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/*
* c_microstrip.cpp - coupled microstrip class implementation * * Copyright (C) 2002 Claudio Girardi <claudio.girardi@ieee.org> * Copyright (C) 2005, 2006 Stefan Jahn <stefan@lkcc.org> * * This program is free software; you can redistribute it and/or modify * it under the terms of the GNU General Public License as published by * the Free Software Foundation; either version 2 of the License, or (at * your option) any later version. * * This program is distributed in the hope that it will be useful, but * WITHOUT ANY WARRANTY; without even the implied warranty of * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU * General Public License for more details. * * You should have received a copy of the GNU General Public License * along with this package; see the file COPYING. If not, write to * the Free Software Foundation, Inc., 51 Franklin Street - Fifth Floor, * Boston, MA 02110-1301, USA. * */
/* c_microstrip.c - Puts up window for coupled microstrips and
* performs the associated calculations * Based on the original microstrip.c by Gopal Narayanan */
#include <cmath>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include "c_microstrip.h"
#include "microstrip.h"
#include "transline.h"
#include "units.h"
C_MICROSTRIP::C_MICROSTRIP() : TRANSLINE(), h( 0.0 ), // height of substrate
ht( 0.0 ), // height to the top of box
t( 0.0 ), // thickness of top metal
rough( 0.0 ), // Roughness of top metal
w( 0.0 ), // width of lines
w_t_e( 0.0 ), // even-mode thickness-corrected line width
w_t_o( 0.0 ), // odd-mode thickness-corrected line width
l( 0.0 ), // length of lines
s( 0.0 ), // spacing of lines
Z0_e_0( 0.0 ), // static even-mode impedance
Z0_o_0( 0.0 ), // static odd-mode impedance
Z0e( 0.0 ), // even-mode impedance
Z0o( 0.0 ), // odd-mode impedance
c_e( 0.0 ), // even-mode capacitance
c_o( 0.0 ), // odd-mode capacitance
ang_l_e( 0.0 ), // even-mode electrical length in angle
ang_l_o( 0.0 ), // odd-mode electrical length in angle
er_eff_e( 0.0 ), // even-mode effective dielectric constant
er_eff_o( 0.0 ), // odd-mode effective dielectric constant
er_eff_e_0( 0.0 ), // static even-mode effective dielectric constant
er_eff_o_0( 0.0 ), // static odd-mode effective dielectric constant
w_eff( 0.0 ), // Effective width of line
atten_dielectric_e( 0.0 ), // even-mode dielectric losses (dB)
atten_cond_e( 0.0 ), // even-mode conductors losses (dB)
atten_dielectric_o( 0.0 ), // odd-mode dielectric losses (dB)
atten_cond_o( 0.0 ), // odd-mode conductors losses (dB)
aux_ms( nullptr ){ m_Name = "Coupled_MicroStrip"; Init();}
C_MICROSTRIP::~C_MICROSTRIP(){ delete aux_ms;}
/*
* delta_u_thickness_single() computes the thickness effect on * normalized width for a single microstrip line * * References: H. A. Atwater, "Simplified Design Equations for * Microstrip Line Parameters", Microwave Journal, pp. 109-115, * November 1989. */double C_MICROSTRIP::delta_u_thickness_single( double u, double t_h ){ double delta_u;
if( t_h > 0.0 ) { delta_u = (1.25 * t_h / M_PI) * ( 1.0 + log( ( 2.0 + (4.0 * M_PI * u - 2.0) / ( 1.0 + exp( -100.0 * ( u - 1.0 / (2.0 * M_PI) ) ) ) ) / t_h ) );
} else { delta_u = 0.0; } return delta_u;}
/*
* delta_u_thickness() - compute the thickness effect on normalized * width for coupled microstrips * * References: Rolf Jansen, "High-Speed Computation of Single and * Coupled Microstrip Parameters Including Dispersion, High-Order * Modes, Loss and Finite Strip Thickness", IEEE Trans. MTT, vol. 26, * no. 2, pp. 75-82, Feb. 1978 */void C_MICROSTRIP::delta_u_thickness(){ double e_r, u, g, t_h; double delta_u, delta_t, delta_u_e, delta_u_o;
