Beyond the Bode-Fano Bound: Wideband Impedance Matching for Short Pulses Using Temporal Switching of Transmission-Line Parameters

Impedance matching is one of the most important practices in wave engineering as it enables one to maximize the power transfer from the signal source to the load in the wave system. Unfortunately, it is bounded by the Bode-Fano criterion which states that, for any passive, linear, and time-invariant matching network, there is a stringent trade-off between the matching bandwidth and efficiency, implying severe constraints on various electromagnetic and acoustic wave systems. Here, we propose a matching paradigm that overcomes this issue by using a temporal switching of the parameters of a metamaterial-based transmission line, thus revoking the time-invariance assumption underlying the Bode-Fano criterion. Using this scheme we show theoretically that an efficient wideband matching, beyond the Bode-Fano bound, can be achieved for short-time pulses in challenging cases of very high contrast between the load and the generator impedances, and with significant load dispersion, situations common in, e.g., small antenna matching, cloaking, and with applications for ultrawideband communication, high resolution imaging, and more.

Figure 1

(a) A generator excites a short-time baseband signal propagating toward a dispersive load with $R_L=5\mathrm{\Omega }$, and $C_L=3.14\mathrm{nF}$ (${\tau }_L=R_L{C}_L=15.7\mathrm{ns}$). The signal bandwidth is $\mathrm{BW}=100\mathrm{MHz}\to T_p=62.8\mathrm{ns}$. Thus, $\overline{T}={\tau }_L/T_p=0.25<1$, indicating a highly dispersive load [12, Sec. 5]. (b) Assuming a passive LTI matching network, the Bode-Fano bound on the source-load transmission efficiency ${\eta }_{\mathrm{BF}}$ for the model discussed in (a) is shown as a function of the energy ratio (ER) between the excited pulse energy within the matching network passband, and the total excited energy (see (Ref. [12], Sec. 5)). Optimal matching in this case yields ${\eta }_{\mathrm{BF}}\approx 0.15$ at $\mathrm{ER}\approx 0.65$. (c) The transmission and group delay of a nearly optimal, passive, LTI matching network based on a 7’th order Chebyshev design. Its bandwidth, $\sim 40\mathrm{MHz}$ yields the optimal value $\mathrm{ER}=0.65$. In its passband, the network is highly dispersive, implying, besides spectrum chopping also an unavoidable additional distortion of the transmitted signal. (d) A distortionless, though suboptimal, matching can be achieved by discarding the matching network in (a) and instead optimizing on $Z_c$ and $R_g$ to maximize energy delivery. The efficiency in this case, as a function of the contrast $\rho =R_L/R_g$ is shown by the red line, which is always below the Bode-Fano limit. However, using the switched TL scheme discussed in this Letter, the matching efficiency can significantly exceed the bound as indicated by the blue line and with minimal signal distortion, as discussed below.

Figure 2

(a) Physical layout of the switched TL system. (b) A possible realization of the switched TL as a discrete periodic structure of parallel varactor diodes and banks of switched series inductors. (c) For $Z_L=R_L$. The ratio between the TL parameters required for the optimal realization versus load-generator contrast $\rho$: $Z_{c2}/Z_{c1}$ in blue line and $v_2/v_1$ in brown line. (d) Continuous line, the optimal efficiency (with switching); dashed line, the efficiency for the “no switching” case.

Figure 3

(a) The optimal value of $x=Z_c/R_g$ as a function of the contrast, $\rho$, and the amount of energy balance $\delta$. (b) Equivalent circuit model for $0.5<\delta <1$ and high generator-load contrast $\rho$. (c) The optimal value, the efficiency, $\eta$ as a function of $\rho$ and $\delta$.