When I check the alternator at first it worked?

Do not disconnect any cable with the engine running!

Are you honestly a mechanic or did you just list that? Every year I did a demonstration on the generator test bench for my students to show what can happen when you do that. It was real easy for the voltage to reach over 35 volts. That will destroy any computer on the vehicle, the generator's internal diodes and built-in voltage regulator, and any light bulbs that are turned on.

The thinking is that if you disconnect either cable and the engine stays running, the generator must be working but a lot of them will stop working due to the voltage regulator responding to the dips in the "ripple" voltage being produced. That will make a perfectly good generator appear to be bad so that test is not valid.

If a mechanic is caught pulling this stunt he will typically get one verbal warning. For the second offense he will be fired. It is that big a deal.

Some generators respond to the high points in the ripple. That momentary higher voltage goes right back to the field winding and creates a stronger magnetic field. That stronger electromagnet creates a higher output voltage which again creates a stronger electromagnet. It is a vicious circle and voltage can keep on rising until something gives out. The main thing that smooths out that ripple so it does not affect the voltage regulator or the generator is the battery.

Three things are needed to generate the output current. They are a magnet, (electromagnet, in this case), a coil of wire, and most importantly, movement between them. That is why the belt needs to make it spin. One thing that can save you from doing damage by removing a battery cable is not raising engine speed. Generators are relatively inefficient at low engine speeds and their output voltage is less likely to rise to dangerous levels, as long as you do not raise engine speed.

One other thing to keep in mind is batteries give off explosive hydrogen gas. Regardless if your generator is working or not there is going to be a big spark when you remove a battery cable with the engine running. Either the generator's current will be recharging the battery, and that can be up to 20 amps, or the battery is going to be supplying the car's electrical systems, and that can easily be over 30 amps. That kind of current is going to create a big spark when a connection is broken or reconnected. Small arc welders run as low as 40 to 60 amps and look at the sparks they create. The reason we do not hear about more battery explosions is because people are careful to not disconnect the cables when there is current flowing through them. It is also why there are huge warning labels on all battery chargers to be sure they are turned off before connecting or disconnecting them from the battery.

Another common generator problem is one defective diode out of the six. You will lose exactly two thirds of the generator's capacity but system voltage will remain normal or it could even be just a little high from the voltage regulator responding to the greatly increased dips in the ripple voltage.

It is always a good idea to wear safety glasses when working around car batteries, but if you still insist on removing a cable while the engine is running, a face shield makes more sense, and have plenty of water on hand to wash any acid off the vehicle's paint.

Ford used to have a really nice generator design that allowed testing right on the back of the unit. Only Chrysler alternators are easier to diagnose. Unfortunately the engineers do not really care about ease of service on GMs and many other brands.

The way you tell if the charging system is working is to measure the battery voltage while the engine is running. It must be between 13.75 and 14.75 volts. There still could be a bad diode though. You need a professional load tester to test for that. Ripple will be very high and the most output current you will get will be one third of the generator's design value. That is not enough to meet the demands of the electrical system under all conditions so the battery will have to make up the difference, until it runs down. You still could have a generator problem. Do a load load-test first, but keep in mind at the mileage you listed, worn brushes are to be expected. Those will always cause an intermittent loss of charging for weeks or months before it becomes permanent. To be valid, any testing has to be done while the problem is occurring.