e_r = m_parameters[EPSILONR_PRM]; u = m_parameters[PHYS_WIDTH_PRM] / m_parameters[H_PRM]; /* normalized line width */ g = m_parameters[PHYS_S_PRM] / m_parameters[H_PRM]; /* normalized line spacing */ t_h = m_parameters[T_PRM] / m_parameters[H_PRM]; /* normalized strip thickness */
if( t_h > 0.0 ) { /* single microstrip correction for finite strip thickness */ delta_u = delta_u_thickness_single( u, t_h ); delta_t = t_h / ( g * e_r ); /* thickness correction for the even- and odd-mode */ delta_u_e = delta_u * ( 1.0 - 0.5 * exp( -0.69 * delta_u / delta_t ) ); delta_u_o = delta_u_e + delta_t; } else { delta_u_e = delta_u_o = 0.0; }
w_t_e = m_parameters[PHYS_WIDTH_PRM] + delta_u_e * m_parameters[H_PRM]; w_t_o = m_parameters[PHYS_WIDTH_PRM] + delta_u_o * m_parameters[H_PRM];}
/*
* compute various parameters for a single line */void C_MICROSTRIP::compute_single_line(){ if( aux_ms == NULL ) aux_ms = new MICROSTRIP();
/* prepare parameters for single microstrip computations */ aux_ms->m_parameters[EPSILONR_PRM] = m_parameters[EPSILONR_PRM]; aux_ms->m_parameters[PHYS_WIDTH_PRM] = m_parameters[PHYS_WIDTH_PRM]; aux_ms->m_parameters[H_PRM] = m_parameters[H_PRM]; aux_ms->m_parameters[T_PRM] = 0.0;
//aux_ms->m_parameters[H_T_PRM] = m_parameters[H_T_PRM];
aux_ms->m_parameters[H_T_PRM] = 1e12; /* arbitrarily high */ aux_ms->m_parameters[FREQUENCY_PRM] = m_parameters[FREQUENCY_PRM]; aux_ms->m_parameters[MURC_PRM] = m_parameters[MURC_PRM]; aux_ms->microstrip_Z0(); aux_ms->dispersion();}
/*
* filling_factor_even() - compute the filling factor for the coupled * microstrips even-mode without cover and zero conductor thickness */double C_MICROSTRIP::filling_factor_even( double u, double g, double e_r ){ double v, v3, v4, a_e, b_e, q_inf;
v = u * ( 20.0 + g * g ) / ( 10.0 + g * g ) + g * exp( -g ); v3 = v * v * v; v4 = v3 * v; a_e = 1.0 + log( ( v4 + v * v / 2704.0 ) / ( v4 + 0.432 ) ) / 49.0 + log( 1.0 + v3 / 5929.741 ) / 18.7; b_e = 0.564 * pow( ( ( e_r - 0.9 ) / ( e_r + 3.0 ) ), 0.053 );
/* filling factor, with width corrected for thickness */ q_inf = pow( ( 1.0 + 10.0 / v ), -a_e * b_e );
return q_inf;}
/**
* filling_factor_odd() - compute the filling factor for the coupled * microstrips odd-mode without cover and zero conductor thickness */double C_MICROSTRIP::filling_factor_odd( double u, double g, double e_r ){ double b_odd = 0.747 * e_r / ( 0.15 + e_r ); double c_odd = b_odd - ( b_odd - 0.207 ) * exp( -0.414 * u ); double d_odd = 0.593 + 0.694 * exp( -0.562 * u );
/* filling factor, with width corrected for thickness */ double q_inf = exp( -c_odd * pow( g, d_odd ) );
return q_inf;}
/*
* delta_q_cover_even() - compute the cover effect on filling factor * for the even-mode */double C_MICROSTRIP::delta_q_cover_even( double h2h ){ double q_c;
if( h2h <= 39 ) { q_c = tanh( 1.626 + 0.107 * h2h - 1.733 / sqrt( h2h ) ); } else { q_c = 1.0; }
return q_c;}
/*
* delta_q_cover_odd() - compute the cover effect on filling factor * for the odd-mode */double C_MICROSTRIP::delta_q_cover_odd( double h2h ){ double q_c;
if( h2h <= 7 ) { q_c = tanh( 9.575 / ( 7.0 - h2h ) - 2.965 + 1.68 * h2h - 0.311 * h2h * h2h ); } else { q_c = 1.0; }
return q_c;}
/**
* er_eff_static() - compute the static effective dielectric constants * * References: Manfred Kirschning and Rolf Jansen, "Accurate * Wide-Range Design Equations for the Frequency-Dependent * Characteristic of Parallel Coupled Microstrip Lines", IEEE * Trans. MTT, vol. 32, no. 1, Jan. 1984 */void C_MICROSTRIP::er_eff_static(){ double u_t_e, u_t_o, g, h2, h2h; double a_o, t_h, q, q_c, q_t, q_inf; double er_eff_single; double er;
er = m_parameters[EPSILONR_PRM];
/* compute zero-thickness single line parameters */ compute_single_line(); er_eff_single = aux_ms->er_eff_0;
