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                                                                       EXHIBIT A

     THIS WARRANT AND THE COMMON STOCK ISSUABLE UPON THE EXERCISE HEREOF
     HAVE NOT BEEN REGISTERED UNDER THE SECURITIES ACT OF 1933.  THEY MAY
     NOT BE SOLD, OFFERED FOR SALE, PLEDGED, HYPOTHECATED, OR OTHERWISE
     TRANSFERRED EXCEPT PURSUANT TO AN EFFECTIVE REGISTRATION STATEMENT
     UNDER THE SECURITIES ACT OF 1933, OR AN OPINION OF COUNSEL
     SATISFACTORY TO THE COMPANY THAT REGISTRATION IS NOT REQUIRED UNDER
     SUCH ACT OR UNLESS SOLD PURSUANT TO RULE 144 UNDER SUCH ACT.



                        WARRANT TO PURCHASE COMMON STOCK
                                       OF
                                 CONDUCTUS, INC.
                          VOID AFTER SEPTEMBER 10, 2003
NO. ___

          This Warrant is issued to ___________ or its registered assigns
("Holder") by CONDUCTUS, INC., a Delaware corporation (the "Company"), on
September 11, 1998 (the "Warrant Issue Date").  This Warrant is issued pursuant
to the terms of that certain Series B Preferred Stock and Warrant Purchase
Agreement dated as of September 11, 1998 (the "Purchase Agreement") by and
between the Company and Holder.

          1.   PURCHASE SHARES.  Subject to the terms and conditions hereinafter
set forth, the Holder is entitled, upon surrender of this Warrant at the
principal office of the Company (or at such other place as the Company shall
notify the Holder in writing), to purchase from the Company up to _____________
(________) fully paid and nonassessable shares of Common Stock of the Company,
as constituted on the Warrant Issue Date (the "Common Stock").  The number of
shares of Common Stock issuable pursuant to this Section 1 (the "Shares") shall
be subject to adjustment pursuant to Section 8 hereof.

          2.   EXERCISE PRICE.  The purchase price for the Shares shall be
$2.70, as adjusted from time to time pursuant to Section 8 hereof (the "Exercise
Price").

          3.   EXERCISE PERIOD.  This Warrant shall be exercisable, in whole or
in part, during the term commencing on the date six months following the Warrant
Issue Date and ending at 5:00 p.m. on September 10, 2003; provided, however,
that in the event of (a) the closing of the Company's sale or transfer of all or
substantially all of its assets, or (b) the closing of the acquisition of the
Company by another entity by means of merger, consolidation or other transaction
or series of related transactions, resulting in the exchange of the outstanding
shares of the Company's capital stock such that at least 50% of the voting power
of the Company is transferred, this Warrant shall, on the date of such event, no
longer be exercisable and become null and void.  In the event of a proposed
transaction of the kind described above, the Company 

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shall notify the  Holder at least fifteen (15) business days prior to the 
consummation of such event or transaction.

          4.   METHOD OF EXERCISE.  While this Warrant remains outstanding and
exercisable in accordance with Section 3 above, the Holder may exercise, in
whole or in part, the purchase rights evidenced hereby.  Such exercise shall be
effected by:

               (a)  the surrender of the Warrant, together with a duly executed
copy of the form of Notice of Election attached hereto, to the Secretary of the
Company at its principal offices; and

               (b)  the payment to the Company of an amount equal to the
aggregate Exercise Price for the number of Shares being purchased.

          5.   NET EXERCISE.  In lieu of exercising this Warrant pursuant to
Section 4, the Holder may elect to receive, without the payment by the Holder of
any additional consideration, shares of Common Stock equal to the value of this
Warrant (or the portion thereof being canceled) by surrender of this Warrant at
the principal office of the Company together with notice of such election, in
which event the Company shall issue to the Holder a number of shares of Common
Stock computed using the following formula:

                         Y (A - B)
                         ---------
                    X =      A

     Where:         X =  The number of shares of Common Stock to be issued to
                    the Holder pursuant to this net exercise;

                    Y =  The number of Shares in respect of which the net issue
                    election is made;

                    A =  The fair market value of one share of the Common Stock
                    at the time the net issue election is made;

                    B =  The Exercise Price (as adjusted to the date of the net
                    issuance).

