diff --git a/.gitignore b/.gitignore index aaf348a..f7672dc 100644 --- a/.gitignore +++ b/.gitignore @@ -1,2 +1,3 @@ .ipynb_checkpoints .spyproject +.pdf diff --git a/archive/lrftubes_doc_latest.pdf b/archive/lrftubes_doc_latest.pdf index 915ccf1..ff5112f 100644 Binary files a/archive/lrftubes_doc_latest.pdf and b/archive/lrftubes_doc_latest.pdf differ diff --git a/img/Bruel_Kjaer_fig1.png b/img/Bruel_Kjaer_fig1.png new file mode 100644 index 0000000..c159858 Binary files /dev/null and b/img/Bruel_Kjaer_fig1.png differ diff --git a/lrftubes.lyx b/lrftubes.lyx index d3db5f1..5c834aa 100644 --- a/lrftubes.lyx +++ b/lrftubes.lyx @@ -5553,7 +5553,7 @@ p=\frac{C_{1}\exp\left(-i\Gamma x\right)+C_{1}\exp\left(-i\Gamma x\right)}{r_{0} \begin_layout Standard \begin_inset Note Note -status collapsed +status open \begin_layout Plain Layout If we assume @@ -6258,6 +6258,33 @@ Z_{c,0}=\frac{\omega\rho_{0}}{\left(1-f_{\nu}\right)S_{f,0}\Gamma_{0}} \end_layout +\begin_layout Standard +and +\begin_inset Formula $\Gamma$ +\end_inset + + is defined in eq. + +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:Gamma" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + and can be approximated as +\begin_inset Formula $\Gamma\approx k$ +\end_inset + + under some undefined and guessed circumstances AANVULLEN, in which +\begin_inset Formula $k=\frac{\omega}{c_{0}}$ +\end_inset + + . +\end_layout + \begin_layout Section Prismatic lined circular duct \end_layout @@ -7918,7 +7945,7 @@ Radiation impedance of a baffled piston \begin_inset Formula $\pi a^{2}$ \end_inset - + cross sectional area [m^2] \end_layout \begin_layout Standard @@ -8269,6 +8296,19 @@ in which can be 'measured' by averaging it over the port's boundary. \end_layout +\begin_layout Standard +\begin_inset Note Note +status open + +\begin_layout Plain Layout +TO DO: redraw image and list what approximations are used +\end_layout + +\end_inset + + +\end_layout + \begin_layout Standard \begin_inset Float figure wide false @@ -8345,10 +8385,6 @@ Electrical and mechanical model of the speaker \end_inset -\end_layout - -\begin_layout Plain Layout - \end_layout \end_inset @@ -10925,7 +10961,7 @@ T_{21} & T_{22} \end{array}\right]\left\{ \begin{array}{c} p_{o}\\ Q_{o} -\end{array}\right\} , +\end{array}\right\} ,\label{eq:transfer_matrix_COMSOL} \end{equation} \end_inset @@ -10977,6 +11013,455 @@ LookupModel . \end_layout +\begin_layout Chapter +Measuring the transmission matrix using the four microphone method +\end_layout + +\begin_layout Standard +Based on Brüel Kjaer - Transmission loss in impedance tube.pdf in /home/anne/next +cloud/wip_redusone/2021-Steegmuller/measurement_setup +\end_layout + +\begin_layout Standard +Modifications: volume flow U instead of velocity v; impedance Z instead + of characteristic impedance z; transfer functions Hir instead of cross + correlations (?). +\end_layout + +\begin_layout Standard +\begin_inset Note Note +status open + +\begin_layout Plain Layout +TO DO: +\end_layout + +\begin_layout Plain Layout +draw own image image +\end_layout + +\begin_layout Plain Layout +fix citation +\end_layout + +\begin_layout Plain Layout +Transfer matrix according to our own definition instead of the definition + of Bruel & Kjaer = definition of COMSOL +\end_layout + +\begin_layout Plain Layout +Consistently use Q or U for volume flow? Also in text above about COMSOL. +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +The transfer matrix of a device can be measured using a four microphone + setup as shown in figure +\begin_inset CommandInset ref +LatexCommand ref +reference "fig:meas_transmatrix_4mic" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +. + The microphones record acoustic pressure and plane waves are assumed. + In the following equations, time dependency +\begin_inset Formula $\exp(+j*\omega*t)$ +\end_inset + + is not shown. +\end_layout + +\begin_layout Standard +\begin_inset Float figure +wide false +sideways false +status open + +\begin_layout Plain Layout +\align center +\begin_inset Graphics + filename img/Bruel_Kjaer_fig1.png + lyxscale 50 + width 80text% + +\end_inset + + +\begin_inset Caption Standard + +\begin_layout Plain Layout +Experimental setup to measure the transfer matrix, using the four microphone + method +\end_layout + +\end_inset + + +\begin_inset CommandInset label +LatexCommand label +name "fig:meas_transmatrix_4mic" + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +The transfer matrix coefficients are calculated based on sound pressure + +\begin_inset Formula $p$ +\end_inset + + and volume velocity +\begin_inset Formula $U$ +\end_inset + +, as related by equation +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:transfer_matrix_COMSOL" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +. + Note that this definition is different than the definition used in LRFtubes + and therefore +\begin_inset Formula $T$ +\end_inset + + should be inverted for further use. + Subscrips +\begin_inset Formula $i$ +\end_inset + + and +\begin_inset Formula $d$ +\end_inset + + refer to +\begin_inset Formula $x=0$ +\end_inset + + and +\begin_inset Formula $x=d$ +\end_inset + + respectively. + There are two equations and four unknowns, so two sets of measurements + are required. + The second set, indicated by superscript +\begin_inset Formula $*$ +\end_inset + +, must be performed with a different acoustic termination. + Together this results in four equations for four unknowns. +\end_layout + +\begin_layout Standard +\align left +\begin_inset Formula +\begin{equation} +\left\{ \begin{array}{c} +p_{i}\\ +Q_{i} +\end{array}\begin{array}{c} +p_{i}^{*}\\ +Q_{i}^{*} +\end{array}\right\} =\left[\begin{array}{cc} +T_{11} & T_{12}\\ +T_{21} & T_{22} +\end{array}\right]\left\{ \begin{array}{c} +p_{o}\\ +Q_{o} +\end{array}\begin{array}{c} +p_{o}^{*}\\ +Q_{o}^{*} +\end{array}\right\} ,\label{eq:transfer_matrix-double} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +Solving for +\begin_inset Formula $T$ +\end_inset + + yields: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +\left[\begin{array}{cc} +T_{11} & T_{12}\\ +T_{21} & T_{22} +\end{array}\right]=\frac{1}{p_{d}Q_{d}^{*}-p_{d}^{*}Q_{d}}\left[\begin{array}{cc} +p_{i}Q_{d}^{*}-p_{i}^{*}Q_{d} & -p_{i}p_{d}^{*}+p_{i}^{*}p_{d}\\ +Q_{i}Q_{d}^{*}-Q_{i}^{*}Q_{d} & -p_{d}^{*}Q_{i}+p_{d}Q_{i}^{*} +\end{array}\right] +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $p$ +\end_inset + + and +\begin_inset Formula $Q$ +\end_inset + + at +\begin_inset Formula $x=0$ +\end_inset + + and +\begin_inset Formula $x=d$ +\end_inset + + can be calculated from travelling +\begin_inset Formula $A$ +\end_inset + +, +\begin_inset Formula $B$ +\end_inset + +, +\begin_inset Formula $C$ +\end_inset + + and +\begin_inset Formula $D$ +\end_inset + +. + The calculation of their second measurement counterparts +\begin_inset Formula $*$ +\end_inset + + goes analogously and uses +\begin_inset Formula $A^{*}$ +\end_inset + +, +\begin_inset Formula $B^{*}$ +\end_inset + +, +\begin_inset Formula $C^{*}$ +\end_inset + + and +\begin_inset Formula $D^{*}$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +p_{i}=A+B +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +Q_{i}=\frac{A-B}{Z_{0}} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +p_{d}=C\cdot e^{-jkd}+D\cdot e^{jkd} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +Q_{d}=\frac{C\cdot e^{-jkd}-D\cdot e^{jkd}}{Z_{0}} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +in which +\begin_inset Formula $Z_{0}=\frac{z_{0}}{S}$ +\end_inset + + is the impedance of an infinite duct, with +\begin_inset Formula $z_{0}$ +\end_inset + + the characteristic impedance and +\begin_inset Formula $S$ +\end_inset + + the cross-sectional area, +\begin_inset Formula $j=\sqrt{-1}$ +\end_inset + +, +\begin_inset Formula $k$ +\end_inset + + the wavenumber. + Travelling waves +\begin_inset Formula $A$ +\end_inset + +, +\begin_inset Formula $B$ +\end_inset + +, +\begin_inset Formula $C$ +\end_inset + + and +\begin_inset Formula $D$ +\end_inset + + can be calculated from transfer functions +\begin_inset Formula $H_{ir}$ +\end_inset + + from reference signal +\begin_inset Formula $r$ +\end_inset + +, as sent to the loudspeaker, to the recorded signal of microphone +\begin_inset Formula $i$ +\end_inset + +. + The calculation of their second measurement counterparts +\begin_inset Formula $*$ +\end_inset + + goes analogously and uses +\begin_inset Formula $H_{ir}^{*}$ +\end_inset + +. +\begin_inset Formula +\begin{equation} +A=\frac{j\left(H_{1r}\cdot e^{jkx_{2}}-H_{2r}\cdot e^{jkx_{1}}\right)}{2\sin\left(k\left(x_{1}-x_{2}\right)\right)} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +B=\frac{j\left(H_{2r}\cdot e^{-jkx_{1}}-H_{1r}\cdot e^{-jkx_{2}}\right)}{2\sin\left(k\left(x_{1}-x_{2}\right)\right)} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +C=\frac{j\left(H_{3r}\cdot e^{jkx_{4}}-H_{4r}\cdot e^{jkx_{3}}\right)}{2\sin\left(k\left(x_{3}-x_{4}\right)\right)} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +D=\frac{j\left(H_{4r}\cdot e^{-jkx_{3}}-H_{3r}\cdot e^{-jkx_{4}}\right)}{2\sin\left(k\left(x_{3}-x_{4}\right)\right)} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Note Note +status open + +\begin_layout Plain Layout +\begin_inset Formula $\sqrt{G_{rr}}$ +\end_inset + + has been removed from the equations because Caspers thinks that +\begin_inset Formula $H_{ir}$ +\end_inset + + refers to the cross spectrum instead of the transfer function. + If the transfer function is used, then +\begin_inset Formula $\sqrt{G_{rr}}$ +\end_inset + + shall be left out. +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Note: if no reference signal has been recorded, the reference signal can + be set to the signal captured by microphone 1. + The equations have no way to figure out whether the loudspeaker really + was driven by such a signal. + Then a requirement is that all microphones are recorded simultaneously + and with synchronized ADC clocks. +\end_layout + \begin_layout Chapter IEC Coupler impedances \end_layout diff --git a/lrftubes.pdf b/lrftubes.pdf index ff5112f..c4ead2d 100644 Binary files a/lrftubes.pdf and b/lrftubes.pdf differ