| (%i1) | load("/home/vcaeken/Desktop/rfMaxima.mac")$ |
Waveguide Loss (TE-10 Mode)
(See "A W-band waveguide fabricated using selective laser melting, by K. Van Caekenberghe et al.,
Wiley Microwave and Optical Technology Letters, vol. 54, no. 11, p. 2572–2575, Nov. 2012)
1 Constants
| (%i2) |
%epsilon_0:8.854e-12$ %mu_0:4*%pi*1e-7$ %eta:float(sqrt(%mu_0/%epsilon_0)); c:float(sqrt(1/(%epsilon_0*%mu_0))); k(f):=2*%pi*f/c; |
2 Design Characteristics
2.1 Material Parameters
| (%i7) |
%sigma_Au:41000410$ %sigma_Cu:5.8e7$ %sigma_Ti6Al4V:5.8e5$ R_q:1.25e-6$ |
2.2 Waveguide Dimensions
WR-28 26.5-40 GHz 0.28 x 0.14 inch
WR-10 75-110 GHz 0.1 x 0.05 inch
WR-3 220-325 GHz 0.034 x 0.017 inch
| (%i11) |
a:0.1*0.0254$ b:0.05*0.0254$ t:0.01*0.0254$ f_0:94e9$ |
3 Complex Propagation Constant, Wave Impedance
| (%i15) |
%lambda_c:2*a; f_c:c/(2*a); %delta(%mu,%sigma,%omega):=sqrt(2/(%omega*%mu*%sigma)); R_s(%delta,%sigma):=1/(%sigma*%delta); define(%alpha(%mu,%sigma,f),block( [%delta], %delta:%delta(%mu,%sigma,2*%pi*f), ratsimp(R_s(%delta,%sigma)/(a^3*b*%beta(f)*k(f)*%eta)*(2*b*%pi^2+a^3*k(f)^2)))); %beta(f):=sqrt(k(f)^2-(%pi/a)^2); Z_TE10(f):=k(f)*%eta/%beta(f); |
3.1 Figures of Merit without Surface Roughness
| (%i22) |
wxplot2d(%delta(%mu_0,%sigma_Ti6Al4V,2*%pi*f*1e9), [f,1,100], [logx], [legend,false], [xlabel,"Frequency [GHz]"], [ylabel,"%delta [m]"], [style,[lines,3,11]], [gnuplot_preamble,"set grid"])$ |
| (%i23) |
wxplot2d(8.868*%alpha(%mu_0,%sigma_Ti6Al4V,f*1e9), [f,75,110], [legend,false], [xlabel,"Frequency [GHz]"], [ylabel,"%alpha [dB/m]"], [style,[lines,3,11]], [gnuplot_preamble,"set grid"])$ |
| (%i24) |
wxplot2d(%beta(f*1e9), [f,75,110], [legend,false], [xlabel,"Frequency [GHz]"], [ylabel,"%beta [rad/s]"], [style,[lines,3,11]], [gnuplot_preamble,"set grid"])$ |
| (%i25) |
wxplot2d(Z_TE10(f*1e9), [f,75,110], [legend,false], [xlabel,"Frequency [GHz]"], [ylabel,"Z_TE10 [rad/s]"], [style,[lines,3,11]], [gnuplot_preamble,"set grid"])$ |
3.2 Figures of Merit with Surface Roughness
References:
- Ansoft HFSS 11, HFSS Online Help - Technical Notes, p 16-84
| (%i26) |
K_w(%delta,R_q):=1+exp(-(%delta/(2*R_q))^1.6); %sigma_c(%sigma,K_w):=%sigma/K_w^2; |
| (%i28) |
f_c; float(Z_TE10(f_0)); float(%delta(%mu_0,%sigma_Ti6Al4V,2*%pi*f_0)); float(R_s(%delta(%mu_0,%sigma_Ti6Al4V,2*%pi*f_0),%sigma_Ti6Al4V)); float(8.868*%alpha(%mu_0,%sigma_Ti6Al4V,f_0)); float(K_w(%delta(%mu_0,%sigma_Ti6Al4V,2*%pi*f_0),R_q)); float(%sigma_Ti6Al4V_rough:%sigma_c(%sigma_Ti6Al4V,K_w(%delta(%mu_0,%sigma_Ti6Al4V,2*%pi*f_0),R_q)))$ float(R_s(%delta(%mu_0,%sigma_Ti6Al4V,2*%pi*f_0),%sigma_Ti6Al4V_rough)); float(8.868*%alpha(%mu_0,%sigma_Ti6Al4V_rough,f_0)); |
| (%i37) |
wxplot2d([%sigma_Ti6Al4V,%sigma_c(%sigma_Ti6Al4V,K_w(%delta(%mu_0,%sigma_Ti6Al4V,2*%pi*f),R_q))], [f,1e6,1e11], [logx], [legend,false], [xlabel,"Frequency [Hz]"], [ylabel,"%sigma [S/m]"], [style,[lines,3,1],[lines,3,2]], [gnuplot_preamble,"set grid"])$ |
| (%i38) |
wxplot2d([8.868*%alpha(%mu_0,%sigma_Ti6Al4V,f*1e9),8.868*%alpha(%mu_0,%sigma_c(%sigma_Ti6Al4V,K_w(%delta(%mu_0,%sigma_Ti6Al4V,2*%pi*f*1e9),R_q)),f*1e9)], [f,75,110], [legend,false], [xlabel,"Frequency [GHz]"], [ylabel,"%alpha [dB/m]"], [style,[lines,3,11]], [gnuplot_preamble,"set grid"])$ |
4 S-Parameters
| (%i39) |
l:0.05$ define(S_WG(f), ABCD2S(block( Z_0:Z_TE10(f), ABCD_LossyTL( %alpha(%mu_0,%sigma_c(%sigma_Ti6Al4V,K_w(%delta(%mu_0,%sigma_Ti6Al4V,2*%pi*f),R_q)),f)+%i*%beta(f), l, Z_TE10(f)))))$ |
| (%i41) |
PlotSmithChart([ parametric, realpart(S_WG(f)[1,1]), imagpart(S_WG(f)[1,1]), [f,75e9,110e9], [nticks,100]]); |
| (%i42) |
PlotSmithChart([ parametric, realpart(S_WG(f)[2,1]), imagpart(S_WG(f)[2,1]), [f,75e9,110e9], [nticks,100]]); |
