Optimal Excitation and Readout of Resonators Used as Wireless Passive Sensors

The general solution $i_H\left(t\right)$ of the homogeneous part of Equation (4) is given by:The actual values of the two complex constants $C_1$ and $C_2$ of the homogeneous solution result from the boundary conditions. The two solutions of the homogeneous differential equation are called natural oscillations at the natural angular frequencies of the resonator. When the resonator is stimulated, it produces damped free oscillations with the damped natural angular frequencies $\pm {\omega }_d$, whose amplitudes decrease in proportion to $e^{-\alpha t}$, where $0<\alpha <{\omega }_0$ was assumed. The amplitudes drop to $e^{-\pi }\approx 4\%$ after Q oscillations. The spectrum of a decaying damped resonator is described by a Lorentz curve.

The currents of the two damped natural oscillations induce voltages across the elements of the circuit. The voltage $u_{AH}$ across $R_A$ due to the natural oscillations is given by Due to the real nature of the source resistance, the antenna’s current and voltage are in phase. The power $P_{out}$, which is taken from the damped natural oscillations in the source resistance of the antenna, is radiated back to the reader unit via the antenna Half of the energy stored in the natural oscillation is radiated back to the reader unit via the antenna due to electrical matching.

With a forced periodic excitation by the voltage $u_0\left(t\right)$ with constant amplitude $U_0$ the steady-state current $i_{St}\left(t\right)$ results in (see Appendix A, Equation (A45))

When the open-circuit voltage $u_0$ is generated in the feeding point of the antenna due to picking up of a readout signal, then the steady-state voltage $u_{AG}$ across the source resistance of the antenna at the forced frequency $\omega$ will be given by the complex voltage divider After inserting the abbreviations of Equation (3) and simplifications (see Equation (A48)), we obtain

Equations (11) and (13) essentially reflect the same relationship, with Equation (11) being derived from a solution of the differential equation and Equation (13) from the steady state, but this is not surprising for linear time-invariant systems. From Equation (13), we obtain the frequency response $H_A\left(\omega \right)$ of the source resistance of the antenna, which is connected in series to the resonator with

The frequency response can be inverse Fourier transformed to calculate the impulse response of the source impedance of the antenna (see Appendix B, Equation (A65)) For $t=0$, this results in a value for $h_A\left(0\right)$ of Figure 3 shows exemplary the frequency response $H_A\left(f\right)$ and the corresponding impulse response $h_A\left(t\right)$ of an example resonator with center frequency of 1 and a loaded quality factor Q of 100. Thereby, electrical matching was assumed, i.e., $R_A=R_D$.

The voltage $u_C$ of the capacitor C and the current i in the coil L correspond to the stored energy in the resonator. Therefore, the values of the voltage $u_C$ and of the coil current i must be continuous by any change in the externally applied voltage $u_0\left(t\right)$. The current i, on the other hand, also defines the voltage $u_A\left(t\right)$ at the source resistance, which, therefore, must remain continuous with any change in the externally applied voltage. The two damped natural oscillations must compensate any discontinuity in the forced oscillations due to the externally applied voltage $u_0\left(t\right)$. Since both the voltages across the resistances and the voltage across the capacitor C must remain continuous, any discontinuity in the external open-circuit voltage $u_0\left(t\right)$ is entirely applied at the coil. These boundary conditions are required for a direct solution of the differential equation, while they will be automatically fulfilled when solving the differential equation by convolving with the impulse response.

The actual voltages in the circuitry consist of both the generated voltages at the forced frequency $\omega$ due to the external voltage $u_0$ and the voltages induced due to the currents of the two natural oscillations $\pm {\omega }_d$. The voltage $u_A$ across the source resistance of the antenna corresponds to both the power which is fed into the resonator by the antenna ($u_{AG}$) and the power which is sent back to the reader ($u_{AH}$). In both cases, $R_A$ acts as a lossless transformer that converts electromagnetic power into electrical power and back. Since both contributions are included in the impulse response, it is, therefore, sufficient for further analysis to concentrate on $u_A$ when calculating the response signal via the impulse response. If we insert Equations (7) and (13) into Equation (17), we will obtain

To calculate the response signal by solving the differential equation while taking the boundary conditions into account, it is more advantageous to start from the voltage across the capacitor $u_C$. The current resulting from Equation (18) must be zero as an initial condition for the stored energy of the coil. Integration of this current, therefore, does not result in any further initial condition for the capacitor voltage. The voltage $u_C$ results analogously from the externally generated voltage $u_{CG}$ (see (A49)) and from the voltage $u_{CH}$ induced by the two natural oscillations where the two constants $\widetilde{C_1}$ and $\widetilde{C_2}$ differ from $C_1$ and $C_2$ in Equation (18).

