Chris Angove, Independent Professional Engineer

Chris Angove is a highly experienced and qualified chartered electronics engineer specialising in electrical and electronics engineering. He manages and owns Faraday Consultancy Limited (FCL).

Q-Factor, Self Resonant Frequency and Loss Tangent

In this article we continue to study the properties of lumped, passive, reactive components in circuit designs at RF, microwave and high speed frequencies.

By lumped components, we mean those which are operating at frequencies in which the shortest wavelength in the electrical transmission medium being used is much greater than the component dimensions. As with much of RF and microwave engineering, there are no hard and fast rules but this criterion is usually understood to be greater than about ten times. This simply means that the shortest wavelength must always be at least ten times the maximum dimension of the component shell under consideration, with no further modification applied. If, for example, we took a surface mount inductor formed by a coil of copper wire, this means that we consider the longest dimension of the packaged, coiled component and not the 'unwound' length of wire. Components such as these can be analysed in more minute detail, but that is probably a task for the component manufacturer and not the circuit designer.

For example, the Coilcraft part 2222SQ-90N is a 90 nH air core surface mount inductor with dimensions 5.21 mm × 5.46 mm × 5.69 mm. So the maximum dimension of the component shell is 5.69 mm. Suppose that this was used on a 50 Ω FR4 microstrip circuit with an effective relative permittivity (or effective dielectric constant), keff of 3.5. Some graphs from the Coilcraft datasheet in Figure 1-4 for this component show that its measured Q-factor is reasonably predictable up to about 200 MHz before the curve gets distorted, due probably to parasitic resonance effects. The shortest wavelength, at 200 MHz, is 1.5 m in air or 1/√keff times this value in the microstrip medium, which is 0.803 m. So in this case the longest dimension of the component is actually about 1/141 of a wavelength. Taking this argument a stage further, the datasheet shows that, for the 90 nH component, the Q-factor gets quite unpredictable at the end of its plotted range at 400 MHz. This corresponds to a wavelength in the microstrip of 400 mm, about 70 times the maximum dimension. So even at the maximum test frequencies used by Coilcraft, we are well within what we consider as the 'lumped element' frequency range.

Alternatively, circuit components might be distributed, but we are not considering those here. In distributed components, we deliberately exploit the electrical dimensions as being significant fractions of a wavelength, or indeed several wavelengths. The rules of transmission lines must be applied to accurately predict their behaviour.

Throughout this paper the units adopted are Systeme International (SI) Units. These are summarised in Table 0-1. Some parameter symbols are subscripted to avoid ambiguity.

Table 0-1 Systeme International (SI) units used in this document: parameters, units and symbols
Parameter SI Unit
Name Symbol Used Symbol
Capacitance C farad F
Inductance L henry H
Voltage V volt V
Current I ampere A
Frequency f hertz Hz
Angular Frequency ω radian/second rad/s
Q-factor Q None N/A
Power P watt W
Energy W joule J
Impedance Z ohm Ω
Resistance R ohm Ω
Reactance X ohm Ω
Admittance Y siemen S
Conductance G siemen S
Susceptance B siemen S

Contents

What are Q factor, SRF and loss tangent?

Q-Factor in General
Q Factor for Lumped Components at Low Frequencies
Unloaded Resonant Circuits
Complex Power and Energy
Definition of Q-Factor for a Resonant Circuit
Time Domain Waveforms of the Series Resonant Circuit
Loaded Resonant Circuits
Loss Tangent

What are Q-Factor, SRF and Loss Tangent?

Q-factor (colloquially known as 'quality factor'), self resonant frequency (SRF) and loss tangent (alternatively tan delta, tanδ or dissipation factor) are all parameters which describe how well various passive components and resonant circuits perform as functions of frequency. In RF, microwave and high speed design we need to consider frequency spectra extending perhaps into many gigahertz. We have observed that, as frequencies extend upwards in this way, the behaviour of passive (and active) components depart significantly from what we would have expected, assuming only basic alternating current (AC) theory.

In general, there are many manufacturers of capacitors and inductors but there are far fewer who promote dedicated ranges of high-Q versions of these for targeting RF, microwave and high speed frequencies. High-Q performance represents added value so we can expect to pay more for these because the manufacturer has to direct more resources to design, testing and results analyses.

The following list includes manufacturers who list ranges of high-Q capacitors and inductors, mostly in surface mount format.

