Yeah, pretty much. If you're switching a coil off, there will be a reverse voltage (flyback). You use either a diode, R-C snubber or some other means to absorb it. The best placement for the diode or other suppressor is right at the coil itself to get maximum effectiveness.
Any coil that is suddenly de-energized will make a fly-back reverse voltage spike as its magnetic flux collapses. The flyback voltage can be estimated as follows:
- \$V_{ind} = L_{ind}\frac{dI} {dt}\$
A typical switch turning off a coil will result in a near-instant \$dt\$, resulting in a good-sized spike even for a smallish, low-current relay.
Let's work an example. This 5V relay: https://www.farnell.com/datasheets/16770.pdf
- L = 65mH
- Coil resistance (current @5V) = 130 ohms (38.5mA)
So now we chop the current from 38.5mA to nothing (\$dI\$) in 1ms (\$dt\$):
- V = 65mH * -38.5mA/1ms = -38.4V
Note that at 1ms I'm being generous with \$dt\$ here. It would be on the order of microseconds, which could result in hundreds, if not thousands of volts at the coil - well above what your switching components can stand (FET, BJT, etc.) Even switch contacts, such as from a push button or another relay, can be damaged by flyback-induced arcing and eventually fail, not to mention the spike making its way back to the power supply and nerfing other parts of your system.
(Flyback diodes. Cheap insurance.)
If your coil drive voltage is AC, a diode won't work. You can use an RC snubber, back-to-back Zener diodes, or a purpose-built device called a TVS (transient voltage suppressor).
Regardless of type of spike-catcher you use, the best placement for the suppressor is right at the coil itself to get maximum effectiveness.
As for your proposed circuit, the first relay certainly needs a diode. The second relay probably isn't an issue as the current should have decayed to zero by the time the contacts swing. However, tf that time is short (say, if the relay contacts or the switch chatters) you could still have a problem.