The resonator reacts to every change in the stimulus signal with natural oscillations that ensure the boundary conditions. When these natural oscillations subside, the transition process settles into the steady state. The resonator reacts analogously to the end of stimulation with associated natural oscillations. During the transient process, there is not only a flow of power from the antenna into the resonator but also a return flow from the resonator to the antenna due to the natural oscillations.

At the beginning of the excitation, no current flows. $u_A\left(t\right)$ is then very small, and almost the entire open-circuit voltage is fed into the resonator. The current flow only will build up slowly when the resonator begins to oscillate. Since current and voltage change over time during the transient process, the impedance with which the antenna is loaded also changes. During the decay process, the power flows from the resonator into the antenna and is radiated.

The terminal voltage of the antenna, which is wired to the resonator, is given by $u_0-u_{AG}$. The power fed into the resonator $P_{fed}\left(t\right)$ is given by

The active power is given by its real part, $\mathfrak{R}\left\{P_{fed}\left(t\right)\right\}$, and the reactive power by its imaginary part $\mathfrak{I}\left(P_{fed}\left(t\right)\right)$.

Now that all the necessary formulas are collected to model the resonator connected in series with an antenna, several waveforms can be analyzed that could be used to excite the resonator.

To compare the analytical analyses with numerical ones, simulations were carried out using MATLAB [29]. The same formulas, signals, and parameters of the resonator were used in MATLAB as in the analytical calculation. The different window functions used as stimulation signals were implemented in the time domain. For the analytical calculations, the convolution of the stimulation signals with the impulse response of the resonator were calculated analytically; for Section 3, the relevant differential equation was also solved directly, given in Appendix C. To present the results, the parameters of the example resonator, $f_0=1$ and $Q=100$, were inserted into the formulas obtained and the outputs were displayed graphically.

For the MATLAB results, the convolutions were calculated numerically by Fourier transforming the excitation signals, multiplying them by the transfer function of the resonator as given in Equation (14) and depicted in Figure 3a,b, and then transforming the results back to time domain via IFFT. It is important that the frame data for the numerical simulation are chosen to be sufficiently large in both the time domain and the frequency domain so that aliasing is avoided. In the depicted examples, the resonator has a center frequency of 1 and a quality of 100. To avoid aliasing, the system was modeled with 8192 points and a bandwidth of 10, measured in units of the resonant frequency. This choice ensures sufficient decay of the signals in both the time domain and the frequency domain.

The curves from the numerical calculation lie indistinguishably on the analytically calculated curves in all graphics. To make them visible, the numerically calculated curves were shifted downwards by 0.01 and plotted as red dots. In Figure 3c, the numerically calculated curve was shifted downwards by 1 dB. The simultaneous drawing of the analytically and numerically calculated graphs initially makes it easier to check the analytically calculated formulas.

The numerical simulation can be coded much faster than the analytical calculations, but it only solves this specific example. The analytical formulas, on the other hand, solve the general problem and show the physical processes involved in the transient behavior during excitation and in the generation of the decaying natural oscillations from the excitation spectrum. In addition, they enable optimization of the readout of a resonator.

After the switching on, in range II, we obtain (see Equations (A103) and (A137))

By inserting Equation (14), we obtain We immediately obtain the signal for the steady state with sinusoidal excitation, plus two natural oscillations, which together compensate the current in the coil and the voltage in the capacitor. We always obtain both natural oscillations to satisfy the boundary conditions because the excitation occurs with a CW signal, the natural oscillations, however, are damped oscillations, i.e., the driving frequency $\omega$ is not an eigenvalue of the differential equation. At the beginning, the sum of the natural oscillations at the frequency ${\pm \omega }_d$ are at the same amplitude but opposite sign as the forced oscillation at frequency $\omega$. As the natural oscillations gradually decay, the oscillations transition to the stationary state and more and more energy is stored in the resonator.