Q-Factor in General

Contrary to some people's understanding, including my own, the concept of Q factor was never suggested to represent 'quality' factor, despite the fact that this is precisely what it does represent. It just happened to be a letter left over after the others had been used as symbols for other parameters [8].

There are essentially two definitions of Q-factor that are used in datasheets (2) (6):

In the first case, Q-factor is a measure of how closely the electrical performance of the reactive component compares to an ideal version of the same component over a range of frequencies. Usually, the most predictable behaviour will be in regions of the Q-factor function clear of any apparent resonances, or self resonant frequencies (SRFs). Any SRFs which may exist are in general unwanted and they should be avoided (6)

In the design of resonant or tuned circuits, Q-factor is a measure of the 'sharpness' of the resonance in terms of voltage or current amplitudes within the circuit across a range of frequencies. In this case, there is one Q-factor value for each circuit and loading condition and, again, a higher Q-factor represents a closer to ideal performance. In these cases Q-factors may be unloaded or loaded. Unloaded Q-factors are a useful theoretical concept and consider the resonant circuit completely in isolation, so the spread of Q-factor values against frequency will be fixed. In the real world however, resonant circuits require to be coupled to other circuits and therefore become 'loaded'. Loaded Q-factors are also expressed as functions of frequency but take into account the explicit loading of the resonant circuit. Unloaded Q-factors are always greater than loaded Q-factors.

Later, when we examine the circuit currents and voltages over higher and higher frequencies, we will see some anomalous behaviours which are caused by phenomena known as skin effect and proximity effect (1). These affect both the resistance and inductance parameters and are usually functions of frequency. Others are resonances at self resonant frequencies or SRFs. These are caused by the resonant effects of inherent stray capacitances and inductances both with other strays and with the intended bulk capacitance or inductance. In general, the challenges of dealing with strays are greater with inductors than with capacitors. Even for well designed, high frequency components there will typically be a few series or parallel SRFs present and identified in the associated datasheets. In general, for RF designs these should be avoided by choosing components with no SRFs near the intended operating frequency range.

Sometimes, SRFs may actually be useful if for example they help to reject an unwanted spurious frequency. However, for these type of problems, a better, targeted solution should be found because the positions and behaviours of SRFs were never intended and may not be consistent.

Q Factor for Lumped Components at Low Frequencies

Firstly, we will look at the Q-factors of real capacitors and inductors at relatively low frequencies: those are frequencies at which they may be accurately considered as lumped components. For each case, the equivalent circuit of the real component may be considered either as a series or parallel combination of an ideal capacitor (or inductor) and an ideal resistor. The circuit schematics, phasor diagrams and derived Q-factor formulas for these are shown in Figure 1-1 and Figure 1-2 for capacitors and inductors respectively.

Figure 1-1 The circuit schematic representations, phasor currents, voltages and Q-factor equations for a real capacitor operating at relatively low frequencies clear of any SRFs.
Figure 1-2 The circuit schematic representations, phasor currents, voltages and Q-factor equations for a real inductor operating at relatively low frequencies clear of any SRFs.

In each of the equivalent circuits, the Q-factor symbol 'Q' is subscripted 'C' for the capacitor and 'L' for the inductor. Neither Q-factor changes as a function of the circuit type, series or parallel. Lower case omega (ω) represents the angular frequency in radians per second (rad/s), so ω =2πf, where f is the frequency in hertz (Hz). At one frequency only, the Q-factor may be equivalently described either by the series or parallel version equivalent circuit (10).

Refer initially to the equivalent circuits for the real capacitor in Figure 1-1. An ideal capacitor dielectric is theoretically a perfect insulator. The relative permittivity (or dielectric constant) of the dielectric material also contributes the capacitance value itself: the greater the relative permittivity, the greater is the capacitance that can be achieved in a fixed space. There is usually pressure to reduce the size of components but for capacitors unfortunately, using a higher relative permittivity to achieve this often simultaneously degrades the Q-factor. This creates a less than perfect insulator, one with a finite resistance. In Figure 1-1 the leakage resistance is represented equivalently, either by a high value in parallel or a low value in series with the ideal capacitor, RCP and RCS respectively. Derived from the magnitudes of the voltage and current phasors shown in Figure 1-1 and Figure 1-2, the Q-factor expressions for the series and parallel CR equivalent circuits are given by (1-1) and (1-2) respectively.

