Corrigendum: Analytic model for ultrasound energy receivers and their optimal electric loads (2017 Smart Mater. Struct. 26 085003)

From the wave equation (equation (6)) [3] and with the viscosity variable N (equation (22)), the complex stiffened wave-speed ${v}_{p}^{{st}}$ is [1]

$$\begin{eqnarray}&&{v}_{p}^{{st}}={v}_{p}\left(1+j\displaystyle \frac{\omega \mu }{2\rho {v}_{p}^{2}}\right)={v}_{p}(1+{jN})={v}_{p}\left(1+\displaystyle \frac{j}{2{Q}_{m}}\right),\end{eqnarray} \tag{ 1 }$$

which should be used, instead of the loss-less wave-speed v_p defined in equation (7), to define the stiffened acoustic impedance of the piezoelectric material ${Z}_{p}^{{st}}$ in equation (14) as

$$\begin{eqnarray}&&{Z}_{p}^{{st}}={\rho }_{p}{{Av}}_{p}(1+{jN})={Z}_{p}(1+{jN}).\end{eqnarray} \tag{ 2 }$$

Consequently, we introduce ${Z}_{p}^{{st}}$ in the expressions for the particle displacement $\bar{\alpha }$ and $\bar{\beta }$ (equations (15)) with the changes highlighted in bold to obtain

$$\begin{eqnarray}&&\bar{\alpha }=\displaystyle \frac{-{\bar{F}}_{p}^{f}}{j\omega {Z}_{p}(1+{\bf{jN}})};\ \bar{\beta }=\displaystyle \frac{{\bar{F}}_{p}^{b}}{j\omega {Z}_{p}(1+{\bf{jN}})}\exp (-{{jkt}}_{p}),\end{eqnarray} \tag{ 3 }$$

but we keep the acoustic transmission ${T}_{x}^{y}$ and reflection ${R}_{x}^{y}$ coefficients defined with the loss-less acoustic impedance Z_p as in the original paper (equations (24)).

Thus, the voltage $\bar{V}$ expression in equations (16) and (18) need to be changed accordingly and the final voltage expression is then $\bar{V}$:

$$\begin{eqnarray}\bar{V} & = & \displaystyle \frac{-h}{j\omega {Z}_{p}(1+{\bf{jN}})}({\bar{F}}_{p}^{f}+{\bar{F}}_{p}^{b})(1-\exp (-{{jkt}}_{p}))\\ & & \times \left(\displaystyle \frac{j\omega {Z}_{{el}}{C}_{0}}{1+j\omega {Z}_{{el}}{C}_{0}}\right).\end{eqnarray} \tag{ 4 }$$

Following the same procedure described in the paper, we then reach the corrected force expressions of equations (28):

$$\begin{eqnarray}&&{\bar{F}}_{p}^{f}=\displaystyle \frac{1+{\bf{jN}}}{1+{jN}\tfrac{{{\bf{T}}}_{a}^{p}}{{\bf{2}}}-{{jT}}_{a}^{p}M}\phantom{\rule{}{3.75ex}}(\displaystyle \frac{h}{2}{T}_{a}^{p}\bar{V}{C}_{0}\phantom{\rule{}{3.75ex}})\\ &&\quad +\,\left.{\bar{F}}_{p}^{b}\left({R}_{p}^{a}\exp (-{{jkt}}_{p})\left(1+j\displaystyle \frac{N(1+{{\bf{R}}}_{p}^{a})}{-{\bf{2}}{{\bf{R}}}_{p}^{a}}\right)+j\displaystyle \frac{{T}_{a}^{p}M}{1+{\bf{jN}}}\right)\right);\\ &&{\bar{F}}_{p}^{b}=\displaystyle \frac{1+{\bf{jN}}}{1+{jN}\tfrac{{{\bf{T}}}_{m}^{p}}{{\bf{2}}}-{{jT}}_{m}^{p}M}\\ &&\qquad \times \,\phantom{\rule{}{3.75ex}}({\bar{F}}_{m}^{b}{T}_{m}^{p}+\displaystyle \frac{h}{2}{T}_{m}^{p}\bar{V}{C}_{0}+\phantom{\rule{}{3.5ex}}({\bar{F}}_{p}^{f}{R}_{p}^{m}\exp (-{{jkt}}_{p})\phantom{\rule{}{3.75ex}})\\ &&\qquad \times \,\left.\left.\left(1+j\displaystyle \frac{N(1+{{\bf{R}}}_{p}^{m})}{-{\bf{2}}{{\bf{R}}}_{p}^{m}}\right)+j\displaystyle \frac{{T}_{m}^{p}M}{1+{\bf{jN}}}\right)\right)\end{eqnarray} \tag{ 5 }$$