If the driving frequency is near the angular natural frequency, i.e., ${\omega \approx \omega }_d$, and the quality factor Q will be high, then the first term of the natural oscillations at the frequency ${-\omega }_d$ is by the factor $4Q$ smaller than the second one at the frequency ${+\omega }_d$. Since our driving frequency is $+\omega$, the main part of the induced natural oscillation is at the frequency ${+\omega }_d$. However, a small component at the frequency ${-\omega }_d$ is also required to satisfy the boundary conditions.

If ${\omega \approx \omega }_d$ and $Q\gg 1$, Equation (23) can be approximated (see (A141)) to Since $\omega$ is not equal to ${\omega }_d$ in general, a beat might be obtained from the constant forced oscillation and the decaying natural oscillations, which can be seen in Figure 6c. The two terms will add constructively when their difference in angular phase is $\pi$For example, if the resonator is excited at the 3 dB band edge $\omega -{\omega }_d=\frac{1}{2}{\omega }_{3dB}$, this will result in the optimal excitation length $T_{{\omega }_{3dB}}$ for the driving voltage of Q oscillations, as is chosen in Figure 6b. We see this effect somewhat in Figure 6b, where the response signal, when the excitation signal is switched off, will show an amplitude of 0.36952, i.e., 7% more than we would expect if we only considered their ratio in $H\left(f\right)$. However, if the frequency distance is twice the span, e.g., $\omega -{\omega }_d={\omega }_{3dB}$, the optimal excitation length for the driving voltage is $\frac{1}{2}Q$ oscillations, as can be seen in Figure 6c. In this case, a stimulation half as long or shorter would result in a significantly higher response signal: 0.27187 for a length of driving voltage of $0.5·Q$ oscillations when compared to the shown 0.21465 for Q oscillations. A length of driving voltage of $0.42·Q$ oscillations would finally result in a response signal of 0.27877, as can be seen in Figure 9.

This characteristic can also be explained in the frequency domain: the shorter an excitation signal is in the time domain, the broader the main lobe of its spectrum. Therefore, if the carrier frequency of the interrogation signal moves slightly away from the resonance frequency, the position of the resonance frequency will slide downward along the main lobe of the excitation spectrum. In order to pump as much power as possible into the forced oscillation and thus maximize the amplitude of the decay signal, it is, therefore, advantageous to shorten the interrogation signal and, consequently, broaden the main lobe of the spectrum.

At the beginning of the excitation, the current is very small and it is in phase with the applied voltage. As the excitation progresses, the phase shift between the current in the resonator and the voltage applied to the resonator builds up and reaches the value specified in Equation (A26), as can be seen in Figure 4 for three excitation frequencies. The impedance, seen by the source impedance of the antenna, is in the beginning very high, near open end. Figure 5 shows the evolution of the reflection coefficient during the transient phase. With a stimulation at center frequency, the impedance evolves from open to the matched condition along the real axis. With a stimulation frequency next to center frequency, the impedance evolves from open to its steady state value, with frequencies higher than the resonance frequency in the inductive plane and with frequencies lower than resonance in the capacitive one.

Since the impedance is very high at the start of stimulation, only a small fraction of the power offered by the source is injected into the resonator, as can be seen in Figure 8 according to Equation (20). The active power is shown by the solid black curve and the reactive power by the dashed blue curve. The excitation is performed in Figure 8a at resonance frequency, in Figure 8b at the 3 dB frequency and in Figure 8c at a frequency twice the distance from the resonance. When excited with a resonant frequency, only active power will be transmitted which reaches 100% of the available power after Q oscillations. When excited next to the resonance frequency, an increasing amount of reactive power will be transferred and the active power absorption remains well below 100%.