Q C = 1 ωC R CS
(1-1) The Q-factor for a real capacitor: CR series equivalent circuit at low frequencies.
Q C =ωC R CP
(1-2) The Q-factor for a real capacitor: CR parallel equivalent circuit at low frequencies.

In Figure 1-2, the equivalent circuits for real inductors are also represented by resistance values in series (RLS) or in parallel (RLP) with the ideal inductance element. A significant contribution to RLS comes from the resistance of the wire used to construct the inductor coil. The Q-factor expressions for the series and parallel LR equivalent circuits are given by (6-3) and (6-4) respectively.

Q L = ωL R LS
(1-3) The Q-factor for a real inductor: LR series equivalent circuit at low frequencies.
Q L = R LP ωL
(1-4) The Q-factor for a real inductor: LR parallel equivalent circuit at low frequencies.

As we noted at the beginning of this article, several manufacturers promote ranges of high-Q lumped inductors and capacitors which are specifically designed for use at high frequencies. For example, data sheets by Coilcraft for high frequency inductors typically include graphs of Q-factor and deviation from the nominal values, both against frequency. One example shown in Figure 1-3 is extracted from their datasheet for the product parts 1515SQ, 2222SQ and 2929SQ, covering nominal values from 47 nH to 500 nH. Reading the datasheet in more detail confirms that these are indeed measured results and Coilcraft also provide supporting information including how they were measured with details of test equipment, test jigs and calibration methods. Reliable and measured data like this are essential for rapid and effective RF and microwave circuit design.

Figure 1-3 Coilcraft actual Q and L against frequency for the range of SQ inductors

In Table 1-1 for the Coilcraft 1515SQ 90 nH inductor, at several frequencies I have calculated the values of equivalent circuit series resistance (RLS) and parallel resistance (RLP) with values of Q and frequency taken from the datasheet, using (1-3) and (1-4) respectively.

Table 1-1 Coilcraft inductor 1515SQ, 90 nH, calculated RLS, RLP from measured Q-factors
Frequency (MHz) Q-factor RLS (Ω) RLP (kΩ)
1.0 40 0.014 0.023
5.0 65 0.043 0.184
10.0 80 0.071 0.452
20.0 115 0.098 1.301
50.0 170 0.166 4.807
100.0 215 0.263 12.158

The datasheet reports that the DC resistance of the 90 nH inductor (equivalent to RLS at zero frequency) is 0.0055Ω. As the frequency is increased however, it is clear that the actual value of RLS also increases in some, as yet unknown, function of frequency. So even in well designed, high-Q, real inductors we cannot make any assumptions about constant coil resistance if we need to accurately measure their RF characteristics. This increase in resistance with frequency is caused by the skin effect acting on the copper wire used to construct the inductor coil (1) (4). The skin effect forces the AC current to pass through a thin layer on the surface of the wire which forms the inductor and this reduces with increasing frequency. There may also have been a contribution from a phenomenon known as proximity effect. Proximity effect is caused when adjacent turns of an inductor coil are so close that the current distribution in one turn influences that in the next. This also increases with frequency but unfortunately is not as predictable as skin effect. Increasing the turn spacing does tend to reduce it but this practice also affects other parameters and may increase the dimensions. Even with these limitations, we can see from Figure 1-5 and (6-3) that there is a net and predictable increase in Q-factor for the 90 nH inductor from 1 MHz up to about 200 MHz.

Returning to Figure 1-3, the left plots of Q-factor against frequency all come to peaks towards the high frequency ends. In the plots of the inductance values against frequency on the right, it is clear that the actual inductance values are close to their nominal values for most of the frequency range and then start increasing, broadly at similar frequencies. The most likely explanations for these are self resonant frequencies (SRFs).

Some examples of high frequency data for high Q capacitors from the Murata GJM range are shown in Figure 1-4. Although this example is for a nominal value of 4.7 pF, it is representative of the whole GJM range.

Figure 1-4 Murata data for capacitors from the GJM series showing ESR and Q factor over frequency

The plots on the left are of equivalent series resistance (ESR) for the capacitor against frequency. This parameter is often used for capacitor specifications and is equivalent to RCS which was used in Figure 1-1 and (1-1).

In general, for high-Q lumped components, assuming similar frequencies and reactance magnitudes, capacitors tend to have better Q-factor performance than inductors and they are usually cheaper. Therefore, if a circuit design allows the option of using one or more capacitors instead of an inductor, it should be seriously considered. For example, some forms of active filters achieve inductor properties with capacitors and resistors.

