1 Introduction↩︎

The implementation of Reconfigurable Intelligent Surface (RIS) technology presents a novel way to improve wireless communication performance by providing adaptive degrees of freedom to design radio-frequency (RF) channels. Using an RIS, signals can be steered around obstacles, beamforming gains can be increased to boost signal-to-noise-ratio (SNR) and lower interference, and the overall quality-of-service (QoS) experienced by network users can be greatly enhanced [1]. RIS technology is especially beneficial in challenging RF environments like urban areas and indoor spaces, where it can be difficult to sustain strong and reliable signal transmissions [2].

An RIS consists of a wide range of reconfigurable subwavelength components, an example of which would be metallic patches, that are electronically controlled to manipulate electromagnetic waves [3]. These unit cells have the ability to precisely control how incoming electromagnetic waves behave by dynamically modifying their reflection characteristics in real time using tunable elements such as PIN diodes, varactor diodes, and liquid crystals, among others [4]. RISs can increase signal strength, boost wireless network performance, and open the door for more effective communication systems by optimizing wave reflection and propagation.

RIS designs can have hundreds to thousands of unit cells operating at millimeter-wave and terahertz frequencies, resulting in notable beamforming gains and allowing for highly directed and reconfigurable beams [5], [6]. Although these qualities improve communication, there are several obstacles to overcome when implementing large-scale RIS designs. Most RIS systems depend on external control units, such as field-programmable gate arrays (FPGAs) or microcontrollers, to configure the reconfigurable elements on the RIS surface [7]. For an RIS with \(M\) elements, these control units perform optimization processes involving \(M\) variables and transmit the optimal configuration to each of the \(M\) RIS elements. Thus, each unit cell requires a dedicated connection to receive the appropriate control voltages or currents, leading to complex wiring networks with hundreds of individual signal connections. This not only complicates fabrication but also increases hardware costs and limits practical deployment. Alternative control strategies such as light-based approaches employing photodiodes, scanning lasers, or encoded light images, have been investigated in order to overcome these difficulties [8]–[12]. Despite their potential, these methods face limitations, including susceptibility to environmental factors such as rain and reduced resolution due to the distance between the light source and the RIS.

This paper investigates the performance of the technique presented in [13] and later studied in [14]–[17] as a realistic and effective way to control RIS elements, as shown in Fig. 1. This technique, utilizing varactors for the reconfigurable reflective control, consists of delivering dc biasing voltages to the varactors inside the unit cells that are connected at specific positions, \(x_m\), along a single biasing transmission line (BTL). The desired dc biasing voltages supplied to every RIS element to achieve a specific reflective radiation pattern, \(w(x_m)\), can be obtained from combinations of \(N\) biasing standing waves (BSWs), \(w(x_m, t)\), that are injected into the BTL. The scope of this paper is limited to the study of the performance of the BTL when only a single frequency is injected into the BTL. Further, the extent to which a single reflected beam can be controlled through this single biasing frequency is explored. This technique drastically lowers signaling overhead and hardware complexity by doing away with the requirement for separate wired connections to every unit cell. Advanced RIS functions, including beam creation, polarization control, and channel equalization, are made possible by this straightforward yet efficient control method, which makes it a viable option for large-scale RIS deployments.

2 Biasing Standing Waves↩︎

To use a wave-controlled feed network for the RIS, an analytical model for the voltages was derived from a transmission line analysis of the system. This model determines the control voltages that are applied to the unit cells by the input wave, and thus is necessary for understanding how the unit cells are reconfigured.

The wave-controlled RIS approach involves two key components: a BTL board and an RIS board, as shown in Fig. 1. The BTL board, placed beneath the reflective surface and shielded from RF interference by a ground plane, supports \(N\) BSWs. Rectifiers placed at \(M\) positions on the BTL board generate dc voltages, \(w(x_m)\), from these waves which are used to bias the \(M\) varactors positioned on the RIS board at \(x_m\) for \(m=0,1,\ldots,M-1\). This process modifies the reflected phase distribution, enabling control of the RIS’s scattered radiation pattern [16]. The necessary voltage applied to each varactor, \(w(x_m)\), for a given reflected phase is calculated using an analytical model that accounts for the RIS’s reflection behavior under various dc bias conditions for the varactors [16], [18]. Once the required voltages are determined, they are generated by a periodic time-domain signal comprising a combination of \(N\) BSWs with specified amplitudes \(W_n\), for \(n=1,2,\ldots, N\), that are injected into the BTL through a single input port, along with a base voltage \(W_0\). The inclusion of \(W_0\) simplifies the reconstruction of the voltage distribution, as the varactor’s biasing voltage range typically does not include 0 V. Consequently, the expected voltage distribution pattern includes a dc component.

As seen on the RIS board in Fig. 1, the RIS unit cells are separated by a distance \(d_x\) along the \(x\)-axis. Likewise, the corresponding rectifiers that convert the standing wave amplitudes, \(w_n(x_m,t)\), into dc values for each RIS unit cell are also placed a distance \(d_x\) away from each other along the \(x\)-axis on the BTL board. These, along with the voltage offset \(W_0\), produce the necessary voltages \(w(x_m)\) [16]. The position along the BTL of the \(m\)th rectifier element is consequently defined as \(x_m = md_x\).

The BTL board is not necessarily as simple as that shown in Fig. 1, an example of which is provided in [15]. This BTL must connect with the RIS board at specific positions along the \(x\)-axis contained within the length \(L\), where \(L = (M-1)d_x\). The total length of the BTL along the \(x\)-axis is thus defined as \(L_{\mathrm{tot}} = L+L_{\mathrm{left}}+L_{\mathrm{right}}\), where \(L_{\mathrm{left}}\) and \(L_{\mathrm{right}}\) are additional segments on either end of the BTL that help refine the voltage distribution pattern along the \(M\) elements. The phase velocity of the propagated modes projected along the \(x\)-axis is \(v_\mathrm{b} = c/n_{\mathrm{slow}}\), where \(c\) represents the speed of light and \(n_{\mathrm{slow}} \ge 1\) is a slowness factor determined by the material properties and geometry of the BTL. The injected periodic time-domain signal oscillates at the biasing frequency \(f_\mathrm{b} = v_\mathrm{b}/\lambda_\mathrm{b}\), where \(\lambda_\mathrm{b}=2\pi /k_\mathrm{b}\) is the guided wavelength, and the corresponding propagation wavenumber is \(k_\mathrm{b} = 2\pi f_\mathrm{b}/ v_\mathrm{b}\).