For $t>T$, we add a second voltage with This cancels the external voltage $u_0$ to zero. The same terms of natural oscillations as in range II show up, however, with alternate signs and time and phase shifted because they now start at $t=T$. The phase shows the phase shift of $\omega T$ of the external voltage between 0 and T, and then it increases with ${\omega }_d\left(t-T\right)$. We obtain (see Equations (A109) and (A148)) when the stimulation is switched off

Figure 6 shows in blue the driving voltage and in black the corresponding response signal of the resonator specified in Figure 3 for a driving frequency at resonance frequency at the 3 dB band edge and at twice the 3 dB band edge, which are calculated according to the analytical formulas (22) and (27). Additionally, the result of a numerical calculation with MATLAB is shown. For a driving frequency at twice the 3 dB band edge, a length of driving voltage of less than $0.5·Q$ is preferred, since then the constant forced oscillation and the decaying natural oscillations interfere constructively. The maximum response signal is obtained with $0.42·Q$ oscillations for the driving signal, which result after switching off the driving signal in a response signal of 0.27877, as can be seen in Figure 7.

To visualize the transient and decay response, Figure 9 shows the real parts of the exciting voltage and the corresponding system response for a resonator with a quality factor of 10.

The term for $n=1$ in Equation (27) is dominant for high Q and $\omega \approx {\omega }_d$. If we ignore the small term for $n=0$ of the natural oscillations at the frequency ${-\omega }_d$ and add the small factor $\frac{j{\omega }_d+\alpha }{{\omega -j\alpha +\omega }_d}$ to the remaining natural oscillation, we will obtain for the decay The natural oscillations at the frequency ${\omega }_d$ consist of two terms. The damping of the larger one starts at $t=T$, while the damping of the smaller one has already started at $t=0$. Depending on the phase difference contained in $e^{j\left({\omega -\omega }_d\right)T}$, they add up constructively or destructively.

Equation (28) can also be written as If $\left|\alpha +j\left(\omega -{\omega }_d\right)\right|T\ll 1$, i.e., we load only for a short period of time, then both terms will exhibit nearly the same amplitude and mostly cancel each other. In this case, $u_A\left(t\right)$ increases with increasing TOn the other hand, if $\alpha T>\pi$, then the second term in the brackets of Equation (29) can be neglected and the equation simplifies to The response signal of the resonator according to Equation (27) to a rectangular excitation signal consists of two identical response signals, of which the first two occur at the beginning and the other two with a negative sign at the end of the excitation. Each pair has a Lorentz-shaped spectrum around the natural frequency. The time delay in the response signals from the end of the excitation results in a modulation in the frequency range. Depending on the phase of this modulation, the spectral components add up positively or destructively. Since the time interval between the two prompts and the two delayed response signals is the same as between the rising and falling edges of the excitation, their joint spectrum also has the identical zero distribution.

The common spectrum of all response signals and the forced oscillation together is calculated in the frequency domain by multiplying the $sin\left(\omega \right)/\omega$ function of the spectrum of the excitation signal, which is centered at the carrier frequency, by the Lorentz curve of the spectrum of the resonator. The four response signals alone, therefore, result in a spectrum that corresponds to the “missing” part between the exciting $sin\left(\omega \right)/\omega$ function and the combined spectrum.

The joint spectral power density of the four response signals was taken from the excitation spectrum. However, the response signals from the end of the excitation alone can contain spectral components that were not included in the excitation spectrum if destructive interference of all Lorentz curves leads to a zero point in the frequency domain. If the response signals from the beginning of the excitation already have decayed at the end of the excitation, a response signal at the natural frequency will show up, even if this frequency was not included in the excitation spectrum.

Figure 10 shows such an example. The carrier of the excitation signal was set to the 6 dB corner frequency of the resonator and the length of the excitation to $Q/f_0$. The first zero point of the associated spectrum is, therefore, at the resonance frequency. The top row shows on the left the spectrum of the excitation signal in blue and the Lorentz curve of the resonator in red and on the right the response signal of the resonator to this excitation signal. The bottom row shows on the left the spectrum of the response signal of the resonator after switching off the excitation and on the right the joint spectrum of the response signals from the start and end of the excitation.

