# GUI for Analysis of Parameters, Accurate Design and Optimization of Microstrip Filters
## 2 Mathematical Preamble to Microstrip Filters
### Low Pass Filter Design  
A low-pass filter design with a ladder network can be achieved using the structure shown in Figure 2. The coefficients \(g_{i}\) are computed according to the type of filter, e.g., Chebyshev with \(L_{Ar}\) = 0.1 dB, etc. and these values are listed in filter design books [24].  
Figure 2: Low-pass prototype filter with a ladder network structure.  
The elements in the ladder network must be scaled from the simple value \(g_{i}\) to an impedance value. With a value \(Z_{0}\) for source impedance, the impedance scaling factor is  
\[\gamma_{0}=\frac{Z_{0}}{g_{0}},\]  
and the \(R\), \(L\) and \(C\) elements are scaled with  
\[R_{i}=\gamma_{0}g_{i},\quad L_{j}=\frac{\Omega_{c}}{\omega_{c}}\gamma_{0}g_{j}, \quad C_{k}=\frac{\Omega_{c}}{\omega_{c}}\frac{g_{k}}{\gamma_{0}}, \tag{8}\]  
where \(g_{i}\), \(g_{j}\), and \(g_{k}\) represents resistance, inductance and capacitance for the \(g\) value, respectively. The values of \(g_{0}\) and \(\Omega_{c}\) are normalized to 1. Finally, \(\omega_{c}=2\pi f_{c}\).  
The ladder network structure is set up in microstrip technology as a chain of high-impedance and low-impedance lines, which represent the inductances and capacitances, as shown in Figure 3.  
Once the \(Z_{0}\) value is defined, the width of the source impedance line, \(W_{0}\), must be found. Afterward, with the values of \(L\) and \(C\) computed from Equation (8), it is necessary to propose a value for \(W_{L}\) and \(W_{C}\) then compute their corresponding impedances \(Z_{0L}\) and \(Z_{0C}\), respectively. Here, \(W_{L}\) is the width for the high impedance line (inductance line) and \(W_{C}\) is the width for the low impedance line (capacitance line).  
A first approach to the physical length of the lines is computed with  
\[l_{Li}=\frac{1}{\beta_{L}}\sin^{-1}\biggl{(}\frac{\omega_{c}L_{i}}{Z_{0L}} \biggr{)},\ \ \text{for}\ i=1,3,\ldots,n, \tag{9}\]  
\[l_{Cj}=\frac{1}{\beta_{C}}\sin^{-1}\bigl{(}\omega_{c}C_{j}Z_{0C}\bigr{)},\ \ \text{for}\ j=2,4,\ldots,n, \tag{10}\]  
in which the propagation constants \(\beta_{L}\) and \(\beta_{C}\) are obtained from  
\[\beta_{L}=\frac{2\pi}{\lambda_{gL}},\quad\beta_{C}=\frac{2\pi}{\lambda_{gC}}, \tag{11}\]  
where \(\lambda_{gL}\) and \(\lambda_{gC}\) are the guided wavelengths of the high and low impedance lines.  
The first approach of the physical lengths does not take into the account series reactance of the low-impedance line or the shunt susceptance of the high-impedance lines. Thus, it is necessary to balance (adjust) the lengths to satisfy the set of nonlinear equations  
Figure 3: (a) General structure of the stepped-impedance low-pass microstrip filters. (b) approximation of the \(L-C\) ladder type of low-pass filters.  
\[\text{for }i=1,\ldots,n\] \[\omega_{c}L_{i} = Z_{0L}\sin(\beta_{L}l_{Li})+Z_{0C}\bigg{[}\tan\bigg{(}\frac{\beta_{ C}}{2}l_{C(i-1)}\bigg{)}+\tan\bigg{(}\frac{\beta_{C}}{2}l_{C(i+1)}\bigg{)} \bigg{]},\ \ \forall i\text{ odd}. \tag{12}\] \[\omega_{c}C_{i} = \frac{1}{Z_{0C}}\sin(\beta_{C}l_{Ci})+\frac{1}{Z_{0L}}\bigg{[} \tan\bigg{(}\frac{\beta_{L}}{2}l_{L(i-1)}\bigg{)}+\tan\bigg{(}\frac{\beta_{L} }{2}l_{L(i+1)}\bigg{)}\bigg{]},\ \ \forall i\text{ even}. \tag{13}\]  
Note that the length \(l_{C(i-1)}\) for \(i=1\), or the lengths \(l_{L(i+1)}\) or \(l_{C(i+1)}\) for \(i=n\), are taken as zero; thus, their corresponding terms \(\tan(0)=0\) in the composition of the set of equations. After solving the set of equations for the lengths, the new values of the lengths correspond to the physical lengths for each microstrip line.  
If \(n\) is odd, the filter is symmetric, and the set of Equations (12) and (13) is reduced to \(i=1,\ldots,(n+1)/2\), considering \(l_{Li}\) as \(l_{L(n-i)}\).