INSTRUCTOR WITH SPI HAD ELECTRICAL FAILURE. CONTINUES WITH GREEN LIGHT AND LANDS.

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DEPARTED TOL ARSA NO DIFFICULTIES. STUDENT'S FIRST LESSON. DEMONSTRATED AND HAD STUDENT PERFORM 4 FUNDAMENTALS. UPON COMPLETION; HEADED BACK TO TOL FOR LNDG. COULD NOT RAISE TOL APCH/DEP ON RADIO ON WAY BACK. CONTINUED IN BECAUSE WE NEVER SWITCHED FROM DEP/APCH CTL; AND BELIEVED THEY HAD US ON RADAR; IF THEY WERE NOT RECEIVING OUR XMISSIONS. APCHED TOL FROM THE N (IN NW PRACTICE AREA). RECEIVED STEADY GREEN LIGHT FROM TWR AND MADE R TFC (NORMAL TFC) AND LANDED ON RWY 25. TAXIED OFF; RECEIVED FLASHING GREEN AND PROCEEDED TO PARKING. AFTER SHUTDOWN AND SECURING ACFT; TALKED ON TELEPHONE TO TWR CTLR. WAS TOLD THEY LOST US ON RADAR AND WERE GETTING READY TO SEND A CREW OUT TO LOOK FOR US. RECEIVED A CALL ON THE MORNING OF THE 28TH FROM MR. X; FAA CLE. HE TOLD ME IT WOULD BE WRITTEN UP AS AN OCCURRENCE. AS IT TURNED OUT; THE ALTERNATOR WAS BAD AND THE WARNING LIGHT WAS NOT BRIGHT ENOUGH TO BE SEEN DURING THE DAY. WE HAD USED UP THE BATTERY AND LOST ALL ELECTRICAL PWR. WE ARE WORKING ON THE WARNING LIGHT PROB.