Suppose I have a PCB trace that needs to be impedance matched, but the trace width changes as it travels from its source to its destination. Something like this:
The 5mil trace would have a certain characteristic impedance, whereas the 10mil trace would have another impedance. How do these two different impedances combine? Asked differently, what is the equivalent impedance of this trace?
You need to know the height of the trace between the ground plane, the relative electric permeability and how high the trace is to find the equivalent impedance.
For standard FR4 (pcb material) the electric permeability constant is ~4.4. The height will depend on your PCB stackup, and what layer the ground plane is on.
Source: https://www.eeweb.com/tools/microstrip-impedance
If the transmission lines are not matched, you will get a reflection at the point of mismatch and lose power.
The equation for finding this would be the reflection equation:
Z1 would be the characteristic impedance of the 5mil trace, and Z2 would be the impedance for the 10mil trace (after you calculate it)
Assuming the height above the ground plane is the same for the whole trace, the 5-mil-wide segments will have one characteristic impedance and the 10-mil-wide segment will have a different, lower, characteristic impedance.
The whole trace would not normally be thought of as having a single "equivalent impedance" characteristic.
What you could do is calculate the input impedance, or the S-parameters of the trace, as a function of frequency. To do this, you could use a simulation tool like Keysight ADS, or Smith chart techniques (if you are only interested in the input reflections), or simply sit down and do a bunch of algebra.
I fear you have entirely missed the point of controlled impedance. Your diagram actually looks (from an impedance point of view) like this
simulate this circuit – Schematic created using CircuitLab
where Z0 and Z2 are 5 mil lines, and Z1 is your 10 mil line.
The whole point of using a controlled impedance line is to ensure that the junction of Z2 and ZLOAD does not cause a reflection when the signal hits it. Assuming that, with your board geometry and material properly selected, Z2 and ZLOAD are in fact matched, you now need to consider the rest of the trace.
The junctions of Z0 and Z1, and Z1 and Z2 will BOTH produce reflections, and this is entirely backwards from what you want to do in the first place.
