# Theory of Operation

Characteristics of Capacitors and Inductors

Engineers love to build reality into simple models. Otherwise no practical design would be possible. The same is true for handling capacitors and inductors.

Using the model of lumped, lossless components a capacitor is just characterized by one single parameter which is the capacitance in F (inductance in H for inductors). Picture 1 left side shows the equivalent circuit. For many applications this level of detail is just deep enough.

But even in HAM Radio dealing with low radio frequencies that model is not sufficient. Loss has to be introduced which is done by a parallel or serial resistor to convert some energy into head (equivalent series resistance - ESR). The middle part of picture 1 shows the corresponding equivalent circuits with the serial resistor type.

The quality Q (magnitude) of the components is introduced as the simple ratio between reactance and resistance at a given frequency:

Even if Equation (1) and (2) show that Q is frequency dependent that equivalent circuit still assumes that L, C and the ESR is not frequency dependent. This is a very important assumption as we will see later that all the calculation of L and C in the original AADE design as well as in the LCQ-Meter design is based on that assumption.

Picture 1: Different equivalent circuits of capacitors and inductors

Before we proof if this assumption is correct, let’s have a quick look at the equivalent circuit on the right side of picture 1. It introduces new components for the capacitor to cover parasitic effects like dielectric loss, ohmic resistance and inductance for the connection leads, and magnetic kernel loss and ohmic resistance and capacitance for the connection leads on the inductor.

However, it still does not include frequency dependency of the connection lead resistance or the loss resistance of the magnetic kernel material. In HAM Radio this sometimes plays very much a role but will not be covered in this article (with one exception which is the parasitic capacitance of the LCQ-Meter’s oscillator).

Back to the equivalent circuit from the middle part of picture 1 which is used here. Picture 2, 3 and 4 show some example plots of selected inductors and capacitors done with a vector network analyzer. The plots do show the serial resistance / ESR (Real Z), the reactance (Imag Z), the inductance or capacitance as well as quality Q along a wide frequency range.

The inductance of the 3 uH inductor shown in picture 2 is relatively stable below 15 MHz. From equation 1 we would expect a linear increase of quality Q with frequency. But this is not the case due to parasitic effects. So we cannot simply use a measured Q on one frequency to predict a Q on another frequency. Also Q is not at all located on a smooth curve. Take marker 7 at 8 MHz as an example where Q is measured at 287. Just moving a little bit backwards in frequency will lead to a different Q at 250. With high Q inductors the serial resistance / ESR is quite low and only a few milliohm change will impact Q significantly.

Picture 3 shows the same plot for a 100 uH choke. Above 2 MHz the inductance increases to infinity at around 7 MHz and comes back as a negative inductance which is nothing else as the characteristic of a capacitor. What happens is that parasitic capacitance becomes dominant above 7 MHz and there is a resonance of the inductance with the parasitic capacitance around 7 MHz. Even below 1 MHz the measured inductance is not constant, but the deviation is below 2 %, compared to a 100 KHz measurement. We would have seen the same scenario for the inductor from picture 2 if we go much higher in frequency.

Picture 4 shows a similar plot of a 1000 pF Mica capacitor with leads. The capacitance is relatively stable along the frequency but increases towards 16 MHz by 8 % compared to a measurement at 100 KHz and increases even more going higher in frequency. We can assume a resonance with parasitic inductance if the frequency would be much higher. Another important fact is that the measured Q of the capacitor is at least 1 to 2 decades higher than Q of the inductors. The serial resistance (ESR) even goes below 1 milliohm and care has to be taken if the measurement method is still able to correctly reflect such low resistance.

However, using a high quality capacitor in a tuned circuit will reduce quality Q discussions mainly to the quality of the involved inductor. But do not assume the same high Q for cheap ceramic capacitors!

Picture 2: Inductor 3 uH on T50-2 Amidon toroid core

Picture 3: Inductor 100 uH choke trough the whole model

Picture 4: Capacitor 1000 pF Mica trough the whole model

From the measurements shown some important conclusions can be drawn.

Using a measuring technique for capacitance and inductance which assumes stability of capacitance and inductance along the frequency is limited in accuracy. It makes no sense to expect 0.1 % accuracy for such designs.

Using a too high measurement frequency for isolated component measurement brings in more trouble than it helps. A series of measurements shows that 0.5 MHz to 1 MHz is a good choice.

Many people state that the original AADE design measures below 1 MHz because of this old LM 311 comparator used. But maybe the frequency was cleverly chosen to avoid a lot of trouble with parasitic effects at higher frequencies?

In HAM Radio in the HF frequency range isolated measurement of inductors and capacitors to build tuned circuits doesn’t make always sense. The parasitic components have a too high influence. Therefore the tuned circuit has to be measured as one circuit as done by the LCQ-Meter.

