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IEEE Transactions on Electrical Tnsulation Vol. EI-19 Nos.3 June 1984 179 CHARGE INJECTION AND TRANSPORT IN A LossY CAPACITOR STRESSED BY A MARX GENERATOR M. Zahn Department of Electrical Engineering and Computer Science High Voltage Research Laboratory Massachusetts Institute of Technology Cambridge, MA ABSTRACT A theoretical solution is given electric field and charge density capacitor with a charge injecting being charged by a Marx capacitor Conservation of Charge: VJ+ INTRODUCTION Injected charge of density q and mobility , increases where q the effective ohmic conductivity a to is time and space dependent. Then, open circuit decay curves at moderate voltages can show a dielectric relaxation time that decreases with increasing voltage while at very high voltages the open circuit decay Such anomalous behavcurves can be non-exponential. ior has been measured in highly purified water [1] and would be present in any material with enough net charge to distort the electric field. A drift dominated conduction model is used to solve for the electric field and charge density distributions and the terminal voltage-current behavior of a lossy capacitor where one electrode injects charge. The capacitor is connected to a Marx capacitor bank so that the injected charge decreases as the Marx voltage decays, the decay rate in turn being in part determined by the injected charge. a+lqp| DRIFT DOMINATED CONDUCTION MODEL Gauss's Law: V.(EE)=q -X = q/c Constitutive -- E) St = .T=0a E+q-iE Law: (4) , x=x/Z, v-v/Vo, E=EZ/VO, J=JZ3/(CiVG2) t=-pVOt/Z2, q=qZ2/(CV0) (5) [= - pVO/Z2 Then (l)-(4) reduce to , =() a+ =-a +k aU at (6) % Dt ~X T where :tY) is the total terminal current per unit electrode area. = TDX T equation for the charge (7) q + at an SX 2 X- tJ I I, A. C.:-- X2 t (2) -. a-(J+ Cx It is convenient to nondimensionalize these equations by normalizing all variables to the electrode spacing Z, the initial voltage V and nominal injected charge transit time Zl /(P@) d+IL%+ / D' C:. If B-T"67,.-'94 =0 while integrating (6) between electrodes relates the current and voltage (1) Edx=zv Conduction q c St (3) Differentiating (6) gives density We consider parallel plate electrodes located at x=O and x=l, where the x=O electrode is assumed to be a source of positive charge with constant mobility ip. and intrinThe dielectric has constant permittivity sic ohmic conductivity a. Neglecting edge effects, the electric field and conduction current are x directed and all quantities only depend on x. The governing equations are then Irrotational Electric Field: VxE=O- for the of a electrode bank. v !. 1 79V -+. . t . .. T '-. EE Authorized licensed use limited to: MIT Libraries. Downloaded on January 22, 2009 at 14:00 from IEEE Xplore. Restrictions apply. (8) 1EEE Transactions InSO rF-l}i 1.0 -/ 0.0 on 2.0 q E - Electrical Insulation / - 0.2 A {,t-; El-19l June No.3, 1984 TO i the time that the charge is injected at For 0=O, there is a charge front which propagates towards the opposite electrode and reaches it at time there x=O. = 0.4 Vol. (14) Tin['/('-1)] X The injected chaVe density of (12) is found by evaluating (6) at x=O dko Wo dko (15) {,+ {, + qoko = ,! + -, + Wko2 = ° T d T dt ko -A _.I u.u _ _ 1.0 O.b _ _ _ 2.0 1.6 2.6 3.0 3.5 4.0 with solution Fig. 1: Charge transport trajectories efor an ini aZZy charged but open circuited Zossy capacitor where the Zower x=O eZectrode injects charge pr. portionaZ to the instantaneous ZocaZ eZectric f The demarcation curves emanating from the origi: various vaZues of dieZectrzic reZaxati on shown,for time separate the initiaZ vaZue probZem with S=O)=O, 2(k, O)=1 from the subsequent char(ge V(x, injection probZem. EoN1 ^v 1~~~~1 = I Ir { (16) (2+l/- J exp [t/-c ] -2 I The solutions for field and charge density are then T, 0 flu X E(x, tJ= ]T(1-exp[-t/T])<X1 (exp [- (17) rv-j t l+x Method of Characteristics c{ (2+1/lbexpj/T ]-A%} The partial differential equations (6) and (7) c be converted to ordinary differential equations by jumping into the frame of reference of the moving charges. a 9S+ d:x' dt 't dt d3 ax dt _aq,d! 9q=_(u2 +/l(j" d't 9't {(/l'xp htl ZO<x<T (9) The V(t) open q(t) where = qO {t0) "U Ili (b rk, ni 'IV lb (1+q 0 (t 0 )T) exp[ (t-t 0 )/Tl -q, (.,) T' qo(TO) (1-exp[-W/r"]) (18) circuit