When is this voltage source activated


So I'm doing first order circuit analysis and I stumbled upon a circuit which had this as a voltage source:

enter image description here

When would this voltage source be active?

For example in this circuit:

enter image description here

The current source on the left would be active at t>1.
The one on the right is active at t>0.

What about that voltage source though?

Best Answer

\$u(t)\$ is the Unit Step Function. More than a mathematical function is a tool that allows us to model activations and deactivations in electrical circuits and abrupt changes of state in general. This function is defined as

$$ u(t) = \left\lbrace \begin{matrix} 0 \qquad t<0 \\ 1 \qquad t>0 \end{matrix} \right. $$

For your question, the variable \$t\$ has been transformed by a reflection with respect to \$t = 0\$. This means that

$$ u(-t) = \left\lbrace \begin{matrix} 0 \qquad t > 0 \\ 1 \qquad t < 0 \end{matrix} \right. $$

What does this mean? It means that the power supply has been active from \$t = -\infty\$ to \$t = 0\$, when it was deactivated. In other words, this source should only be taken into account in determining the initial conditions in the analysis of the circuit.