Quality Q does not behave as predicted using simple models as shown in the middle of picture 1 and by equations (1) and (2). Instead measured Q is quality on a very specific frequency. As in most HAM Radio designs the quality question is about quality of a tuned parallel circuit at resonance frequency. Best would be to measure Q in that target environment.

Using very low loss capacitors in a tuned circuit will also allow showing isolated inductor quality Q with an acceptable accuracy.

Measurement of Capacitance and Inductance

There are very different ways to measure capacity and inductance. Some examples are charge time measuring on capacitors, phase change and amplitude measurement between voltage and current but most of these methods are rather difficult to be implement within a low cost design.

An easy methodology is to measure frequency deviation introducing the device under test (DUT) into a tuned circuit. Picture 5 shows a resonant circuit with all unknown component values except a reference capacitor C_REF.

Picture 5: Basic method to measure C and L by frequency deviation

The first step is to get the values for C_FIX and L_FIX (left side of picture 5). C_REF not connected the resonance frequency will be:

With C_Ref now connected the resonance frequency will be:

Dividing equation (3) by (4) gives a term which is not dependent on L_FIX! Solving for C_FIX gives:

Now as we know C_FIX it is easy to calculate L_FIX from (3):

All components of the tuned circuit are known now and the mathematical accuracy is only dependent on the reference capacitor. But don’t spend a fortune on a 0.1 % reference capacitor! Please read the former paragraph with the VNWA plots again!

If an unknown inductance LX is added in series to L_FIX or an unknown capacitance CX is added in parallel to C_FIX they can be easily calculated by using the same formulas if the new resonance frequency fx is measured and C_REF is not connected:

If L_FIX is disconnected on one side and reconnected with shorted measurement leads for future LX measurement or if the measurement leads are attached to C_FIX left open for future CX measurement the calculation for L_FIX and C_FIX using equations (5) and (6) will include those parasitic capacitance or inductance by the measurement leads. Often this activity is called calibration.

This is basically the full mathematic used in all that LC Meters around using a microcontroller to measure the different frequencies.

The methodology is based on the easiest equivalent circuit from the left side of picture 5, even without using loss. One of the next paragraphs will show that this is a valid method as loss of accuracy by introducing change of resonant frequency due to loss is rather low compared to the other effects we have seen from the VNWA plots.

It is also noticeable that all concepts add an unknown inductance in series to L_FIX and an unknown capacitance in parallel to C_FIX. This insures that the new resonance frequency will always be lower than the one from calibration. To stay in the target frequency range all designs use around 50 uH to 100 uH for L_FIX, 500 pF to 1000 pF both for C_FIX and C_REF. Again in principle all values can be used but a clever choice helps to avoid any high frequency troubles discussed earlier.

After a lot of tests and measurements the LCQ-Meter stays with this design for isolated capacitor and inductor measurement as other choices and methodologies did not show any relevant advantage.

Measurement of Capacitance and Inductance in Tuned Circuits

The methodology to measure capacitance and inductance can be taken one step further to measure both values in parallel in a (resonant parallel) tuned circuit.

Picture 6: Method to measure C and L in parallel in tuned circuits

Let’s assume we have a parallel tuned circuit built with LX and CX as shown in picture 6. There is no fix capacitance except a parasitic C_PARA. C_REF is a known reference capacitor but of lower size as in the former paragraph (100 pF) to allow oscillation at much higher frequencies.

If C_PARA would be known we could calculate CX and LX directly with C_REF switched on and off.

C_REF switched off gives fx, C_REF switched on gives fy:

Knowing CX allows going for LX:

But how to find C_PARA which is dependent on the oscillator and PCB design and probably is also frequency depended?

The solution is to measure the resonance frequency at different LX without any CX. Just along left side of picture 5 substituting C_FIX by C_PARA and L_FIX by LX the parasitic capacitance C_PARA can be calculated at a given frequency. The result is a pair of C_PARA and fx values. Doing this with very different LX values to cover a frequency range from some 100 KHz to 30 MHz will lead to a collection of capacitance/frequency pairs.

In the LCQ-Meter design it showed up that C_PARA is non-linear along the frequency. Using some mathematics to find an interpolation function finally let to:

With the current PCB version 2.3 and oscillator design k1 equals 66.0 and k2 equals 3.2. Therefore the parasitic capacitance runs from 10 pF at higher frequencies to 25 pF at lower frequencies.

Decreasing this parasitic capacitance was the key motivation to go from PCB version 1 to version 2. In the former layout the parasitic capacitance was more than double as high. Careful PCB design pays off! Without any relays the parasitic capacitance was measured at 7 – 8 pF which is very close to the transistor datasheets and LT spice simulation.