terminal voltage is then =0p1/tTb]_ -[/- ] d 2 = This last equation ~~~~~to is directly O integrated % ,%, T(1-expT 0 (10) q/. dt ax ~~~Osxs5 T (l-exp [-t/T]l r 1+1/2 T{ (Af+1/T )exp (19) [ t/T ] -A } (1 1) is the injected charge density at time . The problem is not completely specified until we define the charge injection law. We expect the injected charge density to increase with electrode electric field. Although numerical analysis can treat any injection law, we consider a simple linear law which allows analytic solution q(x--O,t)=AE(x=O,t)-* q%o=;rEo ; A=Azlc (12) OPEN CIRCUIT DECAY Fig. 2: Consider an initially charged capacitor which is then open circuited for t>O (J t)=O). The solutions to (9) are .(T)=.O(To) exp[-(Mo)/T1 x (.) =ko (.o ) "Ti - ; exp F( various Open circuit voZtage decay of (19) for v Jues point if T 2>1. of A and T. (13) .-To )/,Tl Authorized licensed use limited to: MIT Libraries. Downloaded on January 22, 2009 at 14:00 from IEEE Xplore. Restrictions apply. There is an infZection Zahn: Charge Injection and Transport in a Lossy Capacitor MARX CAPACITOR CIRCUIT Another typical configuration is to have a Marx capacitor bank with total capacitance Cm connected directly across the test dielectric of area S. We neglect the Marx shunt resistance and assume the test dielectric is lossless (u->-). Then the terminal current is also given by 6mSZ/ m 3m m (20) To allow closed form solutions e also assume space charge limited injection where A- so that 20=0. Then with 2(x=1,)=-i, (8)-(11) reduce to y= m~ av m d' diX + _, rE12 (lim ) (22) 2-T = = (1+Cm ) InitiaZ Injection Problem Approaching the x0=O, T=0 origin from along the axis has (x=0, 0-)=-, while approaching the origin from the t axis has E(x=0, t=0)=O for space charge limited conditions. This disgontinuity results in a family of injected charge at t=0 all with charge density q=l/?. As the charge propagates, Coulombic repulsion causes the charge to separate maintaining a uniform charge densitq and thu, a linear electric field. Using (6) at x=1 with T= and (21) yields for t>tl a single differential equation in the variable TE1 Pt _ (ItE1)2 m m t m i; t (24) et rb -0, . (. ),;,Zo Zo,0=0-,iE( 0)=O 10>0,X(10)=O 0 0 1/r\., = i/(T_ TO) (30) 1~ t[l- rnl In xt/t m dx = E X q +0 2(1+ ) (23) + C v = constant m dt m (29) Ct 2 - m :(tE, = (~i-1) -l - ~ mis El2= dky et d (21) 2(1+2 m lei (25) is found from (26). This solution where ESE1 is valid unttj time t2 wJen the last of the charge injected at t=0 reaches x=1. For later times it is not possible to obtain analytic solutions, and numerical integration is necessary. The charge transport trajectories obtained by solv- ing (24) break the solution into three spatial regions. Initial VaZue Problem Ahead of the charge front, =0, so that the electric field is uniform. The field at x=1 is found using (6) wi th (21) for q=O and T== dT -S - 2(+m) ; 1=0 -* 21 (26) = 1 t-0 t - 2 (1l +C) This solution is valid until the time t1 when the initially injected charge first reaches the opposite electrode 1, 2(1+f ) = , [ l-exp [-3v'(2 (1+3m ))] } (27) m~~~~~ 0.0 The voltage and current for dkl ~~m 2(1+C6) [1 - Ot.tl (28) m ]2 0.5 1.0 1.5 2.0 3.1 2.5 are Fig. 3: The terminqZ current Y(t) of (21) for various values of Cm. AnaZytiz oZutions zre given in the time intervaZs 0:t1T61 and T1-gtW2. NumericaZ inteqgation is required to find the soZu ion for ttt2. The dash d curves show the solution and 12 for each case. When is that of a step voZtage from rest. %o, Authorized licensed use limited to: MIT Libraries. Downloaded on January 22, 2009 at 14:00 from IEEE Xplore. Restrictions apply. Tj IEEE Transactions on Electrical Insulation Vol. EI-19 182 No.3, 0.8 8 0.6 \\\ 0.\\ \\\:\4 [1] \ 2 \ _ t05 0.4 CM= 0 ~ \ N M. Zahn, D. B. Fenneman, S. Voldman, Fig. 4: of Cm 0.5 1.0 1.5 2.0 2.5 and T. Takada, "Charge Injection and Transport in High Voltage Water/Glycol Capacitors", J. Appl. Phys. 54, pp. 315-325 (1983). This paper was presented at the 1st International Conference on Conduction and Breakdown in SoZid Dielectrics, 4-8 JuZy 1983, Toulouse, France. Manuscript was received 21 January 1984. 0.0 1984 REFERENCES 1.0 Q.0 June 3.0 TerminaZ voZtage v t) for various values Authorized licensed use limited to: MIT Libraries. Downloaded on January 22, 2009 at 14:00 from IEEE Xplore. Restrictions apply.