For purposes of this Section 5, the fair market value of one share of Common
Stock as of a particular date shall be determined as follows:  (i) if traded on
a securities exchange or through the Nasdaq National Market, the value shall be
deemed to be the average of the closing prices of the securities on such
exchange over the thirty (30) day period ending three (3) days prior to the net
exercise election; (ii) if traded over-the-counter, the value shall be deemed to
be the average of the closing bid or sale prices (whichever is applicable) over
the thirty (30) day period ending three (3) days prior to the net exercise; and
(iii) if there is no active public market, the value shall be the fair market
value thereof, as determined in good faith by the Board of Directors of the
Company.

                                       2

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          6.   CERTIFICATES FOR SHARES.  Upon the exercise of the purchase
rights evidenced by this Warrant, one or more certificates for the number of
Shares so purchased shall be issued as soon as practicable thereafter (with
appropriate restrictive legends, if applicable), and in any event within thirty
(30) days of the delivery of the subscription notice.

          7.   ISSUANCE OF SHARES.  The Company covenants that the Shares, when
issued pursuant to the exercise of this Warrant, will be duly and validly
issued, fully paid and nonassessable and free from all taxes, liens, and charges
with respect to the issuance thereof.

          8.   ADJUSTMENT OF EXERCISE PRICE AND NUMBER OF SHARES.  The number of
and kind of securities purchasable upon exercise of this Warrant and the
Exercise Price shall be subject to adjustment from time to time as follows:

               (a)  SUBDIVISIONS, COMBINATIONS AND OTHER ISSUANCES.  If the
Company shall at any time prior to the expiration of this Warrant subdivide its
Common Stock, by split-up or otherwise, or combine its Common Stock, or issue
additional shares of its Common Stock as a dividend with respect to any shares
of its Common Stock, the number of Shares issuable on the exercise of this
Warrant shall forthwith be proportionately increased in the case of a
subdivision or stock dividend, or proportionately decreased in the case of a
combination.  Appropriate adjustments shall also be made to the purchase price
payable per share, but the aggregate purchase price payable for the total number
of Shares purchasable under this Warrant (as adjusted) shall remain the same.
Any adjustment under this Section 8(a) shall become effective at the close of
business on the date the subdivision or combination becomes effective, or as of
the record date of such dividend, or in the event that no record date is fixed,
upon the making of such dividend.

               (b)  RECLASSIFICATION, REORGANIZATION AND CONSOLIDATION.  In case
of any reclassification, capital reorganization, or change in the Common Stock
of the Company (other than as a result of a subdivision, combination, or stock
dividend provided for in Section 8(a) above), then, as a condition of such
reclassification, reorganization, or change, lawful provision shall be made, and
duly executed documents evidencing the same from the Company or its successor
shall be delivered to the Holder, so that the Holder shall have the right at any
time prior to the expiration of this Warrant to purchase, at a total price equal
to that payable upon the exercise of this Warrant, the kind and amount of shares
of stock and other securities and property receivable in connection with such
reclassification, reorganization, or change by a holder of the same number of
shares of Common Stock as were purchasable by the Holder immediately prior to
such reclassification, reorganization, or change.  In any such case appropriate
provisions shall be made with respect to the rights and interest of the Holder
so that the provisions hereof shall thereafter be applicable with respect to any
shares of stock or other securities and property deliverable upon exercise
hereof, and appropriate adjustments shall be made to the purchase price per
share payable hereunder, provided the aggregate purchase price shall remain the
same.

               (c)  NOTICE OF ADJUSTMENT.  When any adjustment is required to be
made in the number or kind of shares purchasable upon exercise of the Warrant,
or in the Warrant Price, the Company shall promptly notify the Holder of such
event and of the number of 

                                       3

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shares of Common Stock or other securities or property thereafter purchasable 
upon exercise of this Warrant.

          9.   NO FRACTIONAL SHARES OR SCRIP.  No fractional shares or scrip
representing fractional shares shall be issued upon the exercise of this
Warrant, but in lieu of such fractional shares the Company shall make a cash
payment therefor on the basis of the Exercise Price then in effect.