The equation (11) allows finding the frequency dependent parasitic capacity for the given hardware design and therefore using equations (9) and (10) it allows calculating CX and LX from a resonant parallel tuned circuit.

This seems complicate, but it isn’t. The interpolation function has to be found just once for any given hardware design. Use a table calculation program of your choice, drop in the value pairs, create a chart, let the program calculate some interpolation functions and just look which one best fits the pairs!

Let me add some comments:

This methodology shows the net capacitance and inductance within a tuned circuit. Parasitic capacitance of the inductor will be added to the capacity and vice versa. But this is what is really needed in practice to tune a tuned circuit to a target frequency.

To arrive at the true resonance frequency for CX parallel to LX this frequency has to be calculated and cannot be measured directly due to C_PARA. The actual frequency is always a little bit lower. The LCQ-Meter software takes care about that.

Using this methodology the measured frequencies with and without C_REF can become much different and can increase up to 30 MHz and higher. This can violate the assumption of frequency independence of some component values. However, later on we will see that this methodology delivers appropriate results.

**Measurement of Quality of Inductors and Tuned Circuits**

Equations (1) and (2) describe that quality Q goes back to identifying the loss resistance ESR at a given frequency.

For a parallel tuned circuit there is no difference except that serial ESR from the capacitor and inductor will add up to a new ESR. As the capacitor ESR is quiet low the resulting ESR is mainly made up by the inductor ESR. This Q is called unloaded Q as there is no relevant (loading) connection to the tuned circuit which will attenuate the circuit’s energy.

The net is we need to identify the ESR resistance at a given frequency to calculate Q.

There are many methods to do this job and most of them are covered in literature like [2]. To evaluate a selection towards implementation for a low cost and small device the following example will be used:

An inductor of 10 uH at a frequency of 5 MHz should be evaluated towards quality Q. Let’s assume this inductor has a quality Q of 150 at the given frequency. If this inductor is part of a parallel tuned circuit let’s also assume that the corresponding capacitor (100 pF) has a very high quality. So all measurement methods, standalone or within a tuned circuit, go back to measure the ESR of the inductor. Using equation (1) leads to an ESR resistance of 2.1 Ohm.

The expectation is now to measure a 10 % change of Q which means increasing Q to 165 and decreasing the ESR to 1.9 Ohm. Each of the following methods should be evaluated towards this change, if it is practical to measure any change of voltage, phase, frequency, time or whatever is used in a method to show this change of Q with a low cost and small device design. However, not all methods will be tested as they do not fit to low cost and small device requirements.

**Methodologies Which Require Tuning of Components**

In [2] and many sources in literature and Internet there are methods described which require tuning of components like capacitors or resistors to either move to resonance (Resonance Method) or to compare the unknown inductor to reference components (Bridge Method). These methods will not be used as we don’t want to tune manually or electromechanically any component.

**Methodologies Which Require Tuning of Frequency or a Separate Oscillator**

On the Resonant Method it is possible to stay with fixed components and tune the frequency instead to arrive at resonance.

Frequency tuning is basically also needed with more sophisticated methods like IV Method, RF-IV Method, Network Analysis Method and Auto Balanced Bridge Method which are all described in detail in [2]. They all take a separate (tunable) oscillator to inject some current trough the DUT to measure voltage and/or currents.

A very practical method is described in [4]. The parallel tuned circuit trap method described was used later on to confirm quality Q values for comparison of accuracy.

However, they all do not fit to the low cost approach. The frequency generator (DDS / SI507 / PLL) and other needed parts showed up to be too expensive.

**Measuring Q With Resonance Frequency Deviation**

Every (well nearly every) HAM Radio operator should know the equation to calculate the resonance frequency of a tuned circuit as shown in (12). But this is just valid for the assumption of lossless components.

If we introduce loss by ESR resistance (R_S) the equation shows a slightly lower resonance frequency.

Equation (13) would allow calculating R_S and therefore Q with (1) if all other component values and also the new, lower resonance frequency can be measured.

Picture 7: Measuring Q by frequency deviation

Taking the circuit from picture 7 with 2 well known reference capacitors it is indeed possible to calculate R_S based on 3 frequencies arriving from switched on and off reference capacitors.

The mathematics behind it to find C, L and also R_S is not shown here (lengthy) but it is possible to quickly evaluate the frequency change along our example using (13).