The ESR and Q-factor plots in Figure 1-4 may be related to each other using the equation for QC (1-1). For example, with the 'new material' GJM part, measured Q-factors were 500 and 230 at 500 MHz and 1000 MHz respectively. Using (1-1), the respective calculated RCS are 0.14 Ω and 0.15 Ω.

Unloaded Resonant Circuits

Q-factor may be used to describe the sharpness of resonance of real resonant circuits or tuned circuits (2). There are two basic types of resonant inductor-capacitor (LC) circuits: series and parallel. Schematics of the ideal versions of these are shown in Figure 1-5 (a) and Figure 1-6 (a) respectively.

Figure 1-5 Series resonant circuit: (a) ideal (LC), (b) real with individual loss resistances (RLS and RLC) and (c) real with one equivalent loss resistance (RLCS)
Figure 1-6 Parallel resonant circuit: (a) ideal (LC), (b) real with individual loss resistances (RLP and RCP) and (c) real with one equivalent loss resistance (RLCP)

Figure 1-5 (a) will be recognised as an ideal series resonant circuit comprising an inductor (L) and a capacitor (C) connected in series and shown in isolation. The resonant (angular) frequency (ω0) is given by (1-5):

ω 0 = 1 LC
(1-5) The resonant angular frequency of a series or parallel LC circuit

Considering the series and parallel resonant circuits, Table 1-2 provides the standard formulas for impedance (Z) and admittance (Y) of the circuit elements of inductance, capacitance and resistance.

Table 1-2 A summary of formulas for impedances and admittances for the basic passive circuit elements: resistance, inductance and capacitance.
L (henry, H) C (farad, F) R (ohm, Ω)
Z(Ω) jωL j ωC R
Y(S) j ωL jωC 1 R

In Figure 1-5 (a) and Figure 1-6 (a), no resistance or Q-factor information for L nor C is shown, so the implication is: (a) both Q-factors are infinite and (b) the equivalent series resistances for L and C are both zero. Unfortunately this would not be any use other than as a theoretic concept because neither circuit is connected to anything. Furthermore, in theory, neither circuit would have a resonant frequency or, more precisely, both circuits would have a resonant frequency but it would occupy zero bandwidth.

A more practical resonant circuit, one of which is shown in Figure 1-5 (b) was derived from Figure 1-5(a) but with series resistances RLS and RCS added to represent the additional losses present in the inductor and capacitor respectively. It would have been just as correct to represent the additional losses by parallel loss resistances, RLP and RCP, as shown in Figure 1-6(b) but the series versions are easier to consider for a series resonant circuit. One further simplification is shown in Figure 1-5 (c), where RLCS simply represents the combined loss resistances of RLS and RCS in series so they are added (1-6).

R LCS = R LS + R CS
(1-6) The total loss resistance for the series resonant circuit shown in Figure 1-5 (b) is the sum of the individual loss resistances.

The complex input impedance (ZIN) of the circuit shown in Figure 1-5 (c) is given by (1-7):

Z IN = R LCS +jωL j ωC
(1-7) The impedance of the real series resonant circuit shown in Figure 1-5 (c)

To demonstrate a practical example of a real series resonant circuit, consider the Coilcraft 90 nH inductor that was examined in Section 1.1.1. To choose an arbitrary resonant frequency of 100 MHz would, from (1-5) require a capacitor value of approximately 28 pF: for example, we will choose one from the Murata CJM range. The impedance magnitude for this circuit (1-7) was plotted using Matlab 2021a® across a frequency range 90  to 110 MHz. The result is shown in Figure 1-7, which is normalised to 100 MHz.

Figure 1-7 Impedance magnitude against frequency (normalised to 100 MHz) for a series, real LR circuit covering 90 MHz to 110 MHz.

We understand that the degree to which each reactive component departs from ideal behaviour is expressed by its Q-factor, like shown with the examples in Figure 1-3 and Figure 1-4. In this case we have assumed this to be relatively constant across this small frequency range with the values of QL=220 for the inductor and QC=400 for the capacitor, both measured at 100 MHz.

The impedance magnitude plot in Figure 1-7 shows the classic 'valley' shape predicted in AC theory for a series resonant circuit with the resonant frequency at 100 MHz. At resonance, the impedance magnitude is at a minimum at approximately 0.4 Ω which corresponds to the impedance having no complex component and is equivalent to RLCS from (1-6). This may be confirmed by calculating RLS (0.26 Ω) and RCS (0.14 Ω) using (1-3) and (1-1) respectively and then using these results in (1-6) to give the combined result.