h2 = m_parameters[H_T_PRM]; u_t_e = w_t_e / m_parameters[H_PRM]; /* normalized even_mode line width */ u_t_o = w_t_o / m_parameters[H_PRM]; /* normalized odd_mode line width */ g = m_parameters[PHYS_S_PRM] / m_parameters[H_PRM]; /* normalized line spacing */ h2h = h2 / m_parameters[H_PRM]; /* normalized cover height */ t_h = m_parameters[T_PRM] / m_parameters[H_PRM]; /* normalized strip thickness */
/* filling factor, computed with thickness corrected width */ q_inf = filling_factor_even( u_t_e, g, er ); /* cover effect */ q_c = delta_q_cover_even( h2h ); /* thickness effect */ q_t = aux_ms->delta_q_thickness( u_t_e, t_h ); /* resultant filling factor */ q = ( q_inf - q_t ) * q_c; /* static even-mode effective dielectric constant */ er_eff_e_0 = 0.5 * ( er + 1.0 ) + 0.5 * ( er - 1.0 ) * q;
/* filling factor, with width corrected for thickness */ q_inf = filling_factor_odd( u_t_o, g, er ); /* cover effect */ q_c = delta_q_cover_odd( h2h ); /* thickness effect */ q_t = aux_ms->delta_q_thickness( u_t_o, t_h ); /* resultant filling factor */ q = ( q_inf - q_t ) * q_c;
a_o = 0.7287 * ( er_eff_single - 0.5 * ( er + 1.0 ) ) * ( 1.0 - exp( -0.179 * u_t_o ) );
/* static odd-mode effective dielectric constant */ er_eff_o_0 = ( 0.5 * ( er + 1.0 ) + a_o - er_eff_single ) * q + er_eff_single;}
/**
* delta_Z0_even_cover() - compute the even-mode impedance correction * for a homogeneous microstrip due to the cover * * References: S. March, "Microstrip Packaging: Watch the Last Step", * Microwaves, vol. 20, no. 13, pp. 83.94, Dec. 1981. */double C_MICROSTRIP::delta_Z0_even_cover( double g, double u, double h2h ){ double f_e, g_e, delta_Z0_even; double x, y, A, B, C, D, E, F;
A = -4.351 / pow( 1.0 + h2h, 1.842 ); B = 6.639 / pow( 1.0 + h2h, 1.861 ); C = -2.291 / pow( 1.0 + h2h, 1.90 ); f_e = 1.0 - atanh( A + ( B + C * u ) * u );
x = pow( 10.0, 0.103 * g - 0.159 ); y = pow( 10.0, 0.0492 * g - 0.073 ); D = 0.747 / sin( 0.5 * M_PI * x ); E = 0.725 * sin( 0.5 * M_PI * y ); F = pow( 10.0, 0.11 - 0.0947 * g ); g_e = 270.0 * ( 1.0 - tanh( D + E * sqrt( 1.0 + h2h ) - F / ( 1.0 + h2h ) ) );
delta_Z0_even = f_e * g_e;
return delta_Z0_even;}
/**
* delta_Z0_odd_cover() - compute the odd-mode impedance correction * for a homogeneous microstrip due to the cover * * References: S. March, "Microstrip Packaging: Watch the Last Step", * Microwaves, vol. 20, no. 13, pp. 83.94, Dec. 1981. */double C_MICROSTRIP::delta_Z0_odd_cover( double g, double u, double h2h ){ double f_o, g_o, delta_Z0_odd; double G, J, K, L;
J = tanh( pow( 1.0 + h2h, 1.585 ) / 6.0 ); f_o = pow( u, J );
G = 2.178 - 0.796 * g; if( g > 0.858 ) { K = log10( 20.492 * pow( g, 0.174 ) ); } else { K = 1.30; } if( g > 0.873 ) { L = 2.51 * pow( g, -0.462 ); } else { L = 2.674; } g_o = 270.0 * ( 1.0 - tanh( G + K * sqrt( 1.0 + h2h ) - L / ( 1.0 + h2h ) ) );
delta_Z0_odd = f_o * g_o;
return delta_Z0_odd;}
/**
* Z0_even_odd() - compute the static even- and odd-mode static * impedances * * References: Manfred Kirschning and Rolf Jansen, "Accurate * Wide-Range Design Equations for the Frequency-Dependent * Characteristic of Parallel Coupled Microstrip Lines", IEEE * Trans. MTT, vol. 32, no. 1, Jan. 1984 */void C_MICROSTRIP::Z0_even_odd(){ double er_eff, h2, u_t_e, u_t_o, g, h2h; double Q_1, Q_2, Q_3, Q_4, Q_5, Q_6, Q_7, Q_8, Q_9, Q_10; double delta_Z0_e_0, delta_Z0_o_0, Z0_single, er_eff_single;
h2 = m_parameters[H_T_PRM]; u_t_e = w_t_e / m_parameters[H_PRM]; /* normalized even-mode line width */ u_t_o = w_t_o / m_parameters[H_PRM]; /* normalized odd-mode line width */ g = m_parameters[PHYS_S_PRM] / m_parameters[H_PRM]; /* normalized line spacing */ h2h = h2 / m_parameters[H_PRM]; /* normalized cover height */
Z0_single = aux_ms->Z0_0; er_eff_single = aux_ms->er_eff_0;