Without loss f_r is 5032921 Hz from equation (12) for the 10 uH and 100 pF components. Loss of 2.1 Ohm ESR included equation (13) leads to an f_r of 5032810 Hz. The difference is just 111 Hz. Now we know why equation (12) does work for most cases. If Q is increased by 10 % which means reducing R_S to 1.9 Ohm the new resonance frequency from (13) is calculated to f_r of 5032830 Hz. This is just 20 Hz difference for a 10 % Q change!

This method can be implemented fast and a microcontroller can measure such small frequency differences with high efficiency. This is especially true if measurement is extended to several seconds. But the issue is stability of the oscillator. In a laboratory environment the method worked well, but just having a small breeze of air along the oscillator components changes frequency quiet heavily.

A similar approach can be taken by switching on and off known reference resistors in series to R_S. This will lead to a mathematical equation which has to be solved numerically, but the issue with the small change of frequency will stay and therefore both methods are not used here. I haven’t found that type of measurement for Q in literature anywhere but in principle it works.

Maybe a clever HAM Radio OM finds a way to build a very stable, maybe crystal based LC oscillator which allows measuring those small frequency differences on a constant base?

**Measuring Q by Switching Tuned Circuit Design**

Equation (13) is just valid for parallel tuned circuits. For serial tuned circuits there is no change in frequency introducing loss and therefore (12) is still valid.

This fact can be used to calculate R_S by putting L and C components firstly into a parallel tuned circuit and secondly putting the same components into a serial tuned configuration.

Without showing the mathematics finally it goes back to the same small changes in frequency as shown in the previous paragraph. Also there is no easy design for an oscillator which works with a parallel and serial tuned circuit without introducing changes by different parasitic components. Let’s look for another method which finally made the race.

**Measuring Q by Analyzing the Amplitude Envelope of a Switched Off Oscillator**

All mentioned methods so far work in the steady state of an oscillation. But there is nothing which keeps us away from moving to the time domain and measure transient behavior of a switched off oscillator with a parallel tuned circuit as the frequency relevant component.

Picture 8: Amplitude against time on a tuned circuit

If a parallel tuned circuit consisting of a capacitor and inductor is part of an oscillator design there will be a steady oscillation at the resonance frequency with specific amplitude. Picture 8 shows an oscilloscope plot from the current LCQ-Meter. The blue line is a sinus curve measured across the tuned circuit. It stays at a constant amplitude until time zero on the x –axis.

At that time zero the oscillator, the energy injecting part, is turned off in a way that the tuned circuit is not relevantly attenuated. From zero onwards the diagram shows that the amplitude of the oscillation gets lower and lower as there is loss in the tuned circuit. The time that it takes to go from one level of amplitude to a lower level is directly related to the loss of the tuned circuit.

Explained in a different way if we can measure the amplitude at 2 different points in time and know the resonance frequency we can calculate quality Q.

The voltage across the tuned circuit is given by:

The formula contains the swinging part at the resonance frequency f_r with the cosine term, the maximum amplitude as U_0 and the envelope describing part with the e-function. All with the assumption that loss is concentrated in the inductor ESR resistance R_S. Obviously the envelope level at a given time is depending on R_S and therefore on quality Q. This makes practical sense as we would expect the swing to die faster if there is more loss.

Let’s assume we can measure the envelope with some kind of peak detector at any time. Therefore we can skip the cosine term. The most easy approach to arrive at wanted R_S is to measure the envelope at time zero, which is nothing else than the amplitude U_0 and measure the envelope at a second time t1.

Dividing (16) by (15) and solving for R_S leads to equation (17).

The last step is to use equation (1) with (17) and solve for Q.

The amazing thing is that Q does not depend on any values of the components. These values come in by the resonance frequency which can very easily be measured by a microcontroller. The challenge is to measure the envelope voltage at a given time. Again, a microcontroller is ideal of doing dedicated actions at dedicated times. So the challenge is left about the envelope detector. But there will be a solution. Picture 8 does include a real measurement of the envelope by a peak detector (red line).

A very important fact in regards of quality Q measurement is that this method gets more accurate as Q gets higher, because the decay time will increase with higher Q. This is a big advantage compared to other methods which will get less accurate with higher Q!

Another question is what resolution for voltage measurement is needed to find out small changes in Q like the 10 % increase using the example from before.

Taking our tuned circuit made of the 100 pF, 10 uH and 2.1 Ohm components with an amplitude of 1 Volt at time zero equation (16) predicts an envelope voltage of 0.349 Volt after 10 microseconds. If Q is increased by 10 % which relates to a new ESR resistance of 1.9 Ohm the voltage will be 0.387 Volt. This is 11 % change of voltage within a resolution which can easily be measured by a 10 Bit ADC included in a microcontroller.

The net is even if this Q measurement method looks a little bit strange at the beginning it is the best fit towards acceptable accuracy in a low cost small device approach.