In a paper [The State Plane Co-ordinate System], regarding Lambert’s Conformal Conic Projection:
In order to obtain grid co-ordinates on a Lambert projection, we must
remember that the grid co-ordinate system is a rectangular system,
which is different to the ‘fan-shaped’ appearance of the projected
region. So there will be a grid convergence factor to be allowed for
when undertaking computations in grid-co-ordinates alone.
is to convert the geographical co-ordinates to polar co-ordinates (r,
θ), then convert these to rectangular grid co-ordinates. Consider the
situation in the Northern Hemisphere, using the following diagram.
Here we have also placed a false origin, so that all co-ordinates on
the grid will be positive. As this is a simple additive (and
arbitrary) transformation, we can leave it until the last step.
In the diagram, r = radius of some parallel of latitude φ; r0 = radius
of the parallel, φ0, upon which the true origin of the co-ordinate
system is situated; (x, y) are the grid co-ordinates of the
geographical point (φ, ∆λ); θ = the projection angle for a departure
of ∆λ from the central meridian, which has a longitude of λ0.
Note that the intermediate value θ of this geographic to rectangular conversion is exactly the convergence you seek.
θ = n ∆λ = n (λ – λ0)
It's worth reading the whole (short) paper for all the details:
"The State Plane Co-ordinate System" by Department of Civil and Environmental Engineering and Geodetic Science Geodetic and Geoinformation Science -- Section GS521 Geodetic Control Surveying]