Definition of Trinity: One being, Three persons: One God, the Father, Son, and Holy Spirit (All the time: from beginning to end)
Analogies for the trinity are not lacking. Here are a few examples: H2O, three leaf clover, and one person being a father, son, and helper. Each one of these have defects which, at least supposedly, to heresy. Take the H2O example; God is like H2O because H2O can be water, steam, or ice and in the same way God is one, but also father, son, and Holy Spirit. The problem with this analogy is that H2O manifest differently dependently on external factors, e.g. temperature. In contrast, the trinity is three persons and one being in various external conditions. If we believed the persons of the trinity manifested in different external context, then we would believe the heresy known as modalism.
My tentative suggestion for an analogy is not perfect, but is close enough that the altercations needed are easy to explain. Imagine a normal human being who gains the ability to multiply. Each copy is controlled by their own will and not the original person. However, each person is also intricately connected that if one loses a quality that is essential to them, the other person loses it as well. For example, a possible essential quality to me is my love for philosophical questions. If suddenly, I just accepted what others told me without investigation, most of my friends would not be sure who I was. However, the fact I was born with dark brown hair is not essential, but accidental. If my natural hair color was instead light brown, or black, or blond, I could still be essentially the same. An essentially property of God is his loving nature. Thus, Jesus, Holy Spirit, and the Father all need to be loving, otherwise they cease being God. Back to the normal human who gain a super power. If she multiplied twice, then there would be one essential being that connected them all, but three distinct persons.
The minor modification we need is this: the Trinity, unlike the woman who gained the super power, is necessarily in this condition eternally.