If an electromagnetic wave of power $P_{in}$ is picked up by the antenna between $t=0$ and $t=T$, an open-source voltage $U_0$ will be generated in the resonator circuit (see Equation (1)). Because the electrical matching is at resonance frequency in the steady state, the loaded voltage over $R_A$, $u_A$ is half the driving voltage. The power transferred by the antenna from the incoming electromagnetic wave to the resonator is

After switching off the readout signal $P_{in}$, a response signal with power $P_{out}$ is sent back with

Near resonance, i.e., $\omega \approx {\omega }_d$, the term $\left(1-e^{\left(j{\omega }_d-j\omega -\alpha \right)T}\right)$ increases linearly with T for $\alpha T<1$ and approaches 1 for $\alpha T>\pi$. If we neglect the constant phase rotation $e^{j2\omega T}$ we will obtain for $\alpha T\ge \pi$

In this case, the power $P_{out}$ during decay starts at the same power level as the picked-up power level $P_{in}$. A longer stimulation phase T beyond $\alpha T=\pi$ does not lead to any further increase in the response signal.

For the range I when the amplitude of the driving signal is increased linearly in time, the stimulating voltage is given by The resonator responds to this stimulation with (see Appendix E.2, Equation (A163))

The resulting amplitude of the driven signal now consists of two parts, one of which is increasing linearly in time and one constant part. The term with the angular natural frequency for $n=1$ is dominant for $Q\gg 1$ and $\omega \approx {\omega }_d$. This dominant term of the angular natural frequency can be expressed as a function of the square of $H_A\left(\omega \right)$ (see Equation (A167)). If we ignore all vanishingly small terms, Equation (36) simplifies to We obtain a forced term with two parts: one increasing with time and one constant. Due to the continuity of the switch-on process at $t=0$, the amplitude of the dominant natural oscillation is proportional to the square of $H_A\left(\omega \right)$ and, additionally, it is suppressed by $1/{{\alpha }_{A}T}_1$. This natural oscillation is canceled at $t=0$ by a forced term of the same power and opposite sign.

The amplitude of the driving signal is kept constant in range II. The analysis (see Appendix E.3, Equation (A174)) delivers

The resulting amplitude of the driven signal is constant and shows the same amplitude as in the case where the amplitude is abruptly switched on. We receive two signals at each of the associated damped angular natural frequencies, one starts to decay at $t=0$ and the other at $t=T_1$. The term of the natural oscillations for $n=1$ is dominant for $Q\gg 1$ and $\omega \approx {\omega }_d$. In this case, we can simplify Equation (38) to We obtain two signals oscillating with the angular natural frequency ${\omega }_d$. Depending on their phase difference $e^{j\omega T_1}$, they add up constructively or destructively.

The signals in range III, where the amplitude of the driving signal is decreased linearly in time, is similar to the situation in range I, but now with a decreasing driven signal. The slope of the decreasing is sometimes set faster than the increasing amplitude slope, therefore, a different slope was chosen. The response signals of the systems are (see Appendix E.4, Equation (A183)):

We have a linearly decreasing driven signal and a constant driven signal, which is exactly compensated at ${t=T}_2$ by a damped natural oscillation. At the switching points $t=0$ and ${t=T}_1$, we obtain two natural oscillations, one at $-{\omega }_d$ and one at $+{\omega }_d$, which start at t = 0 and t = $T_1$ and then die out. The natural oscillation again can be expanded as a function of the square of $H_A\left(\omega \right)$. If we keep only the dominating terms for $Q\gg 1$ and $\omega \approx {\omega }_d$, Equation (40) simplifies to (see Equation (A186)) Of most practical interest are the remaining signals after the driving has stopped.

After the end of the stimulation signal, the resonator oscillates on its natural oscillations (see Appendix E.5, Equation (A191))

The linearly decreasing driven part of the signals vanished after switching off the driving signal at $t=T$. The constant driven signals, however, transformed into a decaying natural oscillation at $t=T$. Thus, at each switching point we obtain two natural oscillations, one at $-{\omega }_d$ and one at $+{\omega }_d$, which start at t = 0, t = $T_1$, t = $T_2$ and t = T, and then die out exponentially. Depending on their sign and relative phase shift 0, $e^{j\omega T_1}$, $e^{j\omega T_2}$, $e^{j\omega T}$ they add up constructively or destructively.