This design approach enables modeling of the voltage of the BSW before and after each rectifier. The voltage of the standing wave at any point along the BTL can be derived by applying transmission line theory [19]. In the general case when there are \(N\) biasing modes (degrees of freedom to control the RIS response), the biasing voltage of the standing wave before rectification and assuming a short circuit termination at \(x=-L_{\mathrm{left}}\) is [13], [16]

\[\begin{align} & w(x, t) = W_0 + \\ & \sum_{n=1}^{N}W_n \sin\left( \frac{ n\omega_\mathrm{b} n_{\mathrm{slow}}}{c} \left(x+L_\mathrm{left}\right)\right) \cos\left(n \omega_{\mathrm{b}} t + \phi_n \right). \end{align} \label{zwdspleq}\tag{1}\] Here, \(n\) is the biasing mode index and \(W_n\) is the real-valued amplitude of the standing wave of the \(n\)th biasing mode. The term \(W_0\) is the dc component, which sets the minimum biasing voltage applied to the BTL and shifts the varactor’s operating range. Furthermore, \(\phi_n\) is the temporal phase offset of the \(n\)th biasing mode. This definition generalizes Eq. (7) of Ref. [16], as here the wavelength \(\lambda_\mathrm{b}\) and, consequently, the fundamental frequency \(f_\mathrm{b}\), do not necessarily depend on the total length of the BTL \(L_\mathrm{tot}\).

The \(m\)th rectifier, shown in Fig. 1, acts as a peak detector and is described in detail in Sec. III. Its output voltage is given by

\[\begin{align} & w(x_m) = W_0 + \\ & \max_t \left[\sum_{n=1}^{N}W_n \sin\left( \frac{ n\omega_\mathrm{b} n_{\mathrm{slow}}}{c} \left(x_m+L_\mathrm{left}\right)\right) \cos(n \omega_{\mathrm{b}} t + \phi_n) \right]. \end{align} \label{qayrmgks}\tag{2}\] In [13], [16], we have explored theoretically the use of the BTL with several standing waves. Here, we focus on the experimental characterization of a BTL and on the use of a single BSW to control the pointing angle of a single beam. Accordingly, when using a single BSW of frequency \(f_\mathrm{b}\), ([eq:StdWave95Volt]) reduces to \[\begin{align} & w(x, t) = W_0 + \\ & W_\mathrm{b} \sin\left( \frac{\omega_\mathrm{b} n_{\mathrm{slow}}}{c} \left(x+L_\mathrm{left}\right)\right) \cos\left(\omega_\mathrm{b} t + \phi \right), \label{eq:StdWave95Volt95Single} \end{align}\tag{3}\]

where \(W_\mathrm{b}\) is the amplitude and \(\phi\) is the phase shift. After rectification at each \(x_m\), the dc voltage is \[w(x_m) = W_0 + \left|W_\mathrm{b} \sin\left( \frac{\omega_\mathrm{b} n_{\mathrm{slow}}}{c} \left(x_m+L_\mathrm{left}\right)\right) \right|. \label{eq:BTL95Volt95Retif95Single}\tag{4}\]

3 System Design↩︎

The wave-controlled RIS system is realized here using two separate boards: the RIS board and the BTL board. The RIS board reflects the incident radiation based on a spatial distribution of the phase reflection coefficient across the RIS elements, which is governed by the biasing voltage distribution \(w(x_m)\) provided by the BSW, supported by the BTL board. Other implementations could be realized in a single multilayer board. This biasing pattern is controlled by the signal frequencies and amplitudes injected into the BTL (see Fig. 1). The RIS unit cell is designed to achieve the desired phase response by applying a variable voltage to the varactor. The design includes the unit cell’s geometry, material, and chosen varactor. The BTL board comprises a transmission line with a defined geometry and integrated rectifiers to provide the dc biasing voltage for the unit cells of the RIS. Material specifications play a crucial role in the overall performance of the BTL and are introduced next.

3.1 RIS Unit Cell Design↩︎

The RIS unit cell consists of an array of two rectangular patches arranged in a mirrored configuration as in Fig. 2, mounted on a grounded dielectric substrate (RT5880LZ) with a relative permittivity of \(\epsilon_r = 2\), a loss tangent of \(\tan \delta = 0.0021\), and a thickness of 1.27 mm. Each patch is connected to the cathode of an individual SMV1231-040LF varactor, while the varactor’s anode is grounded via a through-hole connection. The unit cell design is optimized to maximize the phase dynamic range within the industrial, scientific, and medical (ISM) 2.45 GHz band for reverse bias voltages ranging from \(-4\) V to approximately \(-10\) V.

To interface the patch with the BTL board and allow the dc biasing of the two varactors, a coplanar waveguide with ground is implemented on the ground plane of the RIS and connected to the rectified voltage \(w(x_m)\) in the BTL. This line is oriented to avoid disrupting the RIS ground-plane current and is connected to the center of the patch through a metallic via. This configuration allows efficient biasing of the varactors while minimizing the effect on the RIS RF current flow. To further suppress the effect on the RF current at the connection node located at the center of the coplanar waveguide, a 100 pF capacitor is incorporated to ensure proper grounding at RF. This capacitor is also part of the rectifier design that is discussed in Sec. III B. The RIS board comprises 27 identical unit cells with two copper patches, with each unit cell featuring the following dimensions (in mm): \(d_x = 20\), \(A_1 = 18.5\), \(A_2 = 1\), \(A_3 = 0.3\), \(B = 145.4\), \(B_1 = 36.7\), \(B_2 = 1.2\), \(B_3 = 24.8\), \(G_1 = 0.4\), \(G_2 = 0.5\), \(G_3 = 1.8\), \(r = 0.2\), \(P_1 = 4\), \(P_2 = 0.4\), as shown in Fig. 2.

To calculate the reflected phase provided by each RIS element, the analytical circuit model shown in Fig. 3 is used [16], [18]. This model requires only one full-wave simulation of the unit cell over the frequency of operation (assuming the varactor is disconnected) to extract the circuit elements that represent just the patch over the grounded slab, as detailed in Appendix A. For the design presented in Fig. 2, after fine-tuning, the values obtained are \(R_\mathrm{d}=0.17\) \(\Omega\), \(C_\mathrm{d}=0.74\) pF, \(L_\mathrm{d}=1.64\) nH, and \(L_\mathrm{s}=1.60\) nH.

Once these circuit elements are determined, the varactor is incorporated to complete the RIS unit cell model by adding its RF impedance \(Z_\mathrm{v}\) to the equivalent circuit as shown in Fig. 3. The varactor’s impedance is modeled by an equivalent series RLC circuit \[Z_\mathrm{v}=R_{\mathrm{v}}(V) + j\omega L_{\mathrm{v}} + \frac{1}{j\omega C_{\mathrm{v}}(V)},\] where the variable capacitance \(C_{\mathrm{v}}\mathrm{(V)}\) and variable resistance \(R_{\mathrm{v}}\mathrm{(V)}\) are functions of the reverse biasing voltage \(V\), and the constant inductance \(L_{\mathrm{v}}=2.34\) nH models both the package inductance and the parasitic inductance introduced when the varactor is soldered across the unit cell gap. The values of \(C_{\mathrm{v}}\mathrm{(V)}\) and \(R_{\mathrm{v}}\mathrm{(V)}\) are obtained through a parametric sweep simulation in the commercial Keysight Advanced Design System (ADS) software, based on the SPICE model of the varactor provided by the manufacturer. These values are limited to the reverse voltage bias range of \([4\) V, \(15\) V\(]\) and are detailed in Table 1.