For the parallel form of resonant circuit, refer to the circuit schematics shown in Figure 1-6. In this case the reactive elements and loss resistances are all connected in parallel. In parallel circuits, it is often more convenient to consider them in terms of admittances and conductances instead of impedances and resistances. Then the standard reciprocal relationships between them may be applied as required.

For the parallel resonant circuit, RLP and RCP may be calculated using (1-4) and (1-2) respectively, yielding the results RLP = 12.441 kΩ and RCP = 22.736 kΩ. The equivalent resistance RCLP for both may be obtained from the parallel resistance formula (1-8) which comes to 8.041 kΩ.

1 R CLP = 1 R LP + 1 R CP
(1-8) The parallel resistance formula used to obtain RCLP

With the help of the information shown in Table 1-2, the input admittance (YIN) of the parallel tuned circuit shown in Figure 1-6 (c) is given by (1-9) :

Y IN = 1 R LCS +j ωC 1 ωL
(1-9) The parallel resistance formula used to obtain RCLP

For consistency with the series resonant circuit, the input admittance may be converted to input impedance using the reciprocal relationship (1-10):

Z IN = 1 Y IN
(1-10) The reciprocal relationship for input admittance (YIN) and input impedance (ZIN)

After using (1-9) and (1-10) as described, Figure 1-8 is a plot of the magnitude of the input impedance for the parallel resonant circuit against frequency, created again using Matlab 2021a®.

Figure 1-8 Impedance magnitude against frequency for a parallel, real LR circuit

As expected, the real resistance at resonance is about 8.041 kΩ calculated using (1-8)

Complex Power and Energy

We will see shortly that the Q-factor definition for a resonant circuit is related to energy so, to examine this in more detail, consider a real series LC resonant circuit, similar to Figure 1-5(c), which is connected directly to an AC Thevenin constant voltage source as shown in Figure 1-9

.
Figure 1-9 An LC series resonant circuit for a real inductor and capacitor represented by ideal components, L and C in series with a loss resistance (RLCS), derived from (1-6).

In this case a resistor (RLCS) has been added in series to represent the loss resistance of the inductor and capacitor combined. These were calculated from (1-1), (1-3) and (1-6) (2). The input impedance is the complex quantity ZIN, the applied voltage and current phasors are V and I respectively and the peak values of V and I are V0 and I0 respectively.

In circuits like these there is no explicit source or load resistance connected as we might have expected if we were examining, for example, a 2 port network. The actual resistance shown (RLCS) represents the equivalent series loss resistances for the inductor and capacitor combined on account of them being real components and having finite Q-factors. Ideal reactive components do not dissipate any power but they can store energy, either in the magnetic field for an inductor or in the electric field for a capacitor. A resistor can of course dissipate power but cannot store energy.

Expressed as long term average values, the energy stored in a capacitor (WE) and in an inductor (WM) are given by (1-11) and (1-12) respectively.

W E = 1 2 C V 2 = 1 4 C V 0 2
(1-11) The energy stored in the electric field of a capacitor.
W M = 1 2 L I 2 = 1 4 L I 0 2
(1-12) The energy stored in the magnetic field of an inductor.

In this context, a sinusoidal voltage is applied to the series resonant circuit so the voltage across the capacitor (V) and the current through the inductor (I) will both also be sinusoidal. The resulting energy against time functions, we can see from (1-11) and (1-12) will be sine or cosine squared, as the energies must be positive. The squared functions also double the frequency compared to the fundamental applied by the voltage source. This implies that, the energy content transfers on alternate half cycles between the capacitor and the inductor.

The complex input power delivered to the resonator (Pin) is (1-13):

P in = 1 2 I I * Z in = 1 2 I 0 2 Z in = 1 2 I 0 2 R LCS +j ωL 1 ωC
(1-13) The complex power applied to the LR resonant circuit with real components.

In (1-13), I is the complex conjugate of I and I0 is the peak value of the sinusoidal current waveform.

From (1-13), the complex power comprises a real part which we will call Ploss (1-14) and an imaginary part with two components, one from the inductor and one from the capacitor.

P loss = 1 2 I 0 2 R LCS
(1-14) The real part of the complex power calculated for the series resonant circuit.