Figure 11 shows in blue the driving voltage and in black the corresponding response signal of the resonator specified in Figure 3 for a driving frequency at resonance frequency, at the 3 dB band edge and at twice the 3 dB band edge, which are calculated according to the analytical formulas (36), (38) and (42). Additionally, the result of a numerical calculation with MATLAB is shown.

For a driving frequency at the 3 dB band edge, or at twice the 3 dB band edge, shorter stimulations with fewer than Q oscillations lead to higher response signals. Figure 12 shows the response for shorter stimulations at frequencies at the 3 dB band edge and at twice of it.

Surprisingly, the length $\left(T_2-T_1\right)$ of the constant stimulation does not seem to play a direct role in the Equation (42), but only the rate of rise and fall of the stimulation voltage. However, if the length $\left(T_2-T_1\right)$ is chosen too short, the destructive interference that occurs between the individual terms of Equation (42) will lead to a drastic reduction in the overall response. Furthermore, it is astonishing that the denominators of the individual parts contain the rise and fall times $T_1$ and$T-T_2$. Since these can be selected freely, it seems possible that a stronger decay signal might be generated by a clever choice of the rise and fall rates, compared to the case with hard switching on and off of the driving signal. An analysis shows that any possible increase in signal strength by shortening the rise and fall rates is also deteriorated by a destructive interference of the associated terms of natural oscillation.

The term with the first large square bracket will be the dominant one of Equation (42) when $Q\gg 1$ and $\omega \approx {\omega }_d$. When the time for constant stimulation is chosen long enough to fully load the resonator, i.e., $\left(T_2-T_1\right)·\alpha >\pi$, then the decay terms starting at $t=0$ and $t=T_1$ can also be ignored. In this case, Equation (42) simplifies to The two terms might add constructively if their phase difference is $\pi$. However, in this case either the frequency response $H_A\left(\omega \right)$ will be very small (see Appendix E.5, (A198)), i.e., the driving frequency $\omega$ is not within the resonance, or the second term in Equation (44) will already have nearly faded away. The two terms in Equation (44) will, therefore, always add destructively. Increasing $(T-T_2)$ so that the second term in the square bracket can be neglected does not help either, as it also reduces the prefactor of the bracket. On the other hand, if the decrease time $\left(T-T_2\right)$ is chosen fast enough to ensure then we can approximate the exponential function up to the quadratic term and obtain (see Appendix E.5, (A203)) A comparison of the decay signal in the case of applying a trapezoidal window on the readout signal $u_{A,Tapezoidal}\left(t\right)$ to the decay signal in the case of hard switch on and off of the readout signal $u_{A,cut}\left(t\right)$ gives

That is to say, we obtain a decay signal with nearly the same strength as in the case with hard switch on and off, if the amplitude decrease time $\left(T-T_2\right)$ is short enough.

The last sections showed that the level of the decay signal depends crucially on the length of time of the constant excitation and on a sufficiently fast decay of the excitation signal. How does the situation change if we skip the constant loading and use a driving signal according to a Bartlett window. The corresponding loading signal is given by

The response of the system can be calculated by using the results for a trapezoidal loading signal and adopting the times $T_1=T_2=T/2$ (see Appendix E.6). Substituting $T_1=T/2$ into (36) gives the signals during the increasing of the stimulation For the decreasing part, the substitution of $T_1=T_2=T/2$ in Equation (40) results in And, finally, the decaying natural oscillation for $t>T$, when stimulation is completed, results by the substitution of $T_1=T_2=T/2$ into the Equation (42) For the Equations (48)–(50) approximations for $Q\gg 1$ and $\omega \approx {\omega }_d$ can be estimated, which are given in Appendix E.6, Equations (A208) and (A212). Neglecting the vanishing small second term in (50) leads to (see also Equation A215) $u_{A,Bartlett}\left(t\right)$ increases linear with T for small T (see Equation (A220)), and reaches a maximum at $T\alpha =2.5$ (see Equation (A224)) with Figure 13 shows, in blue, the driving voltage and, in black, the corresponding response signal of the resonator specified in Figure 3 for a driving signal which is weighted in the time domain with a triangle function (Bartlett window). The driving frequency was set to resonance frequency at the 3 dB band edge and at twice the 3 dB band edge. The lengths of the Bartlett windows were optimized to maximum response signal at the time when the driving signal was switched off.