As shown in Fig. 3, the RLC varactor model is connected in parallel to the capacitance \(C_{\mathrm{d}}\), and the total equivalent impedance of the RIS is hence given by

\[Z_{\text{RIS}}=\left(R_{\mathrm{d}}+j\omega L_{\mathrm{d}} + Z_\mathrm{v}\parallel\frac{1}{j\omega C_{\mathrm{d}}}\right)\parallel j\omega L_{\mathrm{s}}.\]

Finally, \(Z_{\text{RIS}}\) is used to determine the reflection coefficients, \(\Gamma_m\), of the RIS elements: \[\Gamma_{m} = \frac{Z_{\mathrm{RIS}}-\eta_0}{Z_{\mathrm{RIS}}+\eta_0}, \label{eq:ReflCoef}\tag{5}\] where \(\eta_0=377\) \(\Omega\) is the free-space impedance. The magnitude and phase of \(\Gamma_m\) are denoted by \(R_m(\mathrm{V})\) and \(\alpha_m(\mathrm{V})\), respectively. The values of \(R_m\) and \(\alpha_m\) as functions of the biasing voltage for the unit cell in Fig. 2 are shown in Fig. 4 () for three different frequencies, including the target RF frequency \(f_\mathrm{c}=2.45\) GHz. The voltage \(\mathrm{V}\) is positive as the varactor is connected in reverse, with its anode grounded and its cathode connected to the positive biasing voltage. At \(f_\mathrm{c}=2.45\) GHz, the phase \(\alpha_m\) varies between \(-160^\circ\) and \(+140^\circ\), with the most significant variations occurring between \(4\) and \(10\) V. The lowest reflection magnitude \(R_m\), approximately \(-2.4\) dB, is observed around \(7\) V. Fig. 4 () illustrates the behavior of \(\alpha_m\) and \(R_m\) as functions of the RF frequency and compares the analytical model (solid lines) with full-wave simulations performed using the commercial software CST Studio Suite (dashed lines) for five different biasing voltages. The agreement is generally good across different biasing voltages and frequencies, with particularly strong accuracy at the target frequency \(f_\mathrm{c}=2.45\) GHz (dotted line in blue). Although in practical RIS implementations the control is optimized based on channel estimates, having an analytical model is useful for design and research purposes or to assist the optimization algorithms.

3.2 Biasing Transmission Line Design↩︎

The BTL design is that of a printed meander (or serpentine) transmission line, chosen for its compact size, phase delay control, and ease of fabrication; see Fig. 5. It is made of copper and implemented on a grounded dielectric substrate RO3010 with relative permittivity \(\epsilon_r = 11.2\), \(\tan \delta = 0.0022\), and thickness \(h=0.64\) mm. The BTL board consists of \(27\) identical unit cells, where the two cells at the ends have been modified to include SMA connectors that allow connection of the signal generator and integration of possible external loads, if required, to ensure optimal performance within the desired frequency range. The proposed design details and geometry are displayed in Fig. 5. The dimensions of the BTL unit cell are (in mm): \(d_x = 20\), \(C = 5\), \(E = 7.64\), \(F = 11\), \(G=1.8\), \(H = 50\), \(L_{\mathrm{p}} = 131.42\), \(W = 2.6\).

Each unit cell of the BTL has a path length \(L_{\mathrm{p}}\) and a spatial period \(d_x\) along the \(x\)-axis (the same as for the RIS board), and is connected to a rectifier that extracts the desired dc voltage to bias the varactor located on the RIS board. The BTL signal travels along the microstrip with phase velocity \(c/n_{{\mathrm{eff}}}\) where \(n_{{\mathrm{eff}}} = \sqrt\epsilon_{{\mathrm{eff}}} = 2.94\) is the effective refractive index of the microstrip and \(\epsilon_{{\mathrm{eff}}} = 8.66\) is its effective permittivity, which is approximately calculated as [20] \[\epsilon_{\text{eff}} = \frac{\epsilon_r + 1}{2} + \frac{\epsilon_r - 1}{2} \frac{1}{ \sqrt{1 + 12 h/W}},\] and is considered dispersionless in the frequency range of operation. The characteristic impedance of the microstrip is \(Z_0 = 19.23\) \(\Omega\). In this case, the slowness factor, used in ([eq:StdWave95Volt])-(4 ), is also determined by the path length \(L_{\mathrm{p}}\) and the period \(d_x\), and is given by \(n_{{\mathrm{slow}}} = n_{{\mathrm{geom}}}n_{{\mathrm{eff}}}\), where \(n_{{\mathrm{geom}}} = L_{\mathrm{p}}/d_x = 6.57\) represents the geometric index that allows for projection of the standing wave pattern from the printed meander transmission line onto the \(x\)-axis. Thus, the resulting slowness factor is \(n_{{\mathrm{slow}}} = 19.34\). Although the segments \(L_\mathrm{left}\) and \(L_\mathrm{right}\) of the BTL follow the same meander line geometry as the main section \(L\), these two additional segments can have a different geometry as they are not connected to any rectifier. However, the values of \(L_\mathrm{left}\) and \(L_\mathrm{right}\) must be normalized with respect to the geometric index in order to be used in ([eq:StdWave95Volt])-(4 ). Therefore, the total length of the BTL along the \(x\)-axis is \(L_{\mathrm{tot}} = 540\) mm, with \(L = 520\) mm, and \(L_\mathrm{left} = L_\mathrm{right} = 0.5d_x = 10\) mm, while the total path length of the meander microstrip is \(3548.3\) mm.

Using printed meander microstrip topology for the BTL design offers the significant advantage of increasing the slowness of the waves in the \(x\)-direction, allowing the use of low frequencies for the BSWs. This advantageous feature helps mitigate unwanted interference from the BTL harmonics on the RF system. The scattering parameters have been analyzed to assess the losses and reflection performance of the BTL design. The selected frequency range spans from 0 to 300 MHz, which is much lower than the RF band (here, around 2.45 GHz). When the RIS is used with several standing waves, as in [13], [16], this design allows for a fundamental frequency of approximately 7.2 MHz. This ensures that at least 21 modes are available below 300 MHz, which is a sufficient number of degrees of freedom for RIS control. The frequency \(f_\mathrm{b}\) of the fundamental BSW decreases when the number \(M\) of RIS elements increases. Fig. 6 presents the transmission coefficient of the BTL design (without considering the losses of the 27 rectifiers), obtained from a full-wave simulation in CST Studio Suite. The results show a gradual decrease with frequency due to ohmic and dielectric losses, reaching a minimum value of \(-2.44\) dB at 300 MHz. Although not depicted in the figure, the reflection coefficient with Port 2 matched to its characteristic impedance remains consistently below \(-34\) dB across the entire frequency range, indicating minimal mismatch. The BSWs are formed along the BTL due to an open or short termination at \(x = -L_{\mathrm{left}}\).

