Memristor-The Missing Circuit Element - IEEE Global History Network

IEEE ,EEE TRANSAcrlONS TRANSACTIONS ON CIRCUIT THEORY, VOL. cr-18, CT-18, No.5, NO. SEPTEMBER 1971 507 

Memristor-The Missing Circuit Element 

LEON O. 0. CHUA, SENIOR MEMBER, IEEE 

Abstract-A new two-terminal circuit element-called the memrirtor- memrislorcharacterized 

by a 

relationship between the charge q(I} q(t) == s f'-'" St% i(r} i(7J dr d7 

and the flux-linkage I~------:" 

i 

+ 

i 

+ 

(a) 

(b) 

(c) (cl 

(d) Cd) 

Fig. 1. 

Proposed symbol for memristor and its three basic realizations. 

(a) Memristor and its qrq q-q curve. (b) Memristor basic realization I: 1: 

M-R mutator terminated by nonlinear resistor CR. &t. (c) Memristor 

basic realization 2: M-L mutator terminated by nonlinear inductor 

.c. C. (d) Memristor basic realization 3: M-C mutator terminated by 

nonlinear capacitor e. 

and theflux-linkage 

'P. cp. Out of the six possible combinations 

of these four variables, five have led to well-known relationships 

[I]. [l]. Two of these relationships are already given 

by q(t)=J~ q(t)=JL .. w i(T) dTd 7 and «J(t)= cp(t)=sf. J~ .. m v(T) D(T) dT. d7. Three other rela 

rela- 

relationships 

are given, respectively,. by the axiomatic definition 

of the three classical circuit elements, namely, the resistor 

(defined by a relationship between v and i), the inductor 

(defined by a relationship between 'P cp and i), and the capacitor 

(defined by a relationship between q and v). Only one relationship 

remains undefined, the relationship between 'P 9 

relaand 

q. From the logical as well as axiomatic points of view, 

it is necessary for the sake of completeness to postulate the 

existence ofa a fourth basic two-terminal circuit element which 

R 

I. 

( 

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508 IEEE TRANSACTIONS ON CIRCUIT THEORY, SEPTEMBER 1971 

508 

IEEE TRANSACTIONS ON CIRCUIT THEORY, SEPTEMBER 1971 

= 

TABLEI 

I 

CHARACTERIZATION CHARACTERIZATIONAND ANDREALIZATION REALIZATIONOF OFM-R, M-L, M-L, AND ANDM-C M-CMUTATORS 

MUTATORS 

[~:] [ T(P)] ~::] 

TRANSMISSION 

MATRIX MATRIX 

SYMBOL 

BASH: BASIC 

REALIZATIONS 

TYPE 

‘PE 

AND 

" 

::I = [T(P’][J 

USING CONTROLLED CHARACTERIZATION 

CHARACTERIZATION 

SOURCES 

I 

d£te 

REALIZATION I I REALIZATION 22 

w+gf 

D---a 

‘2 

F--ail 

il 

+ 

[ 1 

(2) + i2+ 

R 

+(!%) -i2+ 

~ ~; “2 

P 0 

+ 

+ 

I 

I ~R,b) = “2 I c 

“2 

0 P 

Uqdt) - 

(li,dt) 

dVp 

“I= dt 

'..",{.:] I;i-[J'~') ,f~ ;i-@,:,') .f~ 

- 1·- 

~ t ! ~ VI : t v 2 

I 0 

dV2 

I~ (/vldt) ~ ~ (/ildtl 1----4 

VI' dT 

dip di 2 

iI i = -7 

Y-R M·R 

l " -dT 

- 

MUTATOR MUTATOR (q.Vl -RvR,iR) 

REALIZATION I I REALIZATION 2 

~ 2 - 

~; 

2 - 

v, :;: ! v 

'.. ",{ 0 

2 VI ~ t v2 

2 P 

di2 

1';.- (/ildtl ~ ~ (/vldt) f.--4 

VI’ v I=-- 

-7 dt 

:] l;i-~~).t \ ;i-[J~~;),{~ 

I 

i I • ~ dt REALIZATION I REALIZATION 2 

M-L 

MUTATOR 

M-C 

MUTATOR 

II i i 2 l 1 2 

+ + + 

+ 

(q,9?)"""'" (IL,cPLl 

~1'/lldt) 