/* even-mode */ er_eff = er_eff_e_0; Q_1 = 0.8695 * pow( u_t_e, 0.194 ); Q_2 = 1.0 + 0.7519 * g + 0.189 * pow( g, 2.31 ); Q_3 = 0.1975 + pow( ( 16.6 + pow( ( 8.4 / g ), 6.0 ) ), -0.387 ) + log( pow( g, 10.0 ) / ( 1.0 + pow( g / 3.4, 10.0 ) ) ) / 241.0; Q_4 = 2.0 * Q_1 / ( Q_2 * ( exp( -g ) * pow( u_t_e, Q_3 ) + ( 2.0 - exp( -g ) ) * pow( u_t_e, -Q_3 ) ) ); /* static even-mode impedance */ Z0_e_0 = Z0_single * sqrt( er_eff_single / er_eff ) / ( 1.0 - sqrt( er_eff_single ) * Q_4 * Z0_single / ZF0 ); /* correction for cover */ delta_Z0_e_0 = delta_Z0_even_cover( g, u_t_e, h2h ) / sqrt( er_eff );
Z0_e_0 = Z0_e_0 - delta_Z0_e_0;
/* odd-mode */ er_eff = er_eff_o_0; Q_5 = 1.794 + 1.14 * log( 1.0 + 0.638 / ( g + 0.517 * pow( g, 2.43 ) ) ); Q_6 = 0.2305 + log( pow( g, 10.0 ) / ( 1.0 + pow( g / 5.8, 10.0 ) ) ) / 281.3 + log( 1.0 + 0.598 * pow( g, 1.154 ) ) / 5.1; Q_7 = ( 10.0 + 190.0 * g * g ) / ( 1.0 + 82.3 * g * g * g ); Q_8 = exp( -6.5 - 0.95 * log( g ) - pow( g / 0.15, 5.0 ) ); Q_9 = log( Q_7 ) * ( Q_8 + 1.0 / 16.5 ); Q_10 = ( Q_2 * Q_4 - Q_5 * exp( log( u_t_o ) * Q_6 * pow( u_t_o, -Q_9 ) ) ) / Q_2;
/* static odd-mode impedance */ Z0_o_0 = Z0_single * sqrt( er_eff_single / er_eff ) / ( 1.0 - sqrt( er_eff_single ) * Q_10 * Z0_single / ZF0 ); /* correction for cover */ delta_Z0_o_0 = delta_Z0_odd_cover( g, u_t_o, h2h ) / sqrt( er_eff );
Z0_o_0 = Z0_o_0 - delta_Z0_o_0;}
/*
* er_eff_freq() - compute er_eff as a function of frequency */void C_MICROSTRIP::er_eff_freq(){ double P_1, P_2, P_3, P_4, P_5, P_6, P_7; double P_8, P_9, P_10, P_11, P_12, P_13, P_14, P_15; double F_e, F_o; double er_eff, u, g, f_n;
u = m_parameters[PHYS_WIDTH_PRM] / m_parameters[H_PRM]; /* normalize line width */ g = m_parameters[PHYS_S_PRM] / m_parameters[H_PRM]; /* normalize line spacing */
/* normalized frequency [GHz * mm] */ f_n = m_parameters[FREQUENCY_PRM] * m_parameters[H_PRM] / 1e06;
er_eff = er_eff_e_0; P_1 = 0.27488 + ( 0.6315 + 0.525 / pow( 1.0 + 0.0157 * f_n, 20.0 ) ) * u - 0.065683 * exp( -8.7513 * u ); P_2 = 0.33622 * ( 1.0 - exp( -0.03442 * m_parameters[EPSILONR_PRM] ) ); P_3 = 0.0363 * exp( -4.6 * u ) * ( 1.0 - exp( -pow( f_n / 38.7, 4.97 ) ) ); P_4 = 1.0 + 2.751 * ( 1.0 - exp( -pow( m_parameters[EPSILONR_PRM] / 15.916, 8.0 ) ) ); P_5 = 0.334 * exp( -3.3 * pow( m_parameters[EPSILONR_PRM] / 15.0, 3.0 ) ) + 0.746; P_6 = P_5 * exp( -pow( f_n / 18.0, 0.368 ) ); P_7 = 1.0 + 4.069 * P_6 * pow( g, 0.479 ) * exp( -1.347 * pow( g, 0.595 ) - 0.17 * pow( g, 2.5 ) );
F_e = P_1 * P_2 * pow( ( P_3 * P_4 + 0.1844 * P_7 ) * f_n, 1.5763 ); /* even-mode effective dielectric constant */ er_eff_e = m_parameters[EPSILONR_PRM] - ( m_parameters[EPSILONR_PRM] - er_eff ) / ( 1.0 + F_e );
er_eff = er_eff_o_0; P_8 = 0.7168 * ( 1.0 + 1.076 / ( 1.0 + 0.0576 * ( m_parameters[EPSILONR_PRM] - 1.0 ) ) ); P_9 = P_8 - 0.7913 * ( 1.0 - exp( -pow( f_n / 20.0, 1.424 ) ) ) * atan( 2.481 * pow( m_parameters[EPSILONR_PRM] / 8.0, 0.946 ) ); P_10 = 0.242 * pow( m_parameters[EPSILONR_PRM] - 1.0, 0.55 ); P_11 = 0.6366 * ( exp( -0.3401 * f_n ) - 1.0 ) * atan( 1.263 * pow( u / 3.0, 1.629 ) ); P_12 = P_9 + ( 1.0 - P_9 ) / ( 1.0 + 1.183 * pow( u, 1.376 ) ); P_13 = 1.695 * P_10 / ( 0.414 + 1.605 * P_10 ); P_14 = 0.8928 + 0.1072 * ( 1.0 - exp( -0.42 * pow( f_n / 20.0, 3.215 ) ) ); P_15 = fabs( 1.0 - 0.8928 * ( 1.0 + P_11 ) * P_12 * exp( -P_13 * pow( g, 1.092 ) ) / P_14 );
F_o = P_1 * P_2 * pow( ( P_3 * P_4 + 0.1844 ) * f_n * P_15, 1.5763 ); /* odd-mode effective dielectric constant */ er_eff_o = m_parameters[EPSILONR_PRM] - ( m_parameters[EPSILONR_PRM] - er_eff ) / ( 1.0 + F_o );}
/*
* conductor_losses() - compute microstrips conductor losses per unit * length */void C_MICROSTRIP::conductor_losses(){ double e_r_eff_e_0, e_r_eff_o_0, Z0_h_e, Z0_h_o, delta; double K, R_s, Q_c_e, Q_c_o, alpha_c_e, alpha_c_o;
