simulate this circuit – Schematic created using CircuitLab
Figure 1. Transformer tap selector.
This isn't a complete answer but will give you some ideas to work on. Whatever electronic solution you come up with is going to need a steady voltage to work with. This makes it awkward to derive from the transformer as you have a full-wave bridge rectifier on it. A commercial product would add an auxiliary winding to the transformer to give, for example, a 12 V supply that doesn't change with the tap selector. Since this is a one-off project we can add a second transformer (ex wall-wart) to generate this.
How it works
- The main supply is derived from XFMR1 via two tap-selecting relays. If both are off the 5 V tap is selected. If MED is energised the 12 V tap is selected. If HIGH is energised the 24 V tap is selected.
- XFMR2 and the bridge form a secondary supply which remains steady.
- R3 and 4 divide the output voltage down to < V+AUX. A 4:1 ratio would be fine. CMP1, a comparitor, compares the divided down output voltage with the reference set by R7. When Vout > Vref the comparitor output will switch low energising the MED relay.
- A second comparitor circuit (not shown) is required for the HIGH relay.
The circuit is messy. Some hysteresis may be required to prevent the relays switching back and forward when close to the transition points. Vaux will droop every time a relay picks and this will upset the reference, etc.
All these problems can be designed out but a switching regulator solves them all with far less complexity.