VI 

gq-: 

~~ di 2 v 

LW 

2 v, v2 

) 

'.C ",{ , 

:] - 12 -"'jjf 

- - .- 

~ ~ V; I 0 

I - I - 

REALIZATION 3 REALIZATtON REALIZATION 4 

VI" V 

(Identical to TcR,(p) TCR,( p) 

“I = “2 2 i l 

1 2 

i 

off ac Type , I CoR C-R MUTATOR) 

1 

2 

. di2 2 + 

II'-dl 

iI=- 

- 

2 

I 

2 

2 - 2 - 

- 

’ : 

v 

,"-- 

i 

(/lldtl ~ .0-- (vI) - 

(q,~)-(jPL.iL) (q,# -WL, iL) 

REALIZATION I REALIZATION 2 

i, i 2 

[ 1 

i 

0 P 

m 

: 

TML2fP” ) 

:] 

2 i, i 2 

.~ '.C ",{ 0 o 

Y 

di, 

v, = dt 

TLR2(p) 

'~R>' (v2' + 

- 2 - 2 I 

v, ~ ,~ t v2 ~~~)§"" 

~ + ! ~ 

_ lI 2 

vl dt) - (il)_ 

VI' - dT 

(Identical to (p) 

ofmf ac Type 2 L-R MUTATOR) 

MUTATOA !I 

ii, l 

• =v2 v2 

.I 

REALIZATION REALIZATlON I REALIZATION 2 

I, 

~ i 2 i, 

~ 1 2 

+ + + + 

(q,tp) - (qc'v c ) 

, 

:f-pq* 

[ 1 

v, (~- ) v 2 VI lIvldt-v, ) v2 

dt "2 

P 0 

- - -l 

I ;- 

'Me k,(P) ",{ , o ( 

~ ~ v~ I 0 

I - I - 

I 

REALIZATION REALlZATlON 3 REALIZATION REALIZATlON 4 

• dV2 (Identical (I&tical to 10 TLRltp) ( p) 

VI 

I 

dt 

ofd a Type II L-R MUTATOR) 

'1 

~ ~ 

i, = - i2 

~ :] 

v, =-i 

2 

(Idtmticdl t0 TCR2 ( p) 

. dV2 d”2 

of,f a Type 2 CoR C-R MUTATOR) 

WTATOF f) 

II il : =r(t Cit 

~[l',{~ 

I +(!!k) i2+ 

!TDq-y + 

(VI)-“! 

;i-@*),,~ 

v, : t v2 VI t : v2 

~@~).,lE i 2 

L 1 

~@", ..f 

i2 

i," - 1 2 

v, : t v 2 

v t + v 2 

(q .'!') -(vc,qc) 

~ (i,) ~ (/Vldt)- 

~ 

i l 

i2 

r. ,- il i2 

blcp= 

~ '.c,"'{: p o 3pq 

“2 

1 ~] 

v, • - i 

- (VI) 

2 (Identical to TCR2 (p) r---4 ~ 

REALIZATION II 

REALIZATION 2 

-D---a 

'I '2 II 

[Jr.)"t 

12 

+ 

(ipI 

+ 

v, “I ; 

vx 

2 VI v2 

+: (/ll 

t/ildt)- 

): 

I 

I----i 

r+- 

~" ,,;t r-"+ ~ ~ 

+ 


CHUA: MEMRISTOR-MISSING CIRCUIT ELEMENT 509 

r-------------,....----~--~---o+Ecc 

l?&l50) 

RJ9lO) 

Q014 

I 4 

R4( IOOK) 

(2N4236) 

(2~4236) L 

+Ecc 

+ 

~j= 

I f I 1 1 h-b--& 

Port I Port 2 

1 AR, 1 SO-IOA JRLq- 

4- 

iZ + 

- k + 

1 

“2 

- 

l~Z- 

I I I 

'------------------........---.4---Ecc 

’ *-kc 

Fig. 2. Practical active circuit realization of type-! type-l M-R mutator based on realization! 1 of Table I. 

