## 2 Theoretical Background  
The conventional Marchand balun is a three-port network, which consists of a \(\lambda/2\) section and two identical \(\lambda/4\) sections. The principle operation of the balun relies on electromagnetic coupling between a \(\lambda/2\) open/short stub and two \(\lambda/4\) short/open stubs. It can be design using even/odd mode analysis of symmetric coupled transmission lines. The parameters that are important in its design are even/odd mode impedances (\(Z_{Oe}\), \(Z_{Oo}\)) and the electrical lengths of the coupled lines. Figure 1 shows coupled lines with ports' assignment (a) and the Marchand balun (b). The corresponding admittance matrix of Figure 1a is given by [27; 28],  
\[Y= \begin{bmatrix}Y_{11}&Y_{12}&Y_{13}&Y_{14}\\ Y_{21}&Y_{22}&Y_{23}&Y_{24}\\ Y_{31}&Y_{32}&Y_{33}&Y_{34}\\ Y_{41}&Y_{42}&Y_{43}&Y_{44}\\ \end{bmatrix}  \tag{1}\]  
If a homogeneous medium is considered for the coupled lines, then, under even and odd mode excitations, one can obtain the following equations,\[Y_{11}=Y_{22}=Y_{33}=Y_{44}=-j(Y_{Oo}+Y_{Oe})\frac{\cot\theta}{2} \tag{2a}\] \[Y_{12}=Y_{21}=Y_{34}=Y_{43}=j(Y_{Oo}-Y_{Oe})\frac{\cot\theta}{2}\] (2b) \[Y_{13}=Y_{31}=Y_{24}=Y_{42}=-j(Y_{Oo}-Y_{Oe})\frac{\csc\theta}{2}\] (2c) \[Y_{14}=Y_{41}=Y_{23}=Y_{32}=j(Y_{Oo}+Y_{Oe})\frac{\csc\theta}{2} \tag{2d}\]  
If \(\theta=90^{o}\) (\(\lambda/4\)), then Equation (2) can be further simplified to,  
\[Y_{11}=Y_{22}=Y_{33}=Y_{44}=0 \tag{3a}\] \[Y_{12}=Y_{21}=Y_{34}=Y_{43}=0\] (3b) \[Y_{13}=Y_{31}=Y_{24}=Y_{42}=\frac{-j(Y_{Oo}-Y_{Oe})}{2}\] (3c) \[Y_{14}=Y_{41}=Y_{23}=Y_{32}=Y_{42}=\frac{j(Y_{Oo}+Y_{Oe})}{2} \tag{3d}\]  
The conditions for ideal balun operations are,  
\[S_{11}=0 \tag{4a}\] \[S_{21}=-S_{31} \tag{4b}\]  
Using Figure 1b and Equations (3) and (4), one can obtain the following design equations for the balun,  
\[Y_{12}=Y_{21}=Y_{34}=Y_{43}=j(Y_{Oo}-Y_{Oe})\frac{\cot\theta}{2}\]0 \[Y_{12}=Y_{21}=Y_{34}=Y_{43}=j(Y_{Oo}-Y_{Oe})\frac{\cot\theta}{2}\]1 (5b) \[Y_{12}=Y_{21}=Y_{34}=Y_{43}=j(Y_{Oo}-Y_{Oe})\frac{\cot\theta}{2}\]2 (5c) \[Y_{12}=Y_{21}=Y_{34}=Y_{43}=j(Y_{Oo}-Y_{Oe})\frac{\cot\theta}{2}\]3 (6a) \[Y_{12}=Y_{21}=Y_{34}=Y_{43}=j(Y_{Oo}-Y_{Oe})\frac{\cot\theta}{2}\]4 (6b) \[Y_{12}=Y_{21}=Y_{34}=Y_{43}=j(Y_{Oo}-Y_{Oe})\frac{\cot\theta}{2}\]5  
where the subscript \(x\) stands for odd or even mode excitation, \(Z_{Ox}\) is the characteristic impedance of the coupled line, \(V_{p,x}\) is the phase velocity, \(\epsilon_{eff,x}\) is the effective dielectric constant, and c is the speed of light in a vacuum. The capacitance \(C_{x}\) can be further expended using \[Y_{12}=Y_{21}=Y_{34}=Y_{43}=j(Y_{Oo}-Y_{Oe})\frac{\cot\theta}{2}\]6  
\[Y_{12}=Y_{21}=Y_{34}=Y_{43}=j(Y_{Oo}-Y_{Oe})\frac{\cot\theta}{2}\]7 \[Y_{12}=Y_{21}=Y_{34}=Y_{43}=j(Y_{Oo}-Y_{Oe})\frac{\cot\theta}{2}\]8  
where \(C_{p}\), \(C_{f}\), \(C_{ga}\), \(C_{gd}\), and \(C_{f^{\prime}}\) are the parallel plate capacitance, fringing capacitance, gap capacitance in air, gap capacitance in a dielectric, and fringing capacitance in even mode, respectively. Equation (5) is valid if \(\theta=90^{\circ}\), which requires even/odd mode phase velocities to be equal [20]. It also shows that the large even/odd mode characteristic impedance ratio is required for the design of the balun, which can result in low phase and amplitude imbalance. The ratio can be increased either by decreasing the odd mode impedance or increasing even mode impedance. The odd mode impedance can be decreased if the odd mode capacitances shown in Figure 2 are increased. Using three asymmetric coupled lines, the odd mode capacitances increase by almost twice the two coupled lines, which helps in equalizing the even/odd mode phase velocities. Thus, it can result in low phase and amplitude imbalance over a wide bandwidth. A strong capacitive coupling is required for an edged coupled balun to have excellent performance, which is proportional to the dielectric material and the area of the coupled lines and inversely proportional to the gap separation between them.