To extract the dc voltage from the BTL standing wave \(w(x_{m},t)\) at each \(x_m\) location, a rectifier circuit composed of a resistor \(R_\mathrm{r}\), a capacitor \(C_\mathrm{r}\), and a diode \(D_\mathrm{r}\) was designed. The capacitor and the resistor are grounded via a through-hole as shown in Fig. 5 (only the microstrip layer is shown, the ground plane underneath is not shown), while the diode is connected between these components and the BTL at position \(x_m\). The ADS time-domain circuit simulator was employed to optimize the design of the circuit, resulting in \(R_\mathrm{r} = 10\) k\(\Omega\) and \(C_\mathrm{r} = 200\) pF. The BAS16GW diode provided by Nexperia was selected for its fast recovery time of 4 ns and low capacitance of 1.5 pF. By choosing a time constant of \(\tau=2\) \(\mu\)s, the rectifier effectively tracks signal peaks while maintaining the impedance \(Z_\mathrm{rect}\) seen by the BTL between a few hundred to 1000 \(\Omega\) for frequencies up to 300 MHz. The impedance \(Z_\mathrm{rect}\) plays a crucial role in the biasing process, as it is connected to the BTL and can influence the voltage distribution along the RIS. However, in this case, the impedance does not significantly load the BTL, and thus does not have a large impact on system performance. Fig. 5 illustrates the rectifier circuit (inset) and its connection to the BTL and the RIS board.

Fig. 7 shows the BTL and its connection to an arbitrary waveform generator (AWG), modeled with an ideal ac power supply \(V_\mathrm{g}\) in series with an impedance \(Z_\mathrm{g}\), a dc power supply \(V_\mathrm{dc}=W_0\), rectifiers, and some decoupling elements, detailing the current flow of the ac and dc signals. Due to the need for the offset voltage \(W_0\), decoupling the ac and dc signals is necessary to prevent short-circuiting and ensure proper current flow along the BTL. To achieve this, a capacitor \(C_\mathrm{f} =1\) \(\mu\)F is connected after the AWG to act as a high-pass filter, while an inductor \(L_\mathrm{f} = 680\) \(\mu\)H is inserted after the dc power supply to serve as a low-pass filter. An additional capacitor \(C_\mathrm{f}\) is connected between the end of the BTL and either a short or open circuit termination. The chosen capacitors provide a capacitive reactance \(X_{C\mathrm{f}} \approx 0.3\) \(\Omega\) at 500 kHz, while the chosen inductor offers an inductive reactance \(X_{L\mathrm{f}} \approx 2.1\) k\(\Omega\) at the same frequency. This setup ensures that ac signals above 500 kHz (i.e., including all biasing modes) flow through the capacitors but not through the inductor, while signals below 500 kHz (i.e., the dc biasing voltage \(W_0\)) flow through the inductor but not the capacitors, avoiding short circuits at the opposite end of the BTL.

Since the varactors are connected in reverse bias, the current through them is negligible. Unlike the dc current, the ac current also reaches the BTL termination through the coupling capacitor \(C_\mathrm{f}\). Therefore, the maximum expected dc current flowing from the dc power supply depends on the number of elements \(M\), the constant voltage \(W_0\) and the resistance \(R_\mathrm{r}\). In this case, assuming that the diode \(D_\mathrm{r}\) is lossless, the expected dc current is \(I_\mathrm{dc} =27(4/10000) = 10.8\) mA.

4 Impact of the Impedances in the dc Pattern↩︎

4.1 Theoretical Analysis and Experimental Results↩︎

To attain the full capabilities of the wave-controlled RIS, the maximum voltage of \(w(x_m)\) measured after the rectifiers must be high enough to enable the widest possible reflection phase range. This voltage amplitude is influenced by the proper selection of the frequency \(f_\mathrm{b}\) and the BTL termination, either short- or open-circuited. For a short-circuited termination at \(x=-L_{\mathrm{left}}\), the input impedance at \(x = L+L_{\mathrm{right}}\) is given by \(Z_{\mathrm{in}} = jZ_0 \mathrm{tan}\left(\kappa\right)\), where \(Z_0\) is the characteristic impedance of the microstrip and \(\kappa= (\omega_\mathrm{b} n_{\mathrm{slow}}/c)L_{\mathrm{tot}}\). In the particular case where \(f_\mathrm{b}=nf_\mathrm{b,0}\), with \(n=1,2,3,...\), and \(f_\mathrm{b,0}\) is chosen such that the wavelength \(\lambda_{\mathrm{b,0}} = c/(n_{\mathrm{slow}} f_{\mathrm{b,0}}) = 4 L_{\mathrm{tot}}\), the input impedance is \(Z_{\mathrm{in},n} = jZ_0\mathrm{tan}\left(n\pi/2\right)\). Under this condition, \(Z_{\mathrm{in},n}\) becomes infinite for odd multiples of \(f_\mathrm{b,0}\) (i.e., \(n=1,3,5,...\)), and zero for even multiples (i.e., \(n=2,4,6,...\)). As a result, the voltage at \(x = L+L_{\mathrm{right}}\) reaches a maximum for odd multiples and a minimum for even multiples of \(f_\mathrm{b,0}\).

In order to experimentally compare the voltage distribution along the BTL for even and odd multiples of \(f_\mathrm{b,0}\), we use the experimental setup shown in Fig. 8, which includes an AWG, power supply, decoupling elements, and BTL. For the BTL design presented here, \(f_\mathrm{b,0} = 7.18\) MHz. After setting \(V_\textrm{g}=10\) V, Fig. 9 shows the voltages \(w(x_m)\) measured with a multimeter at positions \(x_m = md_x\), \(m = 1,2,3,...\), after each rectifier along the BTL terminated in a short circuit, for six BSWs with frequencies that are multiples of \(f_\mathrm{b,0}\). The results reveal a maximum amplitude difference exceeding \(3\) V between odd and even multiples of \(f_\mathrm{b,0}\). However, the minimum values remain close to \(W_0\) in all cases, slightly falling below \(W_0\) due to losses associated with the rectifiers. The voltages at the first and last elements show slight deviations from the expected values, with these deviations increasing at higher frequencies since the measurements were taken half a period \((d_x/2)\) before the ends of the BTL, corresponding to the lengths of \(L_\mathrm{left}\) and \(L_\mathrm{right}\).