We understand that perfect or ideal reactive components do not dissipate any power which is consistent with L and C only being present in the imaginary coefficients. As we are in fact representing real components, defined with finite Q-factors, the power loss in each case will actually be included in Ploss.

PIN can also be expressed in terms of Ploss and the inductor and capacitor energies, WM and WE respectively. This is shown in (1-15), which comes from (1-13) using expressions for WE and WM from (1-11) and (1-12) respectively.

Pin= P loss +j2ω W M W E
(1-15) The complex power into the series resonant circuit, including contributions from the inductor and capacitor energies, WE and WM respectively.

From (1-15) it is clear that the resonant condition is when the capacitive and inductive energies are identical, so WE = WM. By substituting the associated energy equations in (1-11) and (1-12) respectively, we may arrive at the resonance relationship (1-5), which is repeated here (1-16).

ω 0 = 1 LC
(1-16) The resonant angular frequency of a series LC circuit

This also uses the well known AC voltage and current magnitude relationships for reactive components: for an inductor (1-17) and for a capacitor (1-18). As these are assumed to be ideal components, the voltage across an inductor (VL) leads the current through it by π/2 radians and the voltage across a capacitor (VC) lags the current through it by π/2 radians.

V L =IωL
(1-17) The voltage current relationship for an ideal inductor in which, by definition,, VL leads I by π/2.
V C = I ωC
(1-18) The voltage current relationship for an ideal capacitor in which, by definition,, VC lags I by π/2.

Definition of Q-Factor for a Resonant Circuit

Q-factor is a measure of the loss of a resonant circuit and is defined in (1-19). The greater the Q-factor the smaller the loss.

Q=ω Average energy stored Energy loss per second =ω W M + W E P loss
(1-19) The definition of Q-factor for a resonant circuit.

Let us assume, at the resonance frequency, that ω =ω0. At resonance we know that WE = WM. Therefore, the expressions for Q-factor, in terms of each energy component are (1-20) and (1-21).

Q= 2 ω 0 W M P loss
(1-20) Q-factor at resonance as a function of the inductor (magnetic field) energy.
Q= 2 ω 0 W E P loss
(1-21) Q-factor at resonance as a function of the capacitor (electric field) energy.

By substituting the expressions for WM from (1-12) into (1-20) and for WE from (1-11) into (1-21), the Q-factor equations shown in (1-3) and (1-1) may be confirmed.

To examine the behaviour of the Q-factor response curve near resonance, at an angular frequency of ω rad/s, one may assume (1-21).

ω= ω 0 +Δω
(1-21) The angular frequency ω, close to that for the resonant frequency, ω0.

Using (1-7) and substituting (1-21), ZIN may then be expanded to include ω and ω0 (1-22).

Z IN =R+jωL 1 1 ω 2 LC =R+jωL ω 2 ω 0 2 ω 2 =R+j2L ω 0 Δω ω R+j2QR Δω ω 0 =R+j2LΔω
(1-22) An expansion of ZIN, which includes the approximation result from (1-23)
ω 2 ω 0 2 = ω+ ω 0 ω ω 0 =( ω 0 +Δω+ ω 0 )( ω 0 +Δω ω 0 ) =(2 ω 0 +Δω)Δω =2 ω 0 Δω+ Δω 2 2 ω 0 Δω
(1-23) An expansion of ω2 - ω02 used in (1-22)

From (1-13), when the frequency is such that |ZIN|2 = 2R2, then, from (1-22), the real power is one half, or -3 dB, of that at resonance. If BW is the fractional bandwidth, then this is related to ω and ω0 by (1-24).

BW= 2Δω ω 0
(1-24) The fractional bandwidth (BW) related to the small element of angular frequency (Δω) and the resonant angular frequency ω0.

Using the results from (1-22), (1-23) and (1-24) gives the final results in (1-25) and (1-26)

R+jRQ(BW) 2 =2 R 2
(1-25)The half-power resonance condition for a real LC resonant circuit.
BW= 1 Q
(1-26) The unloaded resonant circuit Q factor definition in terms of the fractional bandwidth (BW)

Time Domain Waveforms of the Series Resonant Circuit

To illustrate some of the features of (unloaded) resonant circuits, some examples of AC voltages and currents around the schematic circuit of Figure 1-9 were simulated using Matlab 2020a®. These are shown in Figure 1-10 for what was found to be the resonant frequency of 100.3 MHz and in Figure 1-11 for the lower -3 dB band edge of 99.9 MHz. Similar results were observed for the upper (-3 dB) band edge of 100.6 MHz, but are not shown. The following is a summary of the parameters used.