e_r_eff_e_0 = er_eff_e_0; e_r_eff_o_0 = er_eff_o_0; Z0_h_e = Z0_e_0 * sqrt( e_r_eff_e_0 ); /* homogeneous stripline impedance */ Z0_h_o = Z0_o_0 * sqrt( e_r_eff_o_0 ); /* homogeneous stripline impedance */ delta = m_parameters[SKIN_DEPTH_PRM];
if( m_parameters[FREQUENCY_PRM] > 0.0 ) { /* current distribution factor (same for the two modes) */ K = exp( -1.2 * pow( ( Z0_h_e + Z0_h_o ) / ( 2.0 * ZF0 ), 0.7 ) ); /* skin resistance */ R_s = 1.0 / ( m_parameters[SIGMA_PRM] * delta ); /* correction for surface roughness */ R_s *= 1.0 + ( ( 2.0 / M_PI ) * atan( 1.40 * pow( ( m_parameters[ROUGH_PRM] / delta ), 2.0 ) ) );
/* even-mode strip inductive quality factor */ Q_c_e = ( M_PI * Z0_h_e * m_parameters[PHYS_WIDTH_PRM] * m_parameters[FREQUENCY_PRM] ) / ( R_s * C0 * K ); /* even-mode losses per unith length */ alpha_c_e = ( 20.0 * M_PI / log( 10.0 ) ) * m_parameters[FREQUENCY_PRM] * sqrt( e_r_eff_e_0 ) / ( C0 * Q_c_e );
/* odd-mode strip inductive quality factor */ Q_c_o = ( M_PI * Z0_h_o * m_parameters[PHYS_WIDTH_PRM] * m_parameters[FREQUENCY_PRM] ) / ( R_s * C0 * K ); /* odd-mode losses per unith length */ alpha_c_o = ( 20.0 * M_PI / log( 10.0 ) ) * m_parameters[FREQUENCY_PRM] * sqrt( e_r_eff_o_0 ) / ( C0 * Q_c_o ); } else { alpha_c_e = alpha_c_o = 0.0; }
atten_cond_e = alpha_c_e * m_parameters[PHYS_LEN_PRM]; atten_cond_o = alpha_c_o * m_parameters[PHYS_LEN_PRM];}
/*
* dielectric_losses() - compute microstrips dielectric losses per * unit length */void C_MICROSTRIP::dielectric_losses(){ double e_r, e_r_eff_e_0, e_r_eff_o_0; double alpha_d_e, alpha_d_o;
e_r = m_parameters[EPSILONR_PRM]; e_r_eff_e_0 = er_eff_e_0; e_r_eff_o_0 = er_eff_o_0;
alpha_d_e = ( 20.0 * M_PI / log( 10.0 ) ) * ( m_parameters[FREQUENCY_PRM] / C0 ) * ( e_r / sqrt( e_r_eff_e_0 ) ) * ( ( e_r_eff_e_0 - 1.0 ) / ( e_r - 1.0 ) ) * m_parameters[TAND_PRM]; alpha_d_o = ( 20.0 * M_PI / log( 10.0 ) ) * ( m_parameters[FREQUENCY_PRM] / C0 ) * ( e_r / sqrt( e_r_eff_o_0 ) ) * ( ( e_r_eff_o_0 - 1.0 ) / ( e_r - 1.0 ) ) * m_parameters[TAND_PRM];
atten_dielectric_e = alpha_d_e * m_parameters[PHYS_LEN_PRM]; atten_dielectric_o = alpha_d_o * m_parameters[PHYS_LEN_PRM];}
/*
* c_microstrip_attenuation() - compute attenuation of coupled * microstrips */void C_MICROSTRIP::attenuation(){ m_parameters[SKIN_DEPTH_PRM] = skin_depth(); conductor_losses(); dielectric_losses();}
/*
* line_angle() - calculate strips electrical lengths in radians */void C_MICROSTRIP::line_angle(){ double e_r_eff_e, e_r_eff_o; double v_e, v_o, lambda_g_e, lambda_g_o;
e_r_eff_e = er_eff_e; e_r_eff_o = er_eff_o;
/* even-mode velocity */ v_e = C0 / sqrt( e_r_eff_e ); /* odd-mode velocity */ v_o = C0 / sqrt( e_r_eff_o ); /* even-mode wavelength */ lambda_g_e = v_e / m_parameters[FREQUENCY_PRM]; /* odd-mode wavelength */ lambda_g_o = v_o / m_parameters[FREQUENCY_PRM]; /* electrical angles */ ang_l_e = 2.0 * M_PI * m_parameters[PHYS_LEN_PRM] / lambda_g_e; /* in radians */ ang_l_o = 2.0 * M_PI * m_parameters[PHYS_LEN_PRM] / lambda_g_o; /* in radians */}
void C_MICROSTRIP::syn_err_fun( double* f1, double* f2, double s_h, double w_h, double e_r, double w_h_se, double w_h_so ){ double g, he;
g = cosh( 0.5 * M_PI * s_h ); he = cosh( M_PI * w_h + 0.5 * M_PI * s_h );
*f1 = ( 2.0 / M_PI ) * acosh( ( 2.0 * he - g + 1.0 ) / ( g + 1.0 ) ); *f2 = ( 2.0 / M_PI ) * acosh( ( 2.0 * he - g - 1.0 ) / ( g - 1.0 ) );
if( e_r <= 6.0 ) { *f2 += ( 4.0 / ( M_PI * ( 1.0 + e_r / 2.0 ) ) ) * acosh( 1.0 + 2.0 * w_h / s_h ); } else { *f2 += ( 1.0 / M_PI ) * acosh( 1.0 + 2.0 * w_h / s_h ); }
*f1 -= w_h_se; *f2 -= w_h_so;}
/*
* synth_width - calculate widths given Z0 and e_r * from Akhtarzad S. et al., "The design of coupled microstrip lines", * IEEE Trans. MTT-23, June 1975 and * Hinton, J.H., "On design of coupled microstrip lines", IEEE Trans. * MTT-28, March 1980 */void C_MICROSTRIP::synth_width(){ double Z0, e_r; double w_h_se, w_h_so, w_h, a, ce, co, s_h; double f1, f2, ft1, ft2, j11, j12, j21, j22, d_s_h, d_w_h, err; double eps = 1e-04;