Similarly, for the case where the BTL is terminated in an open circuit at \(x=-L_{\mathrm{left}}\), the input impedance at \(x=L+L_{\mathrm{right}}\) is \(Z_{\mathrm{in}} = jZ_0 \mathrm{cot}\left(\kappa\right)\). By choosing \(f_\mathrm{b}=nf_\mathrm{b,0}\) as before, the input impedance for \(n\)-multiples of \(f_\mathrm{b,0}\) becomes \(Z_{\mathrm{in},n} = jZ_0\mathrm{cot}\left(n \pi/2\right)\). This causes \(Z_{\mathrm{in},n}\) to become infinite for even multiples of \(f_\mathrm{b,0}\) and zero for odd multiples, resulting in a maximum voltage at \(x = L+L_{\mathrm{right}}\) for odd multiples and a minimum for even multiples of \(f_\mathrm{b,0}\). Fig. 10 presents a comparison of the voltages \(w(x_m)\) measured along the BTL with an open-circuit termination. Similar to previous cases, the results reveal a maximum amplitude difference of over \(3\) V between odd and even multiples of \(f_\mathrm{b,0}\), with higher amplitudes for even multiples in this case. As before, the minimum values remain close to \(W_0\) for all cases, while slight deviations are observed in the first and last elements due to the \(d_x/2\) shift before the BTL endpoints.

The difference observed in the amplitude for open- and short-circuited terminations is attributed to the effect of the output impedance \(Z_{\mathrm{g}}\) of the AWG that injects the periodic time-domain signal at \(nf_\mathrm{b,0}\). As an example, Fig. 11 shows the maximum dc biasing voltage \(w_\mathrm{max}=\max\limits_{m}\;w(x_m)=W_\mathrm{b}+W_{\mathrm{0}}\) versus the frequency \(f_\mathrm{b}\) for a few values of the AWG impedance \(Z_{\mathrm{g}}\) from \(Z_0\) (\(19.23\) \(\Omega\), red curve) to \(100\) \(\Omega\) (green curve), assuming a short-circuit termination. These results are obtained using an analytical model of the BTL.

We observe that for frequencies that are odd multiples of \(f_\mathrm{b,0}\), \(w_\mathrm{max}\) remains constant and reaches its largest possible value. On the contrary, for frequencies that are even multiples of \(f_\mathrm{b,0}\), \(w_\mathrm{max}\) varies depending on the impedance of the AWG. This behavior arises because the relationship \(W_\mathrm{b}=(Z_0/Z_{\mathrm{g}})V_{\mathrm{g}}\) applies for even multiples of \(f_\mathrm{b,0}\), while for odd multiples we have \(W_b=V_\mathrm{g}\). Consequently, to maximize \(w_\mathrm{max}\) for both even and odd multiples of \(f_\mathrm{b,0}\), the BTL should be designed with a characteristic impedance \(Z_0\) that matches the AWG impedance \(Z_{\mathrm{g}}\). The mathematical derivations supporting this analysis are provided in Appendix B. In summary, from a practical perspective, two different design criteria can be employed to maximize \(w_\mathrm{max}\) and ensure that it remains constant for all frequencies. The first approach involves selecting \(f_\mathrm{b} = nf_\mathrm{b,0}\) with \(n\) odd, which guarantees maximization regardless of the relationship between \(Z_\mathrm{g}\) and \(Z_0\). For the second approach, the BTL is designed with \(Z_0 = Z_\mathrm{g}\), which ensures maximization of \(w(x_m)\) for every value of \(f_\mathrm{b}\). The second approach offers an additional advantage when multiple BSWs are used [16], as it effectively doubles the number of biasing modes within the same frequency range while also making \(f_\mathrm{b}\) independent of the total length \(L_{\mathrm{tot}}\).

4.2 Comparison with Numerical Model↩︎

In addition to the analytical model, a numerical implementation of the BTL is developed in ADS to extend the analysis. The simulation setup shown in Fig. 12 includes an AWG modeled as an ideal ac source of amplitude \(V_\mathrm{g}=10\) V in series with an impedance of \(Z_\mathrm{g} = 50\) \(\Omega\), an ideal dc power source, and the required decoupling elements. These components are connected to the rightmost unit cell of the BTL after a transmission line segment of length \(L_\mathrm{right}\) with characteristic impedance \(Z_0\). Each unit cell has a rectifier circuit composed of a diode (\(D_\mathrm{r}\)), a resistor (\(R_\mathrm{r}\)), and a capacitor (\(C_\mathrm{r}\)). The diode is modeled by importing the touchstone file provided by the vendor. Unit cells are cascaded and modeled as identical transmission lines of length \(L_\mathrm{p}\), each with the same rectification circuit. On the left side, the unit cell is connected to a transmission line segment of length \(L_\mathrm{left}\), followed by the capacitor “C_f_left” that is either connected to ground, modeling a short-circuit termination as shown in the figure, or left floating, modeling an open-circuit termination. In this ADS configuration, both decoupling capacitors, labeled “C_f_left” and “C_f_right”, are assigned the same value \(C_\mathrm{f}\).

Fig. 13 compares the voltage \(w(x_m)\) evaluated using the ADS model with that from experimental results for two different BSW frequencies: one using an even multiple of \(f_\mathrm{b,0}\) and the other using an odd multiple, both with a short-circuit termination. The results show good agreement between simulation and measurement in the case of the even multiple of \(f_\mathrm{b,0}\), where \(w_\mathrm{max}\) is minimized. The discrepancy between simulation and measurement increases with distance from the short-circuit termination. This can be attributed to the fact that the TL segments that model the BTL in ADS are straight and therefore do not account for geometric effects that are neglected in the model, such as the curvature present in the meanderline design, as well as coupling effects between parallel segments.

Fig. 14 compares the reflection-phase distribution along the RIS, determined using the relationship in Fig. 4 () using both the ADS and measured voltages. As expected, the case with the even multiple \(2f_\mathrm{b,0}\) exhibits good agreement between simulation and measurement, although the phase variation is limited, ranging from approximately \(-150°\) to \(-45°\) due to the relatively narrow range of biasing voltages. In contrast, the odd multiple case, \(5f_\mathrm{b,0}\), shows a much broader phase variation, ranging from about \(-150^\circ\) to \(120^\circ\), but with a more noticeable mismatch between ADS and experimental results for large \(m\). Finally, Fig. 15 illustrates the expected scattered radiation patterns calculated using the field reflection magnitude and phase for each RIS element obtained from both the simulated and measured cases. The scattered pattern is evaluated using the normalized array factor, defined as \[\label{eq:NormAF} F\left(\theta\right)=\frac{1}{M}\sum_{m=0}^{M-1}R_{m}e^{jmkd_x\sin\left(\theta\right)+j\alpha_{m}},\tag{6}\] where \(\alpha_m\) and \(R_m\), previously shown in Fig. 4 () , represent the phase and magnitude of the reflection coefficient for the \(m\)-th unit cell given the applied voltage \(w(x_m)\) as determined by (4 ). For the two selected values \(2f_\mathrm{b,0}\) and \(5f_\mathrm{b,0}\), the resulting radiation patterns concentrate the reflected energy near \(\theta = 0^\circ\) and \(\theta = \pm 32^\circ\). However, the key takeaway is the comparison between the two cases, where, as expected, the even multiple of \(f_\mathrm{b,0}\) yields better agreement.