(a)
(b)
(c)
(d)
(e)
(f)
Figure 1-10 Series real resonant circuit at resonance (100.3 MHz): current (a), resistor voltage (b), capacitor voltage (c), inductor voltage (d), resistor power (e) inductor and capacitor energies (f).

Referring to the waveforms at resonance in Figure 1-10, the peak voltage across R is 1 V, consistent with the phase voltages across L and C 'cancelling' (the same magnitudes at about 140 V but π/2 radians out of phase). The R voltage is in phase with the current. The C voltage lags the R voltage by π/2 radians (approximately equivalent to 2.5 ns) and the L voltage leads the R voltage by π/2 radians. The resistor power peaks to about 2.5 W, consistent with an applied peak voltage of 1 V and series resistance of 0.4 Ω at resonance. The L and C energies alternate every half cycle (5 ns), in each peaking to about 275 nJ.

(a)
(b)
(c)
(d)
(e)
(f)
Figure 1-11 Series real resonant circuit at the -3 dB (lower) band edge (99.9 MHz): current (a), resistor voltage (b), capacitor voltage (c), inductor voltage (d), resistor power (e), inductor and capacitor energies (f). Similar results were observed for the -3 dB (upper) band edge of 100.6 MHz):

Figure 1-11, are the same voltages as shown in Figure 1-10, but at the -3 dB point off resonance, confirmed by the resistor peak power at about 1.25 W compared to 2.50 W at resonance.

Loaded Resonant Circuits

A resonant circuit of the type shown in Figure 1-9 is considered to be unloaded, despite it including a resistor RLCS. That is because RLCS is an aggregate of the loss resistances of L and C which are an integral feature of the imperfect reactive components. If L and C were considered to be ideal, RLCS would be zero and the circuit would not only be completely impractical but virtually impossible to analyse.

Before a resonant circuit such as this is useful it must be connected to something or 'loaded', for example as part of a filter or an oscillator. When loaded, the Q-factor of the, now loaded, resonant circuit is degraded substantially from its unloaded value and it is said to become more heavily 'damped'. The resonant circuit, represented as a two port network, would normally be loaded in some way both at the input and the output. One example using a Thevenin equivalent circuit is shown in Figure 1-12. In this, the input and output ports of the resonant network are 1 and 2 respectively. The input is loaded with a source impedance ZS and the output is loaded with a load impedance ZL. For resonant circuits used in oscillator circuits, sometimes there may be no explicit input port loading, other than a resistor. The (thermal) additive white Gaussian noise (AWGN) generated by the resistor may actually used as the signal source. The low power filtered AWGN at the output of the resonant network would normally then be amplified externally.

Figure 1-12 A typical, conjugately matched, two port network.

In Figure 1-12, the input and output interfacing is achieved using conjugate matching, to ZS = RS + jXS at the input and to ZL = RL + jXL at the output. This requires the input and output network impedances, ZIN and ZOUT to be the conjugates of ZS and ZL respectively as shown.

This is where the RF circuit design challenges begin. For many common transmission line cascaded networks, the matched source and load resistances, RS and RL, are of the order of perhaps 50 Ω or 75 Ω. Values like these would significantly reduce the effective Q-factor of the circuit shown in Figure 1-9 if it was used as shown in Figure 1-12. However, some controlled reduction in Q-factor may be useful if, for example, some tuning capability was required as would be necessary with a voltage controlled oscillator (VCO).

Loss Tangent

Like Q-factor, loss tangent is also a measure of how much the behaviour of a real reactive component, a capacitor or an inductor, departs from that of a perfect component, across a range of frequencies. Loss tangent has the reciprocal behaviour to Q-factor: the smaller the loss tangent is, the closer the reactive component is to its ideal form. Similarly to Q-factor, as the frequency of operation is increased, the high frequency effects, such as changes in resistance due to the skin effect, become apparent (1) (4) (7).

Although loss tangent mathematically applies to both inductors and capacitors, it is more commonly used for capacitors, or more generally, the imperfect dielectric materials that may be modelled using parallel plate capacitors. That is because the most significant degradation in capacitor performance at higher frequencies is dominated by the dielectric properties. A common example used in printed circuit board (PCB) design is bulk, dielectric substrate material such as FR4. In its raw and unetched state, relatively thick layers of FR4 are sandwiched between relatively thin layers of copper. After processing, the copper layers will be etched as required to form various electrical conductor tracks and planes.

