# Ultra-Wideband Bandpass Filters Using Tapered Resonators
## 2 Analysis  
The required analysis for the exponential and linear tapered transmission lines used in the design of UWB bandpass filters will be presented in this section. Like the uniform transmission lines, the TTLs can support standing waves with some practical advantages such as shorter lengths and better form factor. Exponential tapered transmission line (ETTL) resonators can, practically, be used in many microwave passive circuits where decreased size and weight, ease of fabrication, and extended coverage of the microwave spectrum are important [19]. There are many published works that had driven the formulas of the minimum length of different ETTL resonators.  
The minimum length of a short-circuited ETTL shunt resonator can be derived from the reflection coefficient, which is the solution of the first-order differential equation given by  
\[\frac{dr}{dx}-2\gamma r+\left(\frac{k}{2}\right)\left(1-r^{2}\right)=0, \tag{1}\]  
where \(r\) is the reflection coefficient, \(k\) is the tapered ratio, and \(\gamma=j\beta\), where \(\beta\) is the phase constant.  
The minimum length of an exponential short-circuited stub is given by [19]  
\[l_{min}^{exp}=\frac{\tan^{-1}\sqrt{\left(2\beta/k\right)^{2}-1}}{\sqrt{1- \left(k/2\beta\right)^{2}}}\ \frac{\lambda}{2\pi}, \tag{2}\]  
where \(0\leq k/\beta\leq 2\) and \(\beta=2\pi/\lambda\) In addition, \(\lambda\) and \(k\) are the waveguide wavelength at the resonant frequency and the ETTL tapered ratio, respectively.  
Based on the electrical-equivalence principle of tapered transmission lines [20], the minimum length of an LTTL short-circuited stub can be derived directly from its equivalent ETTL stub as follows:  
\[l_{min}^{lin}=\left(e^{kl_{min}^{exp}}-1\right)/m, \tag{3}\]  
where \(m\) is the tapered ratio of the LTTL.  
Assuming \(l_{min}^{lin}\) is equal to \(l_{min}^{exp}\), then the tapered ratio of LTTL can be found by  
\[m=\left(e^{kl_{min}^{exp}}-1\right)/l_{min}^{exp}. \tag{4}\]  
The input impedance of a short-circuited ETTL stub is given by [19]  
\[Z_{int}^{exp}=jZ_{01}\left(\sqrt{1-\left(\frac{k}{2\beta}\right)^{2}}\ \cot\left(\beta l\sqrt{1-\left(\frac{k}{2\beta}\right)^{2}}\ \right)-\frac{k}{2\beta}\right)^{-1}. \tag{5}\]  
Moreover, the susceptance slope parameter is  
\[b=\left.\frac{\omega}{2}\frac{dB}{d\omega}\right|_{\omega=\omega_{0}}, \tag{6}\]where  
\[B=Im\bigg{[}\frac{1}{Z_{in}}\bigg{]}. \tag{7}\]  
For ETTL, the susceptance slope parameter can be derived and approximately stated as  
\[b_{exp}\approx\frac{\beta_{0}l}{2Z_{01}}-\frac{0.85\ k}{4\beta_{0}Z_{01}}. \tag{8}\]  
Therefore, the equivalent lumped inductor and capacitor for the short-circuited ETTL stub are calculated directly as  
\[L_{exp}=\frac{1}{\omega_{0}b_{exp}}, \tag{9}\]  
\[C_{exp}=\frac{b_{exp}}{\omega_{0}}=\frac{1}{L\omega_{0}^{2}}. \tag{10}\]  
Similarly, for LTTL, the input impedance for the short-circuited stub is given by [21]  
\[Z_{in}^{lin}=jZ_{01}\ \frac{J_{1}(a)Y_{1}(b)-J_{1}(b)Y_{1}(a)}{J_{1}(b)Y_{0}(a)-J_ {0}(a)Y_{1}(b)}, \tag{11}\]  
where \(a=\beta/m\) and \(b=a(1+ml)\). \(J_{n}(x)\) and \(Y_{n}(x)\) are the Bessel functions of the first kind and second kind of order \(n\), respectively.  
The susceptance slope parameters can be derived and approximately stated by  
\[b_{lin}\approx\frac{\beta_{0}l}{2Z_{0}}-\frac{0.81\ m}{4\beta_{0}Z_{0}}. \tag{12}\]  
Consequently, the equivalent lumped inductor and capacitor for the short-circuited LTTL stub can be determined as  
\[L_{lin}=\frac{1}{\omega_{0}b_{lin}}, \tag{13}\]  
\[C_{lin}=\frac{b_{lin}}{\omega_{0}}=\frac{1}{L\omega_{0}^{2}}. \tag{14}\]  
Figure 1 describes the length reduction in the ETTL short-circuited parallel resonators versus the tapered ratio.  
Figure 1: Minimum length of short-circuited ETTL shunt resonator.