is characterized by a

510 IEEE TRANSACfIONS TRANSACTIONS ON CIRCUIT THEORY, SEPTEMBER 1971 

r-----I+ 

to-E 

'- ~M202 >-.......--, 

+E 

~ 

E -E 

'-- 

LM202 ~--~-- 

~+ 

R I2 

(IK Pot) 

+E 

R I7 

(IOK.I~) 

I8 

R 

__.......-i{,ijR 

I4 

+E 

(IOK.I~) 

L 

..---- 10 + horizontal hwlZOontol 

to l 

lerminal twmiml of Of 

alcilloscope 

ouilloscop* 

t 

NEXUS 

IO-A '>--+~-~P----R-ll---"""'-'----lf---+--+ I 

+ t 

22+E v.(II=k.!v(TldT 

v,( I= k, jvbldr 

R 5 

_ -CD 

l?r+ 

--(D 

' 

...---''--1- 1 

R~ -E NEXUS 

OOK.I~) SOIO-A -= 

-==- ... 

(IOK.I~) 

-E 

-E 

R 24 

(20KPat) +E 

R23 (22 Meg.l 

L...--. I ta to ground 

lerminal terminal at d 

olei1I0icope 

oscilbscom 

R2/(IOOKl 

R201300Kl 

R 4 

Seriel 

Resillance 

Cli ±C3 

R 

T I R2 I (Sla;;cor A-3801l 

,>OOO! 

WindillQ 

Cenler 

500Sl 

R I R 2 

Tap Winding' 

R 3 

3.300n 

WindillQ 

-. 

+ 

v,(t) vI(I) 

10 to 

‘sine lin. 

*IO”8 wave 

VoltogI voltoge 

gonratot ge_alar 

~ 

c4tl C2 

PARAMETER IAMETER SPECIFICAT0NS 

SPECIFICATIONS 

R, .R2(l/Oltage-divider resillorl. 

see le.11. 

(current (currenl sensing .....ing ruustor. resislor. i;p~cal iyPical 

value: volue' I, 1.10. I), loo, 100, or ooon 1OO0n ). ), 

( series resiSlor. ta be chosen 

by user as il depends on the 

IIOllage amplitude of Ihe 

sine-wave Qenerator), 

~. qgbca* Rglscale factor foelar for mtegmtof, inlegralor, 

should Ihould be ot at b+t least 5K). 5K), 

~, R12 R I2 .• R,s R I3 

(I (I K wtenttomelsr polentiomeler for 

offset offseladjustment adjuslment for 

LM202 lM 202 OP AMPI. 

I. 

R7' R~s RIB ,R22 •R 22 

(trmwnmg (trimmingieststor relislorfor 

NEXUS SO-IOA SO-lOA OP AMP, 

typtool Iypical voluo: value, 20K). 20Kl. 

C I ,C2 cp .c3.cg ,C (nsutrollzotlon 

3 

,C 4 (neulrallzallon 

capocltors. capacitors, sea ,.. te*t text) 1 

C , CT C ( factor Integrator, 

5 7 

(scale foclor for inlegralor, 

seetmt1. 

1..1), 

R S 

OK Pol,) 

+E 

-E 

R e (100K) 

NEXUS 

SQIO-A 

+ 

-E 

All 

(20K Pot) +E 

R IO 

(22Meg.) 

R g (300K) 

1 

I = 

L 

r 

I 

to ta + vwtlcd verlical 

terminal 

of 

oscillo*copr oaci1I0icope 

t 

vi{ t )*k, ]ilr)dr 

--o 

k ze!?-s 

X 

%C5 

to ground 

'-- 10 grOOM 

twmiml lerminalof 

oscilloscope oscillalcope 

%E (( power supply voltag.e. vollage, *% I5 15 

Yolts volllrtth wilhrespect reopec! to 10qound). 

ground), 

Fig. 3. 

Complete schematic diagramof of memristor tracerfor for tracing thepq CP-q curveof of a memristor. 


. 