One way of measuring the high frequency dielectric properties of a piece of unetched FR4 material, double sided with copper planes, is to approximate it to a parallel plate capacitor. An example is shown in Figure 1-13. The copper planes act like plates or electrodes and may be connected to a transmission/reflection measuring instrument such as a vector network analyzer (VNA). Uncertainties caused by fringing effects of the electric field in the dielectric may be minimised by choosing a thickness (plate separation) that is small compared to the electrode dimensions. The VNA must be set up to measure across an appropriate frequency range and calibrated accurately.

Figure 1-13 The classic parallel plate capacitor with a dielectric comprising, not a perfect insulator, but a slightly lossy dielectric material

An ideal capacitor would have electrodes formed with perfect conductors and a dielectric material formed from a perfect insulator. However, in this real example, neither of course would be perfect. Furthermore, the electrical characteristics of both the electrodes and dielectric will vary with frequency, similarly to how we examined the Q-factor for a real capacitor in Section 1.1.1. Refer also to the parallel form of equivalent circuit in Figure 1-1 and the expression for Q-factor (QC) in (1-2).

One way to study the loss tangent for a capacitor is to take a quasi-static approach. This will typically be adequate up to a few hundred megahertz but will start to fail moving into microwave frequencies. In this, again we represent the real capacitor as an ideal capacitor connected in parallel with a resistor (RCP) which represents the 'leakage' resistance. The leakage resistance will be partly a function of frequency caused, not only by the skin effect in the electrodes, but also with contributions from the absorptive properties of the dielectric material. The rectangular geometry formed by the dielectric is consistent with that for the parallel plate capacitor and a block of bulk conductive material forming the leakage resistance.

The following equations represent the quasi-static relationships. (1-27) is the capacitance of the parallel plate capacitor, (1-28) is the resistance related to the conductivity and resistivity of the dielectric material.

C= ε 0 ε r A t
(1-27) The capacitance (C) for a parallel plate capacitor at low frequencies ignoring fringing effects.
R= t σA = ρt A
(1-28) The resistance (R) of a rectangular block of bulk conductive material.

Using (1-27), (1-28) and the fact that the loss tangent (tanδ) is the reciprocal of the associated Q factor, as shown in Figure 1-1, tanδ is given by (1-29).

tanδ= 1 ωCR = σ ω ε 0 ε r
(1-29) The equation for loss tangent, or tanδ, in terms of angular frequency and the other parameters.

The symbols used in equations (1-27), (1-28) and (1-29) are listed below:

A further examination of the phasor diagram included in Figure 1-1 shows that the loss tangent is defined as the ratio of the AC (current) magnitudes iRCP/iC where iRCP is the current through the leakage resistance and iC is the current through the ideal capacitor component. Therefore, the smaller the value of loss tangent, the closer to ideal will be the performance of the capacitor. The 'ideal' capacitor will have an 'ideal' dielectric comprising a perfect insulator. A perfect insulator will have zero conductivity (σ=0), but finite relative permittivity (εr ≠ 0).

Thinking more generally about imperfect insulators and referring again to (1-29), at a fixed frequency, the condition for a good insulator is σ ⟨⟨ ε0εr.

References

  1. Hall, Stephen H., Heck, Howard L.; skin depth
  2. Pozar, David M. (Q-factor and resonators)
  3. Pozar, David M. (op. cit.); (Conductivities and Dielectric Constants); p 687.
  4. Pozar, David M. (op. cit.); (skin depth); pp. 18 - 19.
  5. Ramo, Whinnery and Van Duzer
  6. Kraus, John D. and Carver, Keith R.
  7. Hall, Stephen H., Heck, Howard L. (op. cit.); pp. 256 - 257.
  8. Green, Estill I..
  9. US Patent 1,628,983: Electrical Network; K. S. Johnson; Electrical Networks;
  10. Williams, Arthur B., Taylor, Fred J, Third Edition;
  11. Kraus, John D. and Carver, Keith R. (op. cit.); Parallel Plate Capacitors; pp. 49 - 50
  12. Kraus, John D. and Carver, Keith R. (op. cit.); Conductivity Equation; pp. 114 - 117
  13. Pozar, David M. (op. cit.); Conjugate Matching; pp. 78 - 79.