The correct voltage expression across the attached Z_el electric load (equation (30)) is

$$\begin{eqnarray}&&\bar{V}=({{jC}}_{0}({\rm{\Theta }}-1){\bar{F}}_{m}^{b}h\omega {Z}_{{el}}{T}_{m}^{p})((1+{\rm{\Theta }}+{N}^{2}){T}_{p}^{a}\\ &&\quad +\,({\rm{\Theta }}-1)(1+{jN}){T}_{a}^{p})/\phantom{\rule{}{3.75ex}}({Z}_{p}\omega (j-{C}_{0}\omega {Z}_{{el}})\\ &&\quad \times \,\phantom{\rule{}{3.75ex}}(\displaystyle \frac{{(N-j)}^{2}}{2}((j+N){T}_{p}^{m}-{{jT}}_{m}^{p})((j+N){T}_{p}^{a}-{{jT}}_{a}^{p})\\ &&\quad -\,{{\rm{\Theta }}}^{2}(2+{{jNT}}_{a}^{p})(2+{{jNT}}_{m}^{p})\phantom{\rule{}{3.75ex}})\\ &&\quad +\,2{C}_{0}{h}^{2}({\rm{\Theta }}-1)\displaystyle \frac{({Z}_{a}+{Z}_{m}){Z}_{p}}{({Z}_{a}+{Z}_{p})({Z}_{m}+{Z}_{p})}\\ &&\quad \left.\left((1+{\rm{\Theta }}+{N}^{2})+2(1+{jN})({\rm{\Theta }}-1)\displaystyle \frac{{Z}_{p}}{{Z}_{a}+{Z}_{m}}\right)\right),\end{eqnarray} \tag{ 6 }$$

from which the dissipated power W_el at Z_el can be obtained by developing equation (34). The shortened general form of W_el compared to equation (51) is again

$$\begin{eqnarray}&&{W}_{{el}}=\displaystyle \frac{{{\rm{\Delta }}}_{1}{Z}_{{el}}^{{Re}}}{{{\rm{\Delta }}}_{2}({Z}_{{el}}^{{Re}2}+{Z}_{{el}}^{{Im}2})+{{\rm{\Delta }}}_{3}{Z}_{{el}}^{{Im}}+{{\rm{\Delta }}}_{4}{Z}_{{el}}^{{Re}}+{{\rm{\Delta }}}_{5}},\end{eqnarray} \tag{ 7 }$$

in which the Δ variables are new and represent the material coefficients. However, we do not show their full expressions in this corrigendum due to their length.

The corrected equivalent acoustic impedance of the receiver (equation (45)) is

$$\begin{eqnarray}&&{Z}_{{rec}}={Z}_{p}(1+{\bf{jN}})\displaystyle \frac{({\bar{F}}_{p}^{f}{{\rm{\Theta }}}^{-1}+{\bar{F}}_{p}^{b})+\tfrac{h\bar{V}}{j\omega {Z}_{{el}}}}{-{\bar{F}}_{p}^{f}{{\rm{\Theta }}}^{-1}+{\bar{F}}_{p}^{b}},\end{eqnarray} \tag{ 8 }$$

and the corrected Π variable (equation (47)) that represents the influence of the viscosity on the zero reflection optimal load is