I 

q ( I) • J i( T) dT 

-CI) 

I 

0 p(t)?:.O and the memristor is obviously passive. To prove the 

converse, suppose that there exists a point qo q. such that 

M(qo)O e> 0 such that M(qo+.6.q)



t 

q(tl q.(t) -ji(TldT 

=ii(r)dr 

-CD -m 

t 

9I(tl -jv(TldT 

-CD 

(0). (a ). Simple Memristor Voltage-Divider Circuit 

v( t I, volts 

i(t), mo 

t, 1, msec. 

rp, Q, milliweber 

milli- 

t,msec. t , ( b 1 Horuontal Scole:2.66 milli-weber per division, 

Vertical Stole: 5 p coul per division. 

(c( c I.Horizontol hlorizontol Scale' Scale: 2 msec. per division. 

Vertical Scale' Scale:5 5 mo per division (upper trocel. trace). 

101lOits lOvolts per division (lower trocel. trace). 

v( t I. volts 

i(t), mo 

'''~'~l!,,''''''':'''':'''':'- 

i -t -+-J - 1 

_+.~ ~ -+- I J- 

'P, p. milJiweber 

milli- 

(d ).Horizontol Mlorizontol Scale' Scala: 2.66 milli-weber per division. 

Vertical Scale: 5 ,. p coul per division. 

t, t , msec. 

t, msec. 

(e I.Horizontol LHorizontol Scale' Scale: 5 msec. per division. divisioo. 

Vertical Scale:2 2 mo a per division (upper tracel. tmce). 

5 volts volte per division (lower tracel. trace). 

V( t I, volts 

i(t I,mo 

. . . . ,:) . . 

J=~A·.····.····.· 

(jJ, p, milliweber 

milli- 

( f I.Horizontal Mlorizontol Scale' Scale: 2.66 milli-weber per division. division, 

Vertical Scale' Scale: 5,. 5 p coul per division. 

(g g I.Horizontal ).Horizontol Scale, Scale:5 5 msec. per division. 

Vertical Scole'5 Scale:5 mo per division (upper trocel. trace). 

5 volts wits .per division (lower trocel. trace). 

Fig. 5. 

Voltage and current waveforms associated with simple memristor circuit corresponding to a sinusoidal input 

signal [(c) and (e)] and a triangular input signal [(g)], r(g)], respectively. 

t,msec. 

t,msec. 

Theorem 2: Closure Theorem 

(Kirchhoff voltage law) equations: 

A one-port containing only memristors is equivalent to a 

memristor. 

Proof: If we let ij, ii, Vj, vj, qj, and 'Pj vj denote the current, voltage, 

charge, and flux-linkage of the jth memristor, where j= I, 1, 

2, 2;.., ... , b, and if we let i and v denote the port current and 

port voltage of the one-port, then we can write (n-I) 1) independent 

KCL (Kirchhoff current law) equations (assuming 

the network is connected); namely, 

inde- 

b 

cxjoi CvjOi + 2 L cxjkik ajkik = 0, 

j=l,2,.*.,n-1 j = 1, ... , n - 1 (6) 

k=l k~l 

@j&J + 5 PjkVk = 0, j=l,2,..., j = 1, 2, ... , b - n + 2 (7) 

k=l k~l 

where (3jk @jk is either I, 1, -I,- 1, or 0. If we integrate each equation 

in (6) and (7) with respect to time and then substitute 

'Pk ‘pk = 'Pk(qk) (pk(qk) for 'Pk pk in the resulting expressions,? expressions,7 we obtain 

b 

& L CXjkqk ffjk@ = Qj - 

CXjoq, ffjoPt j=l,2,***,n-1 j = 1, ... , n - 1 (8) 

k=l 

PjOCp + f: pjk(pk(qk) = *j, j = 1, 27 ’ ’ . , b - n + 2 (9) 

k=l kzl 

b 

{3jOV + L {3jkVk = 0, 

b 

(6) {3jO'P + L (3jk'Pk(qk) = j, j = 1,2, ... , b - n + 2 (9) 

where CXjk ajk is either I, 1, -I,- 1, or 0, b is the total number of 

memristors, and n is the total number of nodes. Similarly, 

we can write a system of (b-n+2) independent KVL 

7 We have assumed for simplicity that the mernristors are chargecontrolled. 

The proof can be easily modified to allow memristors mernristors char 

char- 

chargeacterized 

by arbitrary qr-q e curves. 