5 Reflective Control Using a Single Biasing Frequency↩︎

We study the beam- and null-steering capabilities of the wave-controlled RIS with a single biasing frequency, as visualized in Fig. 16. Although using a single biasing frequency does not allow the complete reflective control permitted by multiple BSWs along the transmission line [16], it will be shown that this strategy can be used for reflective control in the following ways: first for creating nulls at broadside which can, for example, enable designs to minimize the radar cross section, and second for single beam steering, with limitations at large angles. Further possibilities for extending this control will also be discussed.

The radiation pattern plots in the analysis below result from the proposed design assuming the BTL is lossless and impedance matched to the AWG, but the analysis applies in more general settings as well. These assumptions allow (4 ) to be used to determine the exact voltage seen at different points along the BTL. It is important to note that when the BTL is impedance matched to the AWG, the length of the BTL to the right of the last unit cell, \(L_\mathrm{right}\), has no impact on the voltage pattern seen by the element rectifiers.

5.1 Nulling the Specular Reflection↩︎

For many applications, including those described in [21] and [22], it is necessary to reduce the specular reflection of the incident radiation. Maximizing the reflection towards a different direction does not necessarily fully diminish the specular reflection, due to the non-perfect phase gradient along the RIS. The wave-controlled RIS with multiple BSWs described in [16] was able to suppress the specular component by explicitly including it as a constraint in its optimization criteria. The wave-controlled RIS with a single biasing frequency can also aim at reducing specular reflection, but with fewer control parameters.

Fig. 17 (a) shows the RIS reflection at \(\theta=0^\circ\) for the above design due to normal incident illumination against the biasing frequency \(f_\mathrm{b}\) and the wave amplitude \(W_\mathrm{b}\) in (4 ). The dc bias, \(W_0\), is not considered to be a tunable parameter here and is set to 4 V as this is the minimum biasing voltage required for the varactor specified in this design (see Table 1), and increasing the dc bias will reduce the range of controllable phases. From this colormap, it can be seen that for most combinations of frequency and wave amplitude, the specular reflection at \(\theta^\circ=0\) is non-negligible. However, there are combinations of these parameters that result in significantly reduced specular reflection. The region of reduced reflection begins at 2 MHz with \(W_\mathrm{b}=12\) V, and is approximately constant at \(W_\mathrm{b} = 4\) V for frequencies larger than 7 MHz, although the magnitude fluctuates with frequency. The minimal reflections occurring for \(W_\mathrm{b}=4\) V coincide with the approximately linear region of the voltage-to-phase relationship shown in Fig. 4 ().

The region of minimal specular reflection results in the radiation being directed elsewhere, as seen by comparing Figs. 17 (a) and 17 (b) where we observe radiation at \(\theta=\pm 10^\circ\). Fig. 17 (b) shows that the redirected radiation at \(\theta=-10^\circ\) is maximized for frequencies lower than \(10\) MHz without a symmetric peak at \(\theta=10^\circ\); the symmetric peak often arises with beams steered to larger angles [23].

5.2 Single Beam Control Near Broadside↩︎

For single beam control, the ideal phase difference between adjacent RIS elements is \[\label{eq:SingleBeam} \alpha_{m+1}-\alpha_m = -2\pi d_x \sin(\theta_{\mathrm{p}})/\lambda_\mathrm{c},\tag{7}\] where \(\theta_{\mathrm{p}}\) is the desired beam angle (peak radiation direction) and \(\lambda_\mathrm{c}\) is the free-space wavelength of the carrier [24]. Hence, the phase for the element at \(x_m = md_x\) on the BTL is \[\label{eq:SingleBeamElementPhase} \alpha_m = \alpha_0 -2\pi x_m \sin(\theta_{\mathrm{p}})/\lambda_\mathrm{c}.\tag{8}\] For large angles, achieving this phase gradient across the RIS leads to phase wrapping. However, for small angles where \(|\alpha_{M-1}-\alpha_0|<360^\circ\) for which phase wrapping is not required, this linear phase response along the RIS can be created by a single BSW. This method works best when \(L_\mathrm{left}\) is small as it allows for maximum phase variation across the elements. This also follows from the discussion in Sec. IV as the length \(L_\mathrm{right}\) does not impact the voltage pattern seen by element \(M-1\) when the BTL is impedance matched to the AWG.

There are two aspects of this design that limit the linear phase gradient from being exactly reproduced across the board. The first is the nonlinearity of the voltage-to-phase relationship as seen in Fig. 4 (). This problem can be addressed by using a wideband unit cell design, which would have a more linear voltage-to-phase relationship, such as the dual resonant element presented in [25]. The second factor is the sinusoidal nature of the voltage gradient across \(L\). For the short-circuit termination, the voltage gradient is approximately linear when \((2\pi f_\mathrm{b} n_{\mathrm{slow}}(x_{M-1}+L_\mathrm{left}))/c \ll 1\), allowing the small-angle approximation to be applied to (4 ). Beyond this limit and for either termination, linearity diminishes but the phase gradient is still monotonically increasing or decreasing along the BTL while \(\lambda_{\mathrm{b}} > 4(L+L_\mathrm{left})\). This condition still allows for a highly directive beam with relatively low sidelobes and a minor loss of power at progressively larger angles, as seen in Fig. 18. The wavelength \(\lambda_{\mathrm{b}} > 4(L+L_\mathrm{left})\) will limit the maximum allowed \(\theta_\mathrm{p}\) as it corresponds to the case where \(|\alpha_{M-1}-\alpha_0|<360^\circ\). Beyond this limit, when \(\lambda_{\mathrm{b}} < 4(L+L_\mathrm{left})\), the phase gradient is no longer monotonic leading to the generation of other radiation peaks specifically in the \(-\theta_{\mathrm{p}}\) direction.

Due to these limitations, a trade-off exists between the length of the array, \(L\), and the maximum usable angle for beam steering, \(\theta_\mathrm{p}\). Larger arrays will have smaller beamwidths, yet will be more limited in beam steering angles. This is shown in Fig. 18 which illustrates two beam steering examples for both the open- and short-circuit terminations for a wave-controlled RIS with either 27 or 60 elements. At the specified angles, optimal \(f_\mathrm{b}\) and \(W_\mathrm{b}\) are obtained through a search of the parameter space (see table 2), while \(\theta_p=0^\circ\) is maximized when \(W_\mathrm{b}=0\) for any frequency. All of the patterns shown in the figure maintain the \(|\alpha_{M-1} - \alpha_0| < 360^\circ\) requirement since the radiation patterns for larger \(\theta_\mathrm{p}\) would have rapidly decreasing directivity in the desired direction.