$$\begin{eqnarray}&&{\rm{\Pi }}=\displaystyle \frac{{{jh}}^{2}{{\rm{\Theta }}}^{2}\sinh ({{jkt}}_{p})}{4{\omega }^{2}{Z}_{p}{({R}_{p}^{a}-{{\rm{\Theta }}}^{2}{R}_{p}^{m})}^{2}}\phantom{\rule{}{3.75ex}}(\phantom{\rule{}{3ex}}({T}_{a}^{p2}{T}_{p}^{m2}\\ &&\quad +\,(3-2{R}_{p}^{a}+3{R}_{p}^{a2}){T}_{m}^{p2}\phantom{\rule{}{3ex}})\cosh ({{jkt}}_{p})\\ &&\quad -\,2{T}_{a}^{p}{T}_{m}^{p}\phantom{\rule{}{3ex}}({T}_{p}^{a}{T}_{p}^{m}+{T}_{m}^{p}{T}_{a}^{p}+2({R}_{p}^{m}-{R}_{p}^{a})\sinh ({{jkt}}_{p})\phantom{\rule{}{3ex}})\phantom{\rule{}{3.75ex}}).\end{eqnarray} \tag{ 9 }$$

Despite the changes to the zero reflection optimal load expression, the deviation between the old and new predictions by the analytic model is less than 1% in the cases that we discussed in the paper.

The expressions of the power maximization loads (equations (53) and (54)) are

$$\begin{eqnarray}&&\mathrm{Im}[{Z}_{{Pow}}^{{opt}}]=\displaystyle \frac{1}{{C}_{0}^{{\prime} }\omega (1+{\delta }_{{tn}}^{2})}-\displaystyle \frac{{h}^{2}{{\rm{\Lambda }}}_{3}}{{\omega }^{2}{Z}_{p}{{\rm{\Lambda }}}_{1}}+\displaystyle \frac{{{Nh}}^{2}{{\boldsymbol{\Lambda }}}_{{\bf{new}}}^{{\bf{Im}}}}{2{\omega }^{2}{Z}_{p}}\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&\mathrm{Re}[{Z}_{{Pow}}^{{opt}}]=\displaystyle \frac{{\delta }_{{tn}}}{{C}_{0}^{{\prime} }\omega (1+{\delta }_{{tn}}^{2})}+\displaystyle \frac{{h}^{2}{{\rm{\Lambda }}}_{2}}{2{\omega }^{2}{Z}_{p}{{\rm{\Lambda }}}_{1}}+\displaystyle \frac{{{Nh}}^{2}{{\boldsymbol{\Lambda }}}_{{\bf{new}}}^{{\bf{Re}}}}{{\omega }^{2}{Z}_{p}}\end{eqnarray} \tag{ 11 }$$

in which only the ${{\rm{\Lambda }}}_{{new}}^{{Im}}$ and ${{\rm{\Lambda }}}_{{new}}^{{Re}}$ variables multiplied by the viscosity are new:

