# Analysis and Design of Single-Phase Unidirectional Transducers with High Directivity
## 2 Design and Optimization  
The basic idea of an EWC-SPUDT is to place a distributed reflection source in a normal interdigital SAW transducer to enhance the forward acoustic wave and meanwhile reduce that traveling along the backward direction [6]. Scattering matrix is an effective description of the performance of the SAW transducers with well impedance matching, especially when designing filters. The expressions of scattering matrix parameters \(S_{11}\) and \(S_{13}\) are given below. \[S_{11}=P_{11}+\frac{2P_{13}^{2}}{G_{L}+P_{33}} \tag{1}\]  
\[S_{13}=\frac{2\sqrt{G_{L}}P_{13}}{G_{L}+P_{33}} \tag{2}\]  
In Equations (1) and (2), \(P_{11}\), \(P_{13}\), and \(P_{33}\) are the elements of P-matrix. In an interdigital transducer, \(P_{11}\) is the reflection coefficient, \(P_{13}\) describes the excitation efficient, and \(P_{33}\) denotes the transducer admittance, which describes the acoustic and electrostatic currents due to a drive voltage. Their detailed expressions are given in [14; 15; 16]. \(G_{L}\) is the combined effective electrical load conductance which is attached to the SPUDT electrical port. The goal for designing a SPUDT filter is to make a zero \(S_{11}\) under matched conditions and provide a desired magnitude and phase response for \(S_{13}\). From Equation (1), \(G_{L}\) is required to match the SPUDT if found by equating \(S_{11}\) to zero [17]. \[G_{L}=-P_{33}-\frac{2P_{13}^{2}}{P_{11}} \tag{3}\]  
If a lossless SPUDT is tuned using a lossless matching network, its insertion loss under the condition of optimum matching will be related to the following parameters: the unidirectional angle \(\theta\), the product of the reflectivity per unit length, and the effective reflection length \(|\kappa|L\). After substituting P-matrix elements into (2), the insertion loss can  
Figure 1: The geometrical configuration of an EWC-SPUDT cell. be obtained. The minimum insertion loss of the SPUDT occurs when \(\theta\) equals \(-90^{\circ}\), and then its expression evolves into the following form:  
\[\left|S_{13}\right|^{2}=1-\exp(-2|\kappa|L) \tag{4}\]  
A similar result was presented in [17]. Equation (4) implies that the insertion loss can be reduced by increasing the reflectivity of the SPUDT. The directivity \(D\), defined as the ratio between the two acoustic outputs for an interdigital transducer (IDT), is given by \(D=\left|P_{32}/P_{31}\right|\). It can be simplified to (5) at center frequency. \[D=\left|\frac{1+j\exp(j\theta)\text{tanh}(\left|\kappa\right|L/2)}{1+j\exp(- j\theta)\text{tanh}(\left|\kappa\right|L/2)}\right| \tag{5}\]  
where \(j\) is an imaginary unit. When \(D>1\), the wave excitation in the right direction is greater than that in the left direction, while when \(D<1\), the situation is just the opposite. In addition, \(D=1\) means that the transducer is bidirectional. It can be clearly seen from the expression above that \(D\) tends to increase with the increase in \(\left|\kappa\right|L\), and when \(\theta=-90^{\circ}\), \(D\) reaches the maximum value \(\exp(\left|\kappa\right|L)\). In summary, optimizing the EWC-SPDUT cell is to maximize its reflectivity at a unidirectional angle of \(-90^{\circ}\). Next, we will discuss the relationship between the geometry of the EWC-SPUDT cell and its unidirectional angle, as well as its reflectivity combining the coupling-of-modes (COM) theory. The governing equations of COM can be written as [14; 18; 19]:  
\[\left\{ \begin{array}{l}\frac{\delta R(x)}{\delta x}=-i\delta R(x)+i\kappa S(x )+i\alpha V\\ \frac{\delta S(x)}{\delta x}=-i\kappa^{*}R(x)+i\delta S(x)-i\alpha^{*}V\\ \frac{\delta t}{\delta x}=-2i\alpha^{*}R(x)-2i\alpha S(x)+i\alpha CV\end{array} \right. \tag{6}\]  
where \(R(x)\) and \(S(x)\) are the amplitudes of the waves propagating in positive and negative \(x\), respectively. \(i\) is the current on the busbar and \(V\) is the applied voltage. \(\omega\) is the angular frequency. \(\delta=\frac{\omega}{\psi}-\frac{2\pi}{\lambda}\) is a detuning parameter.