          10.  NO STOCKHOLDER RIGHTS.  Prior to exercise of this Warrant, the
Holder shall not be entitled to any rights of a stockholder with respect to the
Shares, including (without limitation) the right to vote such Shares, receive
dividends or other distributions thereon, exercise preemptive rights or be
notified of stockholder meetings, and such Holder shall not be entitled to any
notice or other communication concerning the business or affairs of the Company.
However, nothing in this Section 10 shall limit the right of the Holder to be
provided any notices required under this Warrant or the Purchase Agreement.

          11.  TRANSFERS OF WARRANT.  Subject to compliance with applicable
federal and state securities laws, this Warrant and all rights hereunder are
transferable in whole or in part by the Holder to any person or entity upon
written notice to the Company.  The transfer shall be recorded on the books of
the Company upon the surrender of this Warrant, properly endorsed, to the
Company at its principal offices, and the payment to the Company of all transfer
taxes and other governmental charges imposed on such transfer.  In the event of
a partial transfer, the Company shall issue to the holders one or more
appropriate new warrants.

          12.  SUCCESSORS AND ASSIGNS.  The terms and provisions of this Warrant
and the Purchase Agreement shall inure to the benefit of, and be binding upon,
the Company and the Holders hereof and their respective successors and assigns.

          13.  AMENDMENTS AND WAIVERS.  Any term of this Warrant may be amended
and the observance of any term of this Warrant may be waived (either generally
or in a particular instance and either retroactively or prospectively), with the
written consent of the Company and the Holder. Any waiver or amendment effected
in accordance with this Section shall be binding upon each holder of any Shares
purchased under this Warrant at the time outstanding (including securities into
which such Shares have been converted), each future holder of all such Shares,
and the Company.

          14.  NOTICES.  All notices required under this Warrant and shall be
deemed to have been given or made for all purposes (i) upon personal delivery,
(ii) upon confirmation receipt that the communication was successfully sent to
the applicable number if sent by facsimile; (iii) one day after being sent, when
sent by professional overnight courier service, or (iv) five days after posting
when sent by registered or certified mail.  Notices to the Company shall be sent
to the principal office of the Company (or at such other place as the Company
shall notify the Holder hereof in writing).  Notices to the Holder shall be sent
to the address of the Holder on the books of the Company (or at such other place
as the Holder shall notify the Company hereof in writing).

                                       4

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          15.  ATTORNEYS' FEES.  If any action of law or equity is necessary to
enforce or interpret the terms of this Warrant, the prevailing party shall be
entitled to its reasonable attorneys' fees, costs and disbursements in addition
to any other relief to which it may be entitled.

          16.  CAPTIONS.  The section and subsection headings of this Warrant
are inserted for convenience only and shall not constitute a part of this
Warrant in construing or interpreting any provision hereof.

          17.  GOVERNING LAW.  This Warrant shall be governed by the laws of the
State of California as applied to agreements among California residents made and
to be performed entirely within the State of California.

                                       5

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          IN WITNESS WHEREOF, the Company caused this Warrant to be executed by
an officer thereunto duly authorized.

                                CONDUCTUS, INC.

                                By:
                                   ------------------------------------------
                                   Charles E. Shalvoy, President and Chief
                                   Executive Officer

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                               NOTICE OF EXERCISE


To:  Chief Executive Officer
     Conductus, Inc.
     969 W. Maude Avenue
     Sunnyvale, CA 94086

            The undersigned hereby elects to [CHECK APPLICABLE SUBSECTION]:

________    (a)  Purchase _________________ shares of Common Stock of
            Conductus, Inc., pursuant to the terms of the attached Warrant
            and payment of the Exercise Price per share required under such
            Warrant accompanies this notice;

            OR

________    (b)  Exercise the attached Warrant for [all of the shares]
            [________ of the shares] [CROSS OUT INAPPLICABLE PHRASE]
            purchasable under the Warrant pursuant to the net exercise
            provisions of Section 5 of such Warrant.





                              WARRANTHOLDER:




                              By:
                                  ----------------------------------
                              Name:
                                    --------------------------------
                              Title:
                                     -------------------------------

                    Address:
                              --------------------------------------

                              --------------------------------------

Date:
     ----------------------

Name in which shares should be registered:

------------------------------------------