f1 = f2 = 0; e_r = m_parameters[EPSILONR_PRM];
Z0 = m_parameters[Z0_E_PRM] / 2.0; /* Wheeler formula for single microstrip synthesis */ a = exp( Z0 * sqrt( e_r + 1.0 ) / 42.4 ) - 1.0; w_h_se = 8.0 * sqrt( a * ( ( 7.0 + 4.0 / e_r ) / 11.0 ) + ( ( 1.0 + 1.0 / e_r ) / 0.81 ) ) / a;
Z0 = m_parameters[Z0_O_PRM] / 2.0; /* Wheeler formula for single microstrip synthesis */ a = exp( Z0 * sqrt( e_r + 1.0 ) / 42.4 ) - 1.0; w_h_so = 8.0 * sqrt( a * ( ( 7.0 + 4.0 / e_r ) / 11.0 ) + ( ( 1.0 + 1.0 / e_r ) / 0.81 ) ) / a;
ce = cosh( 0.5 * M_PI * w_h_se ); co = cosh( 0.5 * M_PI * w_h_so ); /* first guess at m_parameters[PHYS_S_PRM]/h */ s_h = ( 2.0 / M_PI ) * acosh( ( ce + co - 2.0 ) / ( co - ce ) ); /* first guess at w/h */ w_h = acosh( ( ce * co - 1.0 ) / ( co - ce ) ) / M_PI - s_h / 2.0;
m_parameters[PHYS_S_PRM] = s_h * m_parameters[H_PRM]; m_parameters[PHYS_WIDTH_PRM] = w_h * m_parameters[H_PRM];
syn_err_fun( &f1, &f2, s_h, w_h, e_r, w_h_se, w_h_so );
/* rather crude Newton-Rhapson; we need this beacuse the estimate of */ /* w_h is often quite far from the true value (see Akhtarzad S. et al.) */ do { /* compute Jacobian */ syn_err_fun( &ft1, &ft2, s_h + eps, w_h, e_r, w_h_se, w_h_so ); j11 = ( ft1 - f1 ) / eps; j21 = ( ft2 - f2 ) / eps; syn_err_fun( &ft1, &ft2, s_h, w_h + eps, e_r, w_h_se, w_h_so ); j12 = ( ft1 - f1 ) / eps; j22 = ( ft2 - f2 ) / eps;
/* compute next step */ d_s_h = ( -f1 * j22 + f2 * j12 ) / ( j11 * j22 - j21 * j12 ); d_w_h = ( -f2 * j11 + f1 * j21 ) / ( j11 * j22 - j21 * j12 );
//g_print("j11 = %e\tj12 = %e\tj21 = %e\tj22 = %e\n", j11, j12, j21, j22);
//g_print("det = %e\n", j11*j22 - j21*j22);
//g_print("d_s_h = %e\td_w_h = %e\n", d_s_h, d_w_h);
s_h += d_s_h; w_h += d_w_h;
/* chech the error */ syn_err_fun( &f1, &f2, s_h, w_h, e_r, w_h_se, w_h_so );
err = sqrt( f1 * f1 + f2 * f2 ); /* converged ? */ } while( err > 1e-04 );
m_parameters[PHYS_S_PRM] = s_h * m_parameters[H_PRM]; m_parameters[PHYS_WIDTH_PRM] = w_h * m_parameters[H_PRM];}
/*
* Z0_dispersion() - calculate frequency dependency of characteristic * impedances */void C_MICROSTRIP::Z0_dispersion(){ double Q_0; double Q_11, Q_12, Q_13, Q_14, Q_15, Q_16, Q_17, Q_18, Q_19, Q_20, Q_21; double Q_22, Q_23, Q_24, Q_25, Q_26, Q_27, Q_28, Q_29; double r_e, q_e, p_e, d_e, C_e; double e_r_eff_o_f, e_r_eff_o_0; double e_r_eff_single_f, e_r_eff_single_0, Z0_single_f; double f_n, g, u, e_r; double R_1, R_2, R_7, R_10, R_11, R_12, R_15, R_16, tmpf;
e_r = m_parameters[EPSILONR_PRM];
u = m_parameters[PHYS_WIDTH_PRM] / m_parameters[H_PRM]; /* normalize line width */ g = m_parameters[PHYS_S_PRM] / m_parameters[H_PRM]; /* normalize line spacing */
/* normalized frequency [GHz * mm] */ f_n = m_parameters[FREQUENCY_PRM] * m_parameters[H_PRM] / 1e06;
e_r_eff_single_f = aux_ms->er_eff; e_r_eff_single_0 = aux_ms->er_eff_0; Z0_single_f = aux_ms->m_parameters[Z0_PRM];
e_r_eff_o_f = er_eff_o; e_r_eff_o_0 = er_eff_o_0;
Q_11 = 0.893 * ( 1.0 - 0.3 / ( 1.0 + 0.7 * ( e_r - 1.0 ) ) ); Q_12 = 2.121 * ( pow( f_n / 20.0, 4.91 ) / ( 1.0 + Q_11 * pow( f_n / 20.0, 4.91 ) ) ) * exp( -2.87 * g ) * pow( g, 0.902 ); Q_13 = 1.0 + 0.038 * pow( e_r / 8.0, 5.1 ); Q_14 = 1.0 + 1.203 * pow( e_r / 15.0, 4.0 ) / ( 1.0 + pow( e_r / 15.0, 4.0 ) ); Q_15 = 1.887 * exp( -1.5 * pow( g, 0.84 ) ) * pow( g, Q_14 ) / ( 1.0 + 0.41 * pow( f_n / 15.0, 3.0 ) * pow( u, 2.0 / Q_13 ) / ( 0.125 + pow( u, 1.626 / Q_13 ) ) ); Q_16 = ( 1.0 + 9.0 / ( 1.0 + 0.403 * pow( e_r - 1.0, 2 ) ) ) * Q_15; Q_17 = 0.394 * ( 1.0 - exp( -1.47 * pow( u / 7.0, 0.672 ) ) ) * ( 1.0 - exp( -4.25 * pow( f_n / 20.0, 1.87 ) ) ); Q_18 = 0.61 * ( 1.0 - exp( -2.13 * pow( u / 8.0, 1.593 ) ) ) / ( 1.0 + 6.544 * pow( g, 4.17 ) ); Q_19 = 0.21 * g * g * g * g / ( ( 1.0 + 0.18 * pow( g, 4.9 ) ) * ( 1.0 + 0.1 * u * u ) * ( 1.0 + pow( f_n / 24.0, 3.0 ) ) ); Q_20 = ( 0.09 + 1.0 / ( 1.0 + 0.1 * pow( e_r - 1, 2.7 ) ) ) * Q_19; Q_21 = fabs( 1.0 - 42.54 * pow( g, 0.133 ) * exp( -0.812 * g ) * pow( u, 2.5 ) / ( 1.0 + 0.033 * pow( u, 2.5 ) ) );