513 

where Qi Qj and _joUQo. Following identical procedure 

and notation as given in [11, [ll, ch. 3], 31, we let Wwa a denote 

the set of independent frequencies and make a small change 

in ocf>a=o(wat). This 

pro- 

6~$,=Li(~,t). perturbation induces a change in the 

action A(t): : 

(14) 

(14) 

But since sintie A(q) = J&(q) fgcp(q) dq, we have 

oA 6A = (cp)(oq) ((p)(Sq) = [Re L: F ~~ TY c@at 

iwat ] 

[ a JWWC2 a 1 

.[Re L: c 5 I a (awajawa)Ciwatocf>a]. (ao,/aw,)ej~‘h 

1 (15) 

’ 

aLI Wa al 

Equating (14) and (15) and taking their time averages, we 

obtain the following Manley-Rowe-like formula relating the 

. ~ _1 

reactwe reactive powers P,=+ Pa=2" I 

Im m (V,Z,*): a 

1*)' a . 

~[ac&/awa] [P&a = 0 . 

(16) 

(16) 

P 

a 


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It is possible to derive a Page-Pantell-like inequality re- relating 

the realpowers of a passive memristor by making use 

of the passivity criterion (&)(64)>0 (ocp)(oq)'?°(Theorem (Theorem 1); namely, 

L 

where Pa=3 Pa== t Re (Val:) VaZz) is the real power at frequency w,. wa. 

Since the procedure for deriving (17) follows again mutatis 

mutandis that given by Penfield [11], 111, it will not be given here 

to conserve space. An examination of (17) shows that gain 

proportional to the frequency squared is likely in a mem- memristor 

upconverter, but that severe loss is to be expected in 

a memristor mixer. It is also easy to show that converting 

efficiencies approaching 100 percent may be possible in a 

memristor harmonic generator. 

So far we have considered only pure memristor networks. 

Let us now consider the general case of a network containing 

resistors, inductors, capacitors, and memristors. The equa- equations 

of motion for this class of networks now take the form 

of a system of m first-order nonlinear differential equations 

in the normal form $=f(x, x=f(x, t) [I], [l], where x is an mX 1 vector 

whose components are the state variables. The number m is 

called the “order "order of complexity” complexity" of the network and is equal 

to the maximum number of independent initial conditions 

that can be arbitrarily specified [1]. [I]. The following theorem 

shows how the order of complexity can be determined by 

inspection. 

Theorem 5: Order of Complexity 

Let N be a network containing resistors, inductors, capacitors, 

memristors, independent voltage sources, and inde 

inde- 

pendent current sources. Then the order of complexity m of 

N is given capaciby 

I 

(17) 

-1 

(18) 

state variables occurs whenever an independent loop con- consisting 

of elements corresponding to those specified in the 

definition of IZ.&~ nM and nLw nLM is present in the network. [We as- assume 

the algebraic sum of charges around any loop (flux- (fluxlinkages 

in any cut set) is zero.] Similarly, a constraint 

among the state variables occms occ\:lrs whenever an independent 

cut set consisting of elements corresponding to those speci- specified 

in the definition ofnM fiM and &CM nCM is present in the network. 

Since each constraint removes one degree of freedom each 

time this situation occurs, the maximum order of complexity 

(bL+bc+bM) must be reduced by one. 

Q.E.D. 

