Although Hohmann transfers use the minimum amount of energy and fuel to reach a new orbit, they also require the most time. For missions with time constraints, a short transfer time can be achieved at the cost of more fuel.
Instead of using an elliptical transfer orbit that just reaches the outer orbit, using a transfer ellipse which extends past the outer orbit will result in faster transfer times. This gives us the general coplanar transfer.
Using more Δv than required to reach the outer orbit will put the satellite in a faster trajectory; however it will also require the satellite to change directions upon arriving at the desired outer orbit.
To find the direction change we must first define the local horizontal: the direction perpendicular to the satellite’s position vector.
The angle between the satellite’s velocity and the local horizontal is called the flight path angle (φ).
For a circular orbit, the satellite’s velocity is perpendicular to its position, thus φ = 0.
Flight path angle can be found by using conservation of angular momentum. Assuming gravity is the only external force acting on the satellites, the specific angular momentum (h) of the satellite is conserved.
h= r v cos(φ)
Also, using the conservation of energy and some algebraic manipulation yields equations for Δv for general coplanar transfers.
Δv12=v12 + vcirc12 - 2v1vcirc1cos(φ1)
Δv22= v22 + vcirc22 - 2v2vcirc2cos(φ2)
Thus, Hohmann transfers are just a special case of coplanar transfers where φ = 0.