r_e = pow( f_n / 28.843, 12 ); q_e = 0.016 + pow( 0.0514 * e_r * Q_21, 4.524 ); p_e = 4.766 * exp( -3.228 * pow( u, 0.641 ) ); d_e = 5.086 * q_e * ( r_e / ( 0.3838 + 0.386 * q_e ) ) * ( exp( -22.2 * pow( u, 1.92 ) ) / ( 1.0 + 1.2992 * r_e ) ) * ( pow( e_r - 1.0, 6.0 ) / ( 1.0 + 10 * pow( e_r - 1.0, 6.0 ) ) ); C_e = 1.0 + 1.275 * ( 1.0 - exp( -0.004625 * p_e * pow( e_r, 1.674 ) * pow( f_n / 18.365, 2.745 ) ) ) - Q_12 + Q_16 - Q_17 + Q_18 + Q_20;
R_1 = 0.03891 * pow( e_r, 1.4 ); R_2 = 0.267 * pow( u, 7.0 ); R_7 = 1.206 - 0.3144 * exp( -R_1 ) * ( 1.0 - exp( -R_2 ) ); R_10 = 0.00044 * pow( e_r, 2.136 ) + 0.0184; tmpf = pow( f_n / 19.47, 6.0 ); R_11 = tmpf / ( 1.0 + 0.0962 * tmpf ); R_12 = 1.0 / ( 1.0 + 0.00245 * u * u ); R_15 = 0.707 * R_10 * pow( f_n / 12.3, 1.097 ); R_16 = 1.0 + 0.0503 * e_r * e_r * R_11 * ( 1.0 - exp( -pow( u / 15.0, 6.0 ) ) ); Q_0 = R_7 * ( 1.0 - 1.1241 * ( R_12 / R_16 ) * exp( -0.026 * pow( f_n, 1.15656 ) - R_15 ) );
/* even-mode frequency-dependent characteristic impedances */ m_parameters[Z0_E_PRM] = Z0_e_0 * pow( 0.9408 * pow( e_r_eff_single_f, C_e ) - 0.9603, Q_0 ) / pow( ( 0.9408 - d_e ) * pow( e_r_eff_single_0, C_e ) - 0.9603, Q_0 );
Q_29 = 15.16 / ( 1.0 + 0.196 * pow( e_r - 1.0, 2.0 ) ); tmpf = pow( e_r - 1.0, 3.0 ); Q_28 = 0.149 * tmpf / ( 94.5 + 0.038 * tmpf ); tmpf = pow( e_r - 1.0, 1.5 ); Q_27 = 0.4 * pow( g, 0.84 ) * ( 1.0 + 2.5 * tmpf / ( 5.0 + tmpf ) ); tmpf = pow( ( e_r - 1.0 ) / 13.0, 12.0 ); Q_26 = 30.0 - 22.2 * ( tmpf / ( 1.0 + 3.0 * tmpf ) ) - Q_29; tmpf = ( e_r - 1.0 ) * ( e_r - 1.0 ); Q_25 = ( 0.3 * f_n * f_n / ( 10.0 + f_n * f_n ) ) * ( 1.0 + 2.333 * tmpf / ( 5.0 + tmpf ) ); Q_24 = 2.506 * Q_28 * pow( u, 0.894 ) * pow( ( 1.0 + 1.3 * u ) * f_n / 99.25, 4.29 ) / ( 3.575 + pow( u, 0.894 ) ); Q_23 = 1.0 + 0.005 * f_n * Q_27 / ( ( 1.0 + 0.812 * pow( f_n / 15.0, 1.9 ) ) * ( 1.0 + 0.025 * u * u ) ); Q_22 = 0.925 * pow( f_n / Q_26, 1.536 ) / ( 1.0 + 0.3 * pow( f_n / 30.0, 1.536 ) );
/* odd-mode frequency-dependent characteristic impedances */ m_parameters[Z0_O_PRM] = Z0_single_f + ( Z0_o_0 * pow( e_r_eff_o_f / e_r_eff_o_0, Q_22 ) - Z0_single_f * Q_23 ) / ( 1.0 + Q_24 + pow( 0.46 * g, 2.2 ) * Q_25 );}
void C_MICROSTRIP::calcAnalyze(){ /* compute thickness corrections */ delta_u_thickness(); /* get effective dielectric constants */ er_eff_static(); /* impedances for even- and odd-mode */ Z0_even_odd(); /* calculate freq dependence of er_eff_e, er_eff_o */ er_eff_freq(); /* calculate frequency dependence of Z0e, Z0o */ Z0_dispersion(); /* calculate losses */ attenuation(); /* calculate electrical lengths */ line_angle();}
void C_MICROSTRIP::showAnalyze(){ setProperty( Z0_E_PRM, m_parameters[Z0_E_PRM] ); setProperty( Z0_O_PRM, m_parameters[Z0_O_PRM] ); setProperty( ANG_L_PRM, sqrt( ang_l_e * ang_l_o ) );
//Check for errors
if( !std::isfinite( m_parameters[Z0_O_PRM] ) || m_parameters[Z0_O_PRM] <= 0.0 ) setErrorLevel( Z0_O_PRM, TRANSLINE_ERROR );
if( !std::isfinite( m_parameters[Z0_E_PRM] ) || m_parameters[Z0_E_PRM] <= 0.0 ) setErrorLevel( Z0_E_PRM, TRANSLINE_ERROR );
if( !std::isfinite( m_parameters[ANG_L_PRM] ) || m_parameters[ANG_L_PRM] <= 0.0 ) setErrorLevel( ANG_L_PRM, TRANSLINE_ERROR );
// Check for warnings
if( !std::isfinite( m_parameters[PHYS_WIDTH_PRM] ) || m_parameters[PHYS_WIDTH_PRM] <= 0.0 ) setErrorLevel( PHYS_WIDTH_PRM, TRANSLINE_WARNING );