IV. AN ELECTROMAGNETIC INTERPRETATION 

OF MEMRISTOR CHARACTERIZATION 

It is well known that circuit theory is a ;llimiting limiting 

special 

casg cas~ 

of electromagnetic field theory. In particular, the char- characterization 

of the three classical circuit elements can be 

given an elegant electromagnetic ekctromagnetic interpretation in terms of 

the quasi-static expansion of Maxwell’s Maxwell's equations [12]. Our 

objective in this section is to give an analogous interpreta- interpretation 

for the characterization of memristors. While this 

interpretation does not prove the physical realizability of a 

“memristor "memristor device” device" without internal power supply, it does 

suggest the strong plausiblity that such a device might some- someday 

be discovqred. discovered. Let us begin by writing down Maxwell’s Maxwell's 

equations in differential form: 

curl E = 

aD 

curl H = J +-+ f8f at 

dB 

curl E = - 

a- Ly -- 

(21) 

aT a7 

where bL br. is the total number of inductors; bc is the total 

number of capacitors; b,ll M is the total number of memristors; 

aD 

curl H = J + a! $ 

(?a 

nM nnl is the number of independent loops containing only 

(22) 

aT 

memristors; liCE /?CE is the number of independent loops containing 

only capacitors and voltage sources; 11 nL.ll LM is the 

where E, H, D, B, and J are functions of not only the posi 

posi- 

connumber 

of independent loops containing only 

inductors 

and memristors; ;nM h,,r is the number of independent cut sets 

containing only memristors;nLJ fiLJ is the number of independent 

cut sets containing only inductors and current 

sources; flCM ric.nr is the number of independent cut sets con 

con- 

indetion 

(x, y, z), but also of a(Y and T. 7. If we were to expand these 

vector quantities as aformala power series in acy and substitute 

them into (21) and (22), we would obtain upon equating the 

coefficients coeficients of an, CP, the nth-order Maxwell's Maxwell’s equatiol1s, equaiions, where 

n=O, 1,2, .... ’ . . . 

taining only capacitors arid and memristors. . 

Many electromagnetic phenomena and systems can be 

Proof: ProCf: It is well known that the order of complexity of an 

satisfactorily analyzed by using only the zero-order and firstorder 

first- 

RLC network is given by m~(bL+bc)-l1cE-fzLJ 

m=(bL+bc)-IzCE-YiLJ [IJ. [l]. It 

Maxwell's Maxwell’s equations; the corresponding solutions are 

follows, therefore, from (1)-(4) (l)-(4) that for an RLC-memristor 

called quasi-staticfields. It has been shown that circuit theory 

network with I1 n, m =nLM=ii,l/=fi= nLlll = i?,,, = i2c.1, cJ1 =0, =O, each nlemristor niemristor belongs to the realm of quasi-static fields and can be studied 

introduces a new state variable and we have m=(bl.+bc 

m=(b,,+bc with the help of 

r the 

. 

following Maxwell's Maxwell’s equations in quasistatic 

form [12]. 

quasi- 

+bM)-ncE-nLJ. +b,+i)--ncg-CiLJ. Observe next that a constraint among the 

1121. 

aB 

at 

(19) 09) 

(20) 

where E and H are the electric and magnetic field intensity, 

D and B are the electric and magnetic flux density, and J 

is the current density. We will follow the approach presented 

in [12][ 121 by defining a "family “family time" time” T=cd, r=at, where a is called 

the "time-rate “time-rate parameter." parameter.” In terms of the new variable T, 

Maxwell's Maxwell’s equations become 

aB


Zero-Order Maxwell’s Maxwell's Equations 

curl Eo EIJ = 0 

where .§J('), 3( .), (B( (R(.), and D(-) a)( .) are one-to-one continuous functions 

from R3 onto R3. Under these assumptions, (26) and 

(23) 

func- 

(23) (27) can be combined to give 

curl Ho o = Jo. 

(24) 

First-Order Maxwell’s Maxwell's Equations 

curl HI = d(E1). (30) 

Observe that (30) does not contain any time derivative. 

aBo 

(25) 

Hence, hence, under any specified boundary condition con,dition appropriate 

cur1 E1 = - a, aT 

for the device, the first-order electric field E1 1 is related to 

aDo 

the first-order magnetic field HI dy by a functional relation; 

; 

aoo 

curl HI = II J1 +--. -I- -. (26) cw namely 

a7 

aT 

The total quasi-static vector quantities are obtained by keeping 

keep- 

EI = f(H,). (31) 

only oniy the first two terms of the formal pdwer power series and atid Oy by If we substitute (31) for E1 1 in (29) and then substitute the in 

in- 

setting a= CY= I; 1; namely, E~Eo+EI, E-Eo+E1, H~Ho+HI, H=H”+Hl, D~bo+DI' 