$$\begin{eqnarray}&&{{\rm{\Lambda }}}_{\mathrm{new}}^{\mathrm{Im}}=\displaystyle \frac{1}{{{\rm{\Lambda }}}_{1}^{2}}\left({{\rm{\Phi }}}^{3}({R}_{p}^{a}+{R}_{p}^{m}-{R}_{p}^{a2}{R}_{p}^{m}-{R}_{p}^{a}{R}_{p}^{m2})\sin \left(\displaystyle \frac{\omega }{{f}_{p}}\right)\right.\\ &&\quad \times \,\left(2{T}_{a}^{p}{T}_{m}^{p}({{\rm{\Phi }}}^{2}+{R}_{p}^{a}{R}_{p}^{m})\sin \left(\displaystyle \frac{\omega }{2{f}_{p}}\right)\right.\\ &&\quad +\,\left.{\rm{\Phi }}\sin \left(\displaystyle \frac{\omega }{{f}_{p}}\right)({R}_{p}^{m}+{R}_{p}^{a}(1+{R}_{p}^{m}({R}_{p}^{a}+{R}_{p}^{m}-4)))\right)\\ &&\quad -\,{{\rm{\Lambda }}}_{1}\left(\displaystyle \frac{{R}_{p}^{a2}+{R}_{p}^{m2}}{2}(1-{{\rm{\Phi }}}^{4})-{R}_{p}^{a2}{R}_{p}^{m}-{R}_{p}^{m2}{T}_{a}^{p}{R}_{p}^{a}\right.\\ &&\quad -\,{{\rm{\Phi }}}^{4}(1-{R}_{p}^{a}-{R}_{p}^{m})\\ &&\quad +\,{\rm{\Phi }}{T}_{a}^{p}{T}_{m}^{p}\left((1-{{\rm{\Phi }}}^{2})({R}_{p}^{a}+{R}_{p}^{m})\cos \left(\displaystyle \frac{\omega }{2{f}_{p}}\right)\right.\\ &&\quad +\,\left.\left.\left.{\rm{\Phi }}(1-{R}_{p}^{a}{R}_{p}^{m})\cos \left(\displaystyle \frac{\omega }{{f}_{p}}\right)\right)\right)\right)\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&{{\rm{\Lambda }}}_{\mathrm{new}}^{\mathrm{Re}}=\displaystyle \frac{{\rm{\Phi }}}{4{{\rm{\Lambda }}}_{1}^{2}}\phantom{\rule{}{3.75ex}}(2{T}_{a}^{p}{T}_{m}^{p}({R}_{p}^{a}+{R}_{p}^{m})\\ &&\quad \times \,\left({{\rm{\Phi }}}^{2}\sin \left(\displaystyle \frac{3\omega }{2{f}_{p}}\right)({{\rm{\Phi }}}^{2}+{R}_{p}^{a2}{R}_{p}^{m2})\right.\\ &&\quad -\,\left.\sin \left(\displaystyle \frac{\omega }{2{f}_{p}}\right)({{\rm{\Phi }}}^{6}+{R}_{p}^{a}{R}_{p}^{m}(2{{\rm{\Phi }}}^{4}+2{{\rm{\Phi }}}^{2}+{R}_{p}^{a}{R}_{p}^{m}))\right)\\ &&\quad +\,\phantom{\rule{}{3ex}}({R}_{p}^{a}{R}_{p}^{m}\phantom{\rule{}{2ex}}(3{R}_{p}^{a2}{R}_{p}^{m}-{R}_{p}^{a2}-{R}_{p}^{a}{R}_{p}^{m}\\ &&\quad -\,{R}_{p}^{m2}(1-{T}_{a}^{p}{R}_{p}^{a}(3+{R}_{p}^{a}))-{T}_{a}^{p}{R}_{p}^{a2}{R}_{m}^{m3}\phantom{\rule{}{3ex}})\\ &&\quad -\,{{\rm{\Phi }}}^{4}\phantom{\rule{}{3.50ex}}({R}_{p}^{a}-{T}_{m}^{p}+{R}_{p}^{m}{R}_{p}^{a}(2-3{R}_{p}^{a}+{R}_{p}^{a2}-3{R}_{p}^{m}\\ &&\quad +\,{R}_{p}^{a}{R}_{p}^{m}+{R}_{p}^{m2})\phantom{\rule{}{3ex}})\phantom{\rule{}{3.75ex}})2{\rm{\Phi }}\sin \left(\displaystyle \frac{\omega }{{f}_{p}}\right)\\ &&\quad +\,{{\rm{\Phi }}}^{3}\sin \phantom{\rule{}{3.75ex}}(\displaystyle \frac{2\omega }{{f}_{p}}\phantom{\rule{}{3.75ex}})\phantom{\rule{}{3.7ex}}({R}_{p}^{a2}(1-2{R}_{p}^{m}-2{R}_{p}^{m3}+{R}_{p}^{m4})\\ &&\quad +\,{R}_{p}^{m2}(1-2{R}_{p}^{a}+4{R}_{p}^{a2}-2{R}_{p}^{a3}+{R}_{p}^{a4})\phantom{\rule{}{3.75ex}})\phantom{\rule{}{3.75ex}})\end{eqnarray} \tag{ 13 }$$

Also in this case there are no significant changes in the predictions presented in the original paper with less than 1% deviation.

In the original paper [3], we compared our analytic model to the KLM model without the stiffened wave-speed and with the acoustic attenuation γ as done in [2], and saw that they are slightly different when the acoustic attenuation is very high. However, if the stiffened wave-speed is also included in the KLM model as in [1], the KLM and the analytic model predictions are equal.