5.3 Single Beam Control for Large Angles↩︎

For the wave-controlled RIS with a single biasing frequency, significant side lobes are created which limit the effectiveness of single beam-steering when the required phase range exceeds \(360^\circ\). However, extending the application of this single-frequency methodology to these larger angles could be useful in applications where lower directivity and large side lobes are acceptable. To study this case, we apply a method for optimizing \(f_b\) and analyze the resulting radiation patterns.

To simplify the optimization of \(f_\mathrm{b}\), the conditions under which the BSW’s phase gradient approximates the slope of the ideal phase gradient are found. This occurs when \(\lambda_\alpha = \lambda_\mathrm{b}/4\), where \(\lambda_\alpha\) is the spatial period of the ideal wrapped phase gradient. The spatial period is defined as \(\lambda_\alpha=|x_{m_1}-x_{m_2}|\), where \(x_{m_{1,2}}\) are positions along the BTL with reflected phases \(\alpha_{m_{1,2}}\) such that \(|\alpha_{m_1}-\alpha_{m_2}|=2\pi\). Using (8 ), we find \(\lambda_\alpha = |\sin{(\theta_\mathrm{p})|/\lambda_\mathrm{c}}\). This yields the following relationship between a given biasing frequency and the beam steering angle: \[\label{eq:DesignedAngle} f_\mathrm{b} = \frac{f_\mathrm{c} \left|\sin{(\theta_{\mathrm{p}})}\right|}{4 n_{\mathrm{slow}}}.\tag{9}\]

Several cases of radiation patterns using four biasing frequencies that are multiples of \(f_{\mathrm{b},0}\) are shown in Fig. 19, each leading to different peak-radiation angles. These radiation patterns illustrate the limitations of the technique discussed in this section. The blue line corresponds to the limit of the near-broadside beam steering when \(|\alpha_{M-1}-\alpha_0|=360^\circ\). The orange and purple lines correspond to biasing frequencies that create symmetric standing wave voltages around the center of the BTL (\(w(x_m) = w(x_{M-1-m})\)) in (4 )) and thus symmetric beam-splitting radiation patterns around \(\theta=0°\). The line in yellow shows that, when the standing wave voltage is not symmetric and \(|\alpha_{M-1}-\alpha_0|>360^\circ\), the radiation pattern has large sidelobes that reduce the directivity towards \(\theta_\mathrm{p}\). This figure also compares the radiation patterns for the current design, shown in Fig. 19 (a), to those of a lossless design with unit cells that have a linear voltage-to-phase relationship for the carrier signal \(f_\mathrm{c}\), shown in Fig. 19 (b). The linear relationship is calculated by \(\alpha_m=360^\circ*(V-V_\mathrm{min}))/(V_\mathrm{max}-V_\mathrm{min})\) where \(V\) is the voltage applied to the unit cell, and \(V_\mathrm{min}=4\) and \(V_\mathrm{max}=15\) are the respective minimum and maximum voltages allowed for the design. These results illustrate that the beam steering method can be improved by using wider bandwidth elements [26].

An alternative approach for beam steering at large angles that uses the same optimization methods is to split the RIS into several segments with distinct BSWs. This method comes with certain tradeoffs [23], but it avoids the creation of undesirable symmetric beams around \(\theta = 0^\circ\). Additionally, the beam steering angle can be extended using multi-resonant unit cells with a phase variation limit larger than \(360^\circ\) [25].

6 Conclusions↩︎

This paper has presented a comprehensive study of the design and modeling of a wave-controlled RIS, providing both analytical and numerical frameworks to characterize its performance. The designed RIS integrates a varactor-tuned unit cell along with a BTL and rectifiers that extract the required biasing voltage for the RIS. The effectiveness of this approach in generating the correct dc bias from a single standing wave frequency is validated through theoretical analysis and experimental verification. Furthermore, key design criteria for optimizing the waveguide voltages are identified, including the selection of an appropriate fundamental biasing frequency and its odd multiples, as well as matching the characteristic impedance of the BTL to the signal generator impedance, effectively doubling the number of biasing modes within the same frequency range. Additionally, this work demonstrates the feasibility of achieving reflective control of the RIS using a single standing wave frequency, paving the way for more efficient and cost-effective implementations of dynamically reconfigurable metasurfaces.

Acknowledgment↩︎

This work is supported by the US National Science Foundation Award ECCS-2030029, and by a National Science Foundation scholarship funded under grant DUE-1930546. The authors thank DS SIMULIA for providing CST Studio Suite.

7 Derivation of the Analytical Model of the RIS Unit Cell↩︎

The realistic RIS circuit model shown in Fig. 3 is used to evaluate the reflection coefficient of the RIS [18], [3]. The equivalent impedance of the RIS, \(Z_\mathrm{eq}\), seen at the surface level by a normally-incident plane wave, assuming that the varactor is not yet connected, is given by \[\label{eq:zin} Z_\mathrm{eq}= \left (R_\mathrm{d}+j\omega L_\mathrm{d}+ \frac{1}{j\omega C_\mathrm{d}}\right )\, || \,j\omega L_\mathrm{s}.\tag{10}\] After some algebraic manipulations, this expression simplifies to \[Z_\mathrm{eq}= \frac{j\omega L_\mathrm{s}\left (1+j\omega R_\mathrm{d}C_\mathrm{d}-\omega^{2}\left (L_\mathrm{d}C_\mathrm{d}\right )\right )}{1+j\omega R_\mathrm{d}C_\mathrm{d}-\omega^{2}C_\mathrm{d}(L_\mathrm{d} + L_\mathrm{s})}.\]

This expression can be reformulated in terms of the so-called electric resonance, \(\omega_\mathrm{e}\), and magnetic resonance, \(\omega_\mathrm{m}\). At \(\omega_\mathrm{e}\), the reflection phase is nearly \(180°\), while when it is near (but not exactly at) \(\omega_\mathrm{m}\), the reflection phase is approximately \(0°\). The magnetic resonance also plays a key role in manipulating the phase of the reflection coefficient and enabling its tunability [27]–[29]. These two resonances are defined as

\[\omega_\mathrm{e}^2 = \frac{1}{C_\mathrm{d} L_\mathrm{d}}, \quad \omega_\mathrm{m}^2 = \frac{1}{C_\mathrm{d}\left (L_\mathrm{d}+L_\mathrm{s}\right )},\]

leading to the final expression for the impedance at the surface of the RIS unit cell:

\[Z_\mathrm{eq}= \frac{j\omega L_\mathrm{s}\left (1+j\omega R_\mathrm{d}C_\mathrm{d}-\left (\omega/\omega_\mathrm{e}\right )^{2}\right )}{1+j\omega R_\mathrm{d}C_\mathrm{d}-\left (\omega/\omega_\mathrm{m}\right )^{2}}. \label{eq:Zeq95we95wm}\tag{11}\]

