Okay, conceptually this is pretty easy, as I think you know.
A DDS chip from AD with sin/cos outputs, appropriate low pass filter, buffer amplifier. Apply a voltage much less than the bias voltage (but high enough to get good SNR) to the sample and measure the current multiplied by the sin and by the cos, low-pass filter the two results and calculate the real and imaginary components of the impedance from the (measured) voltage and (measured) current levels.
You should be able to add the bias voltage in the buffer amplifier, but you might want to capacitively couple the current input to keep the dynamic range of the mixer reasonable.
At 10MHz most precision analog multipliers are running out of steam, so I'd look at Gilbert cell mixers. Unfortunately the low-frequency and DC performance is seldom well specified.
Of course you could simply digitize the data at hundreds of MHz and digitally demodulate it with a fast FPGA, but that would be even more of a project.
The impedance of 0.156uF at 10MHz is only about 0.1 ohm, so the buffer should be able to handle tens of mA at 10MHz and your signal chain has to be happy with ~1mV total signal.
If you have access to a "lock-in amplifier" (the rack mount instrument), look at that to replace a chunk of the work. Same if you have a function generator with quadrature outputs.
I did something similar to characterize magnetic samples (there were some very special requirements) but the frequency had to be as close to DC as possible, so it was simply measured at low swept frequencies and curve-fit extrapolated to even lower frequencies (where there would be no SNR left).
It's not clear to me whether your model is primarily a series R-C or parallel R-C, of course the general measurement of Z gives you a complex number which could be applied to either model.
Your project also reminds me of some interesting work I did on conductivity cell measurements for dialysis water treatment. There were some heuristics involved.
To clarify what Brian Drummond is talking about with "C" or "D" type circuit breakers - "C" type is for general purpose loads. "D" type is for loads with a high inrush current.
Using a "D" type breaker might allow the circuit breaker to restrain from tripping on inrush curent.
Excerpt from the Clipsal 4-series miniature circuit breaker catalogue:
With regards to soft starters;
It depends on the type of soft starter.
Some use an inductor or resistor or auto-transformer in series with the load, to reduce the starting voltage and hence the starting current. The inductor/resistor/auto-transformer is bypassed after a few seconds. This type would likely work well for you.
Electronic soft starters use some kind of semiconductor device (usually thyristors) to do pulse-width modulation of the voltage. This is a bit harsh and your machine may not like it.
Best Answer
Short story: In my opinion, don't change your basic idea covered in your points (1 to 4)
Both "transformer" circuits you have offered-up appear to use types of double balanced mixers that demodulate the audio from the microphone after the microphone has modified the phase shift of an oscillator.
The microphone is excited at the XTAL frequency by either a direct winding (Sennheiser schematic) or via a capacitor (1 pF, C5) in the 2nd diagram. The mixer stage is like the traditional balanced mixer using transformers: -
I'm not going to go into detail how each circuit in the question uses a double balanced mixer similar to the traditional one in my picture. If you want more details, you'll have to study a bit more.
Suffice to say that a double balanced mixer has one fundamental property that makes it useful for "mixing" two RF signals and that is signal multiplication.
And what you get with signal multiplication is an output level that can vary significantly with amplitudes i.e. it can be used as a phase detector but you need to keep signal and reference amplitudes stable or you'll get a phase angle DC output level that is also somewhat made erroneous by amplitude variations in the input signals.
So, to use DBM accurately as a phase detector you need to have a limiter circuit or saturate the mixer.
Given that you are operating at (only) 11 MHz there are a few quite fast EXOR gates that can do the job - after all, you have an oscillator already at CMOS levels and the tank circuit (fed via a resistor) from your oscillator can have an output level that is easily amplified to CMOS levels via a fast schmitt trigger (plenty to choose from) so, in my humble opinion, this is the best route to take.
You are operating at 11 MHz and your base bandwidth might be (say) 50 kHz, so what do you really think will be the problem of jitter. At (say) 50 kHz, there would be 220 cycles of clock jittering a bit this way and that way but if you averaged the effect of the jitter, just how much base band noise is really going to be present?
You can easily simulate this and find out BTW.
Think about all those RF receiver chips that use Gilbert cells to electronically perform multiplication - how much noise do they produce when they demodulate (say) an FM broadcast using quadrature detection? Are these chips all as noisy as hell? No they aren't but, you could say that they are operating at carrier frequencies about ten times higher than 11 MHz so the base band filtering will be ten times better.
But those gilbert cell mixers are dealing with a low level RF signal - a signal that is much much smaller than what appears across your tank. So, what is a gilbert cell?
Its forerunner was an attempt to design an exclusive OR gate which brings us nicely back to my original claim that I really don't think you are going to improve on using an EXOR gate acting as a quadrature detector.
I've /designed built sensitive capacitance probes and the minimum delta signal level was less than 20 femto farads - in other words this change in capacitance could be discerned on the demod output when connected to an oscilloscope.