Hi Colin,
I really dont think you are looking at the relationship between the inductor and capacitor quite right. I say this because you are saying that there can be no back EMF. Thus i have to question how you are interpreting this "back emf" as to what it really is and under what circumstances it can start to appear.
Any inductor that is charged is going to exhibit back emf at some time or another. To put in in a more clear way, once the inductor is charged it's going to want to discharge at some point, and that' s where the back emf becomes the driving force.
To start the capacitor is charged. Then the user closes the external switch. This causes current to flow into the solenoid coil. The coil acts as an inductance which charges up with a positive current where the capacitor is the driving force behind all the action so far.
Next the capacitor starts to run out of energy so it's voltage goes to zero. But the inductor still has energy and current flowing, so it generates a 'back emf' to keep the current flowing even though the cap voltage is zero. This drives the capacitor voltage negative. How far negative it does depends on the value of the capacitance and the value of the inductance.
Note here that the cap voltage does not go through zero just on it's own because it runs out of energy. It gets driven through zero by the inductor at some point.
Another way to look at it is like a step drive voltage into a resonant cap and inductor. A cap in series with an inductor has resonance as im sure you know. This causes a semi sinusoidal waveform to appear across the cap. We know that the current in an inductor driven by a sinusoidal voltage, when it changes from increasing to decreasing (at the peak) the voltage in the cap is going through zero. If there is nothing to stop the cap from going through zero the current in the coil will drive the cap voltage negative, and how negative it goes depends on how much energy is left in the coil, or simply the inductance value. Thus, some inductance values will drive it very negative, and some not as much, but they will always drive it negative unless we get so very lucky that the coil runs out of energy at the same time that the cap does.
The series resistance of course will play a part too if it can act as an over damper. It could eat up all the energy. But there we would be depending on luck.
So the bottom line is that some coils will cause a more severe reverse capacitor voltage than others. If the reverse diode is across the cap, it will clamp that voltage. If the diode is not there, the cap may act like a low resistance in the reverse direction and damp out the energy of the coil, but that's probably not a good idea.
If we were able to specify a specific solenoid for use (such as Acme 12345A) then we could analyze this in that light where we would have only one variable we could look at and determine if it worked or not. But if we dont have the liberty to spec that part ("connect any solenoid you want to the output") then we can not say for sure that there will never be any damaging back emf without the clamp diode.
So to clarify just a little bit more the situation we have here...
If we have a cap in series with an inductor where the inductor has internal series resistance, and the cap has some initial voltage, if we can not spec the inductor (and series resistance) we can not spec the negative undershoot without a clamp diode. It's as simple as that.
It works ok with one inductor but not another.
A scope picture of the cap voltage of an actual circuit working with an actual typical coil would be nice to see right about now. It would not be conclusive however, but it would be interesting.
I really dont think you are looking at the relationship between the inductor and capacitor quite right. I say this because you are saying that there can be no back EMF. Thus i have to question how you are interpreting this "back emf" as to what it really is and under what circumstances it can start to appear.
Any inductor that is charged is going to exhibit back emf at some time or another. To put in in a more clear way, once the inductor is charged it's going to want to discharge at some point, and that' s where the back emf becomes the driving force.
To start the capacitor is charged. Then the user closes the external switch. This causes current to flow into the solenoid coil. The coil acts as an inductance which charges up with a positive current where the capacitor is the driving force behind all the action so far.
Next the capacitor starts to run out of energy so it's voltage goes to zero. But the inductor still has energy and current flowing, so it generates a 'back emf' to keep the current flowing even though the cap voltage is zero. This drives the capacitor voltage negative. How far negative it does depends on the value of the capacitance and the value of the inductance.
Note here that the cap voltage does not go through zero just on it's own because it runs out of energy. It gets driven through zero by the inductor at some point.
Another way to look at it is like a step drive voltage into a resonant cap and inductor. A cap in series with an inductor has resonance as im sure you know. This causes a semi sinusoidal waveform to appear across the cap. We know that the current in an inductor driven by a sinusoidal voltage, when it changes from increasing to decreasing (at the peak) the voltage in the cap is going through zero. If there is nothing to stop the cap from going through zero the current in the coil will drive the cap voltage negative, and how negative it goes depends on how much energy is left in the coil, or simply the inductance value. Thus, some inductance values will drive it very negative, and some not as much, but they will always drive it negative unless we get so very lucky that the coil runs out of energy at the same time that the cap does.
The series resistance of course will play a part too if it can act as an over damper. It could eat up all the energy. But there we would be depending on luck.
So the bottom line is that some coils will cause a more severe reverse capacitor voltage than others. If the reverse diode is across the cap, it will clamp that voltage. If the diode is not there, the cap may act like a low resistance in the reverse direction and damp out the energy of the coil, but that's probably not a good idea.
If we were able to specify a specific solenoid for use (such as Acme 12345A) then we could analyze this in that light where we would have only one variable we could look at and determine if it worked or not. But if we dont have the liberty to spec that part ("connect any solenoid you want to the output") then we can not say for sure that there will never be any damaging back emf without the clamp diode.
So to clarify just a little bit more the situation we have here...
If we have a cap in series with an inductor where the inductor has internal series resistance, and the cap has some initial voltage, if we can not spec the inductor (and series resistance) we can not spec the negative undershoot without a clamp diode. It's as simple as that.
It works ok with one inductor but not another.
A scope picture of the cap voltage of an actual circuit working with an actual typical coil would be nice to see right about now. It would not be conclusive however, but it would be interesting.
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