Numerical values for the circuit elements \(L_\mathrm{d}\), \(C_\mathrm{d}\), and \(R_\mathrm{d}\) are determined by performing a single full-wave simulation of the RIS unit cell without including the varactor. In our case, we use CST Studio Suite to model the RIS under an orthogonally incident plane wave traveling along the \(-z\)-axis, thereby accounting for mutual coupling among the RIS elements. The model also incorporates copper losses and the relevant dielectric material properties. From this simulation, the scattering parameters are extracted, and the impedance \(Z^\mathrm{FW}_\mathrm{eq}\) is computed from the Z-parameters to determine the values of \(\omega_\mathrm{e}\) and \(\omega_\mathrm{m}\). Once \(\omega_\mathrm{e}\) and \(\omega_\mathrm{m}\) are determined, the circuit element values can be obtained as follows:

• The equivalent inductance \(L_\mathrm{s}\) is analytically derived by modeling the dielectric substrate as a transmission line of length \(D\) in the \(z\) direction, with a ground plane termination. Using a Taylor series expansion, the impedance expression \(j\omega L_{\mathrm{s}}=j\sqrt{\mu_0/(\epsilon_0 \epsilon_r)} \tan (\omega\sqrt{\epsilon_r}D /c)\) leads to the approximate value

  \[L_\mathrm{s} \approx \mu_0D.\]

• Given \(L_\mathrm{s}\), the inductance \(L_\mathrm{d}\) is obtained as

  \[L_\mathrm{d}= \frac{L_\mathrm{s}}{\left(\omega_\mathrm{e}/\omega_\mathrm{m}\right)^2-1}.\]

• With \(L_\mathrm{d}\) known, the capacitance \(C_\mathrm{d}\) is determined using

  \[C_\mathrm{d} = \frac{1}{L_\mathrm{d} \omega_\mathrm{e}^2}.\]

• Finally, the resistance \(R_\mathrm{d}\) is obtained from the real part of \(Z^\mathrm{FW}_\mathrm{eq}\) evaluated at \(\omega_\mathrm{m}\) from the full-wave simulation (at this frequency, the imaginary part is independent of \(R_\mathrm{d}\)):

  \[R_\mathrm{d} = \frac{L_\mathrm{s}}{C_\mathrm{d}\left(1+L_\mathrm{d}/L_\mathrm{s}\right)\Re\left(Z^\mathrm{FW}_\mathrm{eq}(\omega_\mathrm{m})\right)}.\]

8 Maximization of \(W_\mathrm{b}\) - Short-circuit Termination↩︎

The amplitude \(W_\mathrm{b}\) of the induced standing wave can be calculated as a function of the voltage \(V_\mathrm{in}\) at the connection of the BTL with the signal generator: \[W_\mathrm{b} =\left|\frac{2V_\mathrm{in}}{e^{-j\kappa} - e^{j\kappa}} \right|,\] where \(\kappa= (\omega_\mathrm{b} n_{\mathrm{slow}}/c)L_{\mathrm{tot}}\). Assuming a BTL terminated in a short circuit, the input impedance of the BTL is \(Z_\mathrm{in} = jZ_0 \mathrm{tan}\left( \kappa\right)\). Furthermore, assuming that at frequency \(f_\mathrm{b}\) the capacitances \(C_\mathrm{f}\) can be approximated as short circuits and the inductance \(L_\mathrm{f}\) as an open circuit, we have \[V_\mathrm{in} = \frac{Z_\mathrm{in}}{Z_\mathrm{in} + Z_{\mathrm{g}}}V_{\mathrm{g}} = \frac{jZ_0\tan(\kappa)}{jZ_0\tan(\kappa) + Z_{\mathrm{g}}}V_{\mathrm{g}}.\] The expression for \(W_\mathrm{b}\) can thus be written as a function of the impedances \(Z_0\) and \(Z_{\mathrm{g}}\): \[W_\mathrm{b} =\left|\frac{2\frac{jZ_0\tan(\kappa)}{jZ_0\tan(\kappa) + Z_\mathrm{g}}V_\mathrm{g}}{e^{-j\kappa} - e^{j\kappa}} \right|. \label{eq:W95n95definition}\tag{12}\]

When considering the case with \(f_\mathrm{b}=nf_\mathrm{b,0}\), the wavelength is \(\lambda_\mathrm{b}=4L_\mathrm{tot}/n\), resulting in \(\kappa=n\pi/2\). When \(f_\mathrm{b}\) is an odd multiple of \(f_\mathrm{b,0}\), we find that \(\kappa=n \pi /2\), so \(\tan(\kappa)=\infty\) and \(e^{-j\kappa} - e^{j \kappa}=-j2\), leading to \(W_\mathrm{b}=V_\mathrm{g}\), which as shown in Fig. 11 is always at its maximum value. However, when \(f_\mathrm{b}\) is an even multiple of \(f_\mathrm{b,0}\), \(\tan(\kappa)=0\) while \(e^{-j\kappa} - e^{j\kappa}=0\), making it necessary to evaluate \(W_\mathrm{b}\) in the limit as \(\kappa\to n\pi/2\) for even \(n\), leading to

\[W_\mathrm{b} = \frac{Z_0}{Z_{\mathrm{g}}}V_{\mathrm{g}}.\]

This result indicates that when \(Z_0 = Z_{\mathrm{g}}\), the amplitude of the standing wave equals \(V_{\mathrm{g}}\), the same value obtained when \(f_\mathrm{b}\) is an odd multiple of \(f_\mathrm{b,0}\). In fact, under this matching condition, the specific value of \(\kappa\) becomes irrelevant. Indeed, evaluating (12 ) with \(Z_0 = Z_{\mathrm{g}}\) gives \[W_\mathrm{b} \Bigr\rvert_{Z_0 = Z_{\mathrm{g}}} =\left|\frac{2\frac{j\tan(\kappa)}{j\tan(\kappa) + 1}V_\mathrm{g}}{e^{-j\kappa} - e^{j\kappa}} \right|.\] This expression simplifies to \[\begin{align} W_\mathrm{b} \Bigr\rvert_{Z_0 = Z_{\mathrm{g}}} &=\left|\frac{-\tan(\kappa)}{j\sin(\kappa)\tan(\kappa) + \sin(\kappa)}V_\mathrm{g} \right| = V_\mathrm{g}. \end{align}\]

References↩︎