor:iC+-uquq0uq0斜率CTo the linear capacitor: q =Cu 2. U_I relationship of the linear capacitor:Ciu+3、Character(1)dynamic character+ECRcRb1Rb2ReC2C1C2输入输出Isolate DC（如放大器电路中电容的耦合作用）i(t) = Cdu(t)dt0tu(t)= u(0)+1Ci()diC+-uu/V0.512345is/ A0t/s1(2)me

5、moryi(t) = Cdu(t)dt0tu(t)= u(0)+1Ci()d0t 1u(t)=u(0)+5105 106dt0= 0.5tu(1)=0.5Vu(t)=u(1)+5105 0dt0=0.5V2t 3u(t)=u(2)+5105 106dt2=0.5+0.5(t2)u(3)=1VExample: a timer+-u(t)2FiSu(0)=01t 2(3)the voltage across a capacitor is continuous.i(t) is bounded 如果i(t)在任一时间都有界，则u(t)在任一时间的变化都是连续的。即在任一时间，电容电压都不可能即时地从一

6、个值跃变到另一个值。特别，如果在t=0时有界，i(t) = Cdu(t)dt0tu(t)= u(0)+1Ci()d0ti(t)u(0+)= u(0)00+u(0+)= u(0)+1Ci(t)dt则00+(4)The capacitor with initial voltageu(0)=U0+-uCi+-uc(0)=0+-uCi+uc-U0u(t)=U0+1C0ti()dSummary：1. Capacitors have memory.2. A capacitor is an open circuit to dc.3. The voltage on a capacitor cannot cha

7、nge abruptly as long as the current remains bounded.i(t) = Cdu(t)dt0tu(t)= u(0)+1Ci()dThe instantaneous power delivered to the capacitor is:The energy stored in the capacitor is:t=-t6. 2 series capacitorsc1c2cn+v1-+v2-+vn-+V-V=v1+v2+vn The equivalent capacitance of n series_connected capacitors is t

8、he reciprocal of the sum of the individual capacitances.6. 3 parallel capacitorsCeqcnc1c2i1i2in+V-ii=i1+i2+i3+in The equivalent capacitance of n parallel_connected capacitors is the sum of the individual capacitances.Discussion: Parallel capacitors with initial voltages:+-iu1u2+-C1C2i2i1u+-(t=0)情况一

9、并联前各电容电压相同情况二 并联前各电容电压不同u1(0) u2(0)u1(0+)=u2(0+)=u (0+)qk(0+) = k=1mqk(0) k=1mi(t) = Cdu(t)dt0tu(t)= u(0)+1Ci()d+-iu1u2+-C1C2i2i1u+-(t=0)(C1+C2)u(0+)=C1u1(0)+ C2u2(0)Conservation of the charge:When the switch is closed, some charge is dumped from one capacitor to another instantaneously, this implie

10、s that an impulse of current flows from one capacitor to another.Nonlinear,time-varying capacitors:C(t)q=C(t)u+-uii=dqdtdudtdC(t)dt=C(t)+ uq = f( u )i=dqdtdf(u)dududt=+-uiiAn practical inductor is usually formed into a cylindrical coil with many turns of conducting wires.An inductor is a passive ele

11、ment designed to store energy in its magnetic field.inductors are widely used in power supplies, transformers, radios, TVs, radars, and electric motors.The element called an inductor is an idealization of the physical inductors.(t)=Li(t)2 Characteristics of the linear time_invariant inductors1. Circ

12、uit theory definition: a two_terminal element will be called an inductor if at any time t, its flux (t) and its current I(t) satisfy a relation defined by a curve in the _I Plane.6.4 INDUCTORLi+u(1). Linear inductor = N flux(电感线圈的磁链)L inductance(自感系数)unit：H (Henry）亨（利）Joseph Henry (1797-1878)，(2). V

13、CR of the linear inductors ：i+u+eu(t)=ddtdi(t)dt=LFaradays induction law:(2)、VCR(3) 、characters*dynamic*memory* the current is continuous as long as the voltage is bounded. 0ti(t)=i(0) +1Lu()du(t)=ddtdi(t)dt=Lu(t) is boundedSpecially，if u(t)is bounded at t=0，i(0+)= i(0)（3）the current through an indu

14、ctor is continuousu(t)=Ldi(t)dt0ti(t)=i(0) +1Lu()d（4）An inductor with initial currentu(t)=Ldi(t)dt0ti(t)=i(0) +1Lu()d+Lii(0)=I0-u+LiiL(0)=0-uI0iL0ti(t)=i(0) +1Lu()dThe instantaneous power delivered to the inductor is:The energy stored in the capacitor is:t=-tSUMMARY:Remark: 1). Inductors have memory

15、. 2). The inductors look like a short circuit to dc. 3) The current in an inductor cannot change abruptly as long as the voltage across it remains bounded.6.5 parallel inductorsi=i1+i2+i3+in The equivalent inducance of n parallel_connected inductors is the sum of the reciprocals of the individual ca

16、pacitances.Leqi6.6 Series InductorsVab=v1+v2+vn The equivalent inductance of n series_connected inductors is the sum of the individual inductances.bA DISSCUSSION : SERIES INDUCTORS WITH INITIAL CURRENTS:Case 1：inductors carry the same initial currentCase 2 ： initial currents are not equal.usR2+-R1L2

17、L1u1u2i1i2+-+-u+-t=0i1(0-) i2(0-)i1(0+) = i2(0+)（m为回路包含电感元件的总数） k(0+) = k(0)k=1k=1mmusR2+-R1L2L1u1u2i1i2+-+-u+-t=0L1i1(0+)L2i2(0+)=L1i1(0) L2i2(0)(L1+L2)i1(0+)=L1i1(0) L2i2(0)Conservation of flux: irrespective of the manner in which the connection is made for the m inductors, the total flux remains

18、fixed.q=C u=L ii(t)=Cdu(t)dtu(t)=u(0)+1C0ti()du(t)=Ldi(t)dti(t)=i(0) +0t1Lu()d(t)=0.5Cu2(t)(t)=0.5Li2(t)电流电压i+-u+-uiLC元件约束方程电压-电流关系连续性储存的能量Comparisons between L,CA real, nonideal capacitor: Ciu+Leakage resistanceLeakage resistance may be as high as 100M.A real, nonideal inductor: LRwCwWinding resistance Rw is usually very smallWinding capacitance Cw is usually very