D=&+D1, verse function of (B( CR( .) from (28) into the resulting expression, 

we obtain 

have been identified as electromagnetic systems whose solu 

solu- 

expres- 

B= B~Bo+BI, Bo+ B1, J-Jo+ J~Jo+lI. JI. The three classical circuit elements 

tions correspond to certain combinations of the zero-order 

D1 = a, o f o [W(B1)] = g(B1). (32) 

and first-order solutions of (23)--(26). (23)

516 1EEETRANSACTIONSON IEEE TRANsACnONS ON CIRCUITTHEORY,SEPTEMBER THEORY, SEPTEMBER 1971 

RI 

i + 

+ 

” T 

v*( t 1 i “0 

i 

I 

v*( t ) 

I----'1 

IO). 

Rz 

l- 

E ---..,..-------: 

v,(t) 

E, 

I 

---_ 

T 

------_______ 

-l 

by 

dq/dt = u&V[RI 

+ Rz + M(q)]. 

Since the variables are separable, the solution is readily found 

to be 

where 

dq/dt = v.(t)/[RI + R 2 + M(q)]. (33) 

q(t) = h- l 0 (Jot v.(r) dr + «'(q(to») (34) 

I h(q) = 6% + R& + u?(q) I 

(35) 

and cp


q 

1 

IIOP.·W3·~3 

IoP.·W2·~ 

041 0 


I 

~-------------, 

+ 

0-- --' 

Zener breakdown vallage: voltage : E, = + l!.zE •, E~=E,=E~=AE Z • E • t>.E 

3 4 R, =R2 =R3 =R4 =R3 = + 

(01. 

5G 8 

9 

------------.-::::>F-+*r-+--+----+-~ V 

2 

(b). 

(e). 

v,(I I 

r------,---------------i'=E~--...--------------r_--~ 

o 

0 -t 

4 

-----!--------------'-------l:--E:.-----------'--_~ 

-._____ -----__ 

(d). Cd). 

, :' 

L.--_ 

.-- I 1-------------:- 

, , 

---------:~~~! i-------------i---------- ! 

------ ------------- ------ -~-~~~ -~==~~~~-~---~~~~~==---- 

-------- -------------i-- 

(el. 

Fig. 9. 

Nine-segment memristor can be used to generate ten-step staircase periodic waveform. 

used in many instruments such as the sampling oscilloscope 

and the transistor curve tracer. 

To simplify discussion, let us consider the design ofa a fourstep 

staircase waveform generator. The output voltage wave 

wave- 

fourform 

shown in Fig. 7(d) suggests that a four-step staircase 

waveform can be generated by driving the circuit in Fig. 

7(a) with a symmetrical square wave, provided that a 

memristor with the 'f'-q cp-q curve shown in Fig. 7(b) is available. 

This memristor can be synthesized by the methods presented 

in Section U. II. In fact, a simple realization is shown in Fig. 

8(a) with a nonlinear resistor (R @ connected across port 2 of 

a type-2 M-.R M-R mutator. This nonlinear resistor is, in turn, 

realized by two back-to-back series Zener diodes in parallel 

with a linear resistor and has a V-I curve as shown in Fig. 

8(b). To obtain the desired

CHUA: MEMRISTOR-MISSING CIRCUIT ELEMENT 

VI. CONCLUDING REMARKS 

VI. CONCLUDING REMARKS 

The memristor has been introduced as the fourth basic. 

circuit element. Three new types ofmutators have been introduced 

for realizing memristors in the form of active circuits. 

An appropriate cascade connection of these mutators and 

those already introduced in [3] can be used to realize higher 

order elements characterized by a relationship between v(m)(t) @j(t) 

introand 

i(n)(t), i@)(t), where v(m)(t) rW(t) (i(n)(t» (P(t)) denotes the mth (nth) time 

derivative of v(t) u(t) (i(t» (i(t)) if m>O (n>O), (n>(j), or the mth iterated 

time integral of v(t) u(t) (i(t» (i(t)) if m

Memristor-The Missing Circuit Element - IEEE Global History Network

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