The purpose of this final part of the discussion on bandspreading is to cover a few miscellaneous points which did not fit into the main discussion, or were somewhat glossed over.
From part one, you may have noticed that if we use a trimmer only, then the required inductance turns out to be very small but tuning linearity is good, and if we use a padder only, then the inductance turns out larger, but the linearity of the tuning is bad. Are there any other things we should be looking at? Yes. The discussion of large inductor/small capacitor versus small inductor/large capacitor comes up every so often, and the correct choice is not always clear. There are a number of factors involved.
From the resonance formula, it's obvious that, in theory, we can pick any arbitrary value for the inductance, and then find a corresponding value of capacitance to resonate at a given frequency, or conversely, pick any arbitrary capacitance, and then find a corresponding inductance to resonate at the given frequency. So then, what are the guidelines for determining the best values?
Let's look at it from an energy point of view.
The energy E stored in an inductor is given by
and the energy stored in a capacitor is given by
where i and v are the instantaneous voltage and current.
If, by some means, we inject energy into an LC tuned circuit, it will oscillate at the resonant frequency given by
and that energy in the LC circuit at any instant is the sum of the energy stored in the inductor and the energy stored in the capacitor. Hence:
with v and i varying sinusoidally with time. At some instant in time, because it is varying sinusoidally, the current in the inductor will be zero and therefore the energy stored in the inductor at that instant must also be zero. Hence, at that instant, all of the energy must be in the capacitor, and therefore, the capacitor will be at its peak voltage. We can therefore determine the energy stored in the circuit by measuring the peak voltage and evaluating the capacitor energy formula. More importantly, what this means is: the peak voltage across the capacitor will be proportional to the square root of the energy circulating in the circuit, and inversely proportional to the capacitance. Hence rearranging the capacitor energy formula (2), we have
If we want to maximize the peak voltage in a tuned circuit, then it is obvious from this that the capacitance value should be as low as possible, and hence the inductance must be large. This is generally the case in radio receivers where the tuned circuit is connected to the high impedance input of a voltage amplifier; or in the case of a crystal radio receiver, to a detector diode. In this situation, we want the highest possible signal voltage to overcome the forward voltage drop of the detector diode.
If we were interested in maximizing the peak current in the circuit, we would take the opposite approach, using a small inductance and large capacitance.
Another important consideration is the maximization of circuit Q (which is the ratio of energy stored in the circuit to the energy dissipated in a given time period).
If there were no losses in the circuit, the energy would continue to oscillate back and forth forever. However, a real circuit has losses. The coil losses will be the most significant, but to a lesser extent, the capacitor will also have losses. These losses can be treated as a lumped resistance value R, and the instantaneous power P dissipated due to these is given by
We see that the losses are proportional to the resistance. So, at first glance, it seems plausible that to minimize the losses we need to minimize the resistance, and that since the resistance is mostly due to the inductor, it would make sense to use the smallest possible inductor. However this appears to be at odds with maximizing the peak voltage discussed above, because as the inductance is decreased, the peak voltage will decrease. But more significantly, with a smaller inductor, the peak current must increase, and since the power loss is proportional to the square of the current, the effect of decreasing resistance will be more than offset by losses due to the increasing current. If the inductor is made small enough, a point will be reached where the losses due to the increased peak current will more than offset the decreased losses from lower resistance. Somewhere before that point, there will be an optimum value, but without knowing the relationship between inductance and the inductor's losses, we can't pinpoint where this occurs. There are a number of formulae, both analytical and empirical which give the losses for various types of coils, and coil manufacturers also provide these data for their products. So, these could be used along with the relationships discussed above, to find the optimum component values.
One final criterion that we will look at (very briefly), is frequency stability. When the tank circuit is used as the frequency determining factor in an oscillator, frequency stability is the determining factor. The theory is a bit involved. One of the best discussions is that given by Jiri Vackar in his 1949 paper LC Oscillators and their Frequency Stability. Summarizing his conclusions: the impedance of the tuned circuit as seen at the input to the amplifier should be as low as possible, and the voltage gain of both the amplifier and feedback network should be as close to one as possible. This is often misinterpreted to mean that the tank Q should be low. However, this is not the case. The Q should be as high as possible as long as the low impedance criterion is met. However, in meeting the low impedance criterion, the inductance generally ends up being low while capacitance is high, and this may result in the overall Q being less than what could be obtained with a High-L, Low-C tank.