if( !std::isfinite( m_parameters[PHYS_S_PRM] ) || m_parameters[PHYS_S_PRM] <= 0.0 ) setErrorLevel( PHYS_S_PRM, TRANSLINE_WARNING );
if( !std::isfinite( m_parameters[PHYS_LEN_PRM] ) || m_parameters[PHYS_LEN_PRM] <= 0.0 ) setErrorLevel( PHYS_LEN_PRM, TRANSLINE_WARNING );}
void C_MICROSTRIP::showSynthesize(){ setProperty( PHYS_WIDTH_PRM, m_parameters[PHYS_WIDTH_PRM] ); setProperty( PHYS_S_PRM, m_parameters[PHYS_S_PRM] ); setProperty( PHYS_LEN_PRM, m_parameters[PHYS_LEN_PRM] );
//Check for errors
if( !std::isfinite( m_parameters[PHYS_WIDTH_PRM] ) || m_parameters[PHYS_WIDTH_PRM] <= 0.0 ) setErrorLevel( PHYS_WIDTH_PRM, TRANSLINE_ERROR );
if( !std::isfinite( m_parameters[PHYS_S_PRM] ) || m_parameters[PHYS_S_PRM] <= 0.0 ) setErrorLevel( PHYS_S_PRM, TRANSLINE_ERROR );
if( !std::isfinite( m_parameters[PHYS_LEN_PRM] ) || m_parameters[PHYS_LEN_PRM] <= 0.0 ) setErrorLevel( PHYS_LEN_PRM, TRANSLINE_ERROR );
// Check for warnings
if( !std::isfinite( m_parameters[Z0_O_PRM] ) || m_parameters[Z0_O_PRM] <= 0.0 ) setErrorLevel( Z0_O_PRM, TRANSLINE_WARNING );
if( !std::isfinite( m_parameters[Z0_E_PRM] ) || m_parameters[Z0_E_PRM] <= 0.0 ) setErrorLevel( Z0_E_PRM, TRANSLINE_WARNING );
if( !std::isfinite( m_parameters[ANG_L_PRM] ) || m_parameters[ANG_L_PRM] <= 0.0 ) setErrorLevel( ANG_L_PRM, TRANSLINE_WARNING );}
void C_MICROSTRIP::show_results(){
setResult( 0, er_eff_e, "" ); setResult( 1, er_eff_o, "" ); setResult( 2, atten_cond_e, "dB" ); setResult( 3, atten_cond_o, "dB" ); setResult( 4, atten_dielectric_e, "dB" ); setResult( 5, atten_dielectric_o, "dB" );
setResult( 6, m_parameters[SKIN_DEPTH_PRM] / UNIT_MICRON, "µm" );}
void C_MICROSTRIP::syn_fun( double* f1, double* f2, double s_h, double w_h, double Z0_e, double Z0_o ){ m_parameters[PHYS_S_PRM] = s_h * m_parameters[H_PRM]; m_parameters[PHYS_WIDTH_PRM] = w_h * m_parameters[H_PRM];
/* compute coupled microstrip parameters */ calcAnalyze();
*f1 = m_parameters[Z0_E_PRM] - Z0_e; *f2 = m_parameters[Z0_O_PRM] - Z0_o;}
/*
* synthesis function */void C_MICROSTRIP::calcSynthesize(){ double Z0_e, Z0_o, ang_l_dest; double f1, f2, ft1, ft2, j11, j12, j21, j22, d_s_h, d_w_h, err; double eps = 1e-04; double w_h, s_h, le, lo;
/* required value of Z0_e and Z0_o */ Z0_e = m_parameters[Z0_E_PRM]; Z0_o = m_parameters[Z0_O_PRM];
ang_l_e = m_parameters[ANG_L_PRM]; ang_l_o = m_parameters[ANG_L_PRM]; ang_l_dest = m_parameters[ANG_L_PRM];
/* calculate width and use for initial value in Newton's method */ synth_width(); w_h = m_parameters[PHYS_WIDTH_PRM] / m_parameters[H_PRM]; s_h = m_parameters[PHYS_S_PRM] / m_parameters[H_PRM]; f1 = f2 = 0;
/* rather crude Newton-Rhapson */ do { /* compute Jacobian */ syn_fun( &ft1, &ft2, s_h + eps, w_h, Z0_e, Z0_o ); j11 = ( ft1 - f1 ) / eps; j21 = ( ft2 - f2 ) / eps; syn_fun( &ft1, &ft2, s_h, w_h + eps, Z0_e, Z0_o ); j12 = ( ft1 - f1 ) / eps; j22 = ( ft2 - f2 ) / eps;
/* compute next step; increments of s_h and w_h */ d_s_h = ( -f1 * j22 + f2 * j12 ) / ( j11 * j22 - j21 * j12 ); d_w_h = ( -f2 * j11 + f1 * j21 ) / ( j11 * j22 - j21 * j12 );
s_h += d_s_h; w_h += d_w_h;
/* compute the error with the new values of s_h and w_h */ syn_fun( &f1, &f2, s_h, w_h, Z0_e, Z0_o ); err = sqrt( f1 * f1 + f2 * f2 );
/* converged ? */ } while( err > 1e-04 );
/* denormalize computed width and spacing */ m_parameters[PHYS_S_PRM] = s_h * m_parameters[H_PRM]; m_parameters[PHYS_WIDTH_PRM] = w_h * m_parameters[H_PRM];
/* calculate physical length */ le = C0 / m_parameters[FREQUENCY_PRM] / sqrt( er_eff_e ) * ang_l_dest / 2.0 / M_PI; lo = C0 / m_parameters[FREQUENCY_PRM] / sqrt( er_eff_o ) * ang_l_dest / 2.0 / M_PI; m_parameters[PHYS_LEN_PRM] = sqrt( le * lo );
calcAnalyze();
m_parameters[ANG_L_PRM] = ang_l_dest; m_parameters[Z0_E_PRM] = Z0_e; m_parameters[Z0_O_PRM] = Z0_o;}
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