## 3 Transformer-Based Quadrature Hybrid  
A quadrature hybrid is a four-port (input, through, coupled and isolate) network composed of two transmission lines, as shown in Figure 3a. When the input port is connected to a signal source, mutual magnetic coupling occurs between two transmission lines, known as inductive coupling. Capacitive coupling is provided by the capacitor between two transmission lines. Inductive and capacitive coupling are both utilized in the quadrature hybrid to achieve matched quadrature signals at the through port and coupled port. Figure 2: Cross-section view of GlobalFoundries 45 nm CMOS SOI metal stacks. Applying even-mode and odd-mode analysis to the quadrature hybrid, as shown in Figure 3b,c, the voltage and current equation of transmission line can be expressed as [21]:  
\[V(z)=V^{+}\cdot(e^{-j\beta l}+\Gamma e^{j\beta l}) \tag{1}\]  
\[I(z)=\frac{V^{+}}{Z_{c}}^{+}\cdot(e^{-j\beta l}-\Gamma e^{j\beta l}) \tag{2}\]  
where \(Z_{c}\) is the characteristic impedance of the transmission line; \(\beta\) is the phase shift of a unit-length transmission line; \(l\) is the electrical length of the transmission line; and \(\Gamma\) is the reflection factor, which is expressed as:  
\[\Gamma=\frac{Z_{0}-Z_{c}}{Z_{0}+Z_{c}} \tag{3}\]  
Figure 3: (**a**) Schematic representation of the transmission line quadrature hybrid, as well as its (**b**) even-mode circuit and (**c**) odd-mode circuit. When the even-mode and odd-mode signals are applied to the transmission lines, the following equations are satisfied due to symmetry:  
\[V_{1}^{e}=V_{3}^{e} \tag{4}\]  
\[V_{2}^{e}=V_{4}^{e} \tag{5}\]  
\[V_{1}^{o}=-V_{3}^{o} \tag{6}\]  
\[V_{2}^{o}=-V_{4}^{o} \tag{7}\]  
where \(V_{1}^{e}\), \(V_{2}^{e}\), \(V_{3}^{e}\) and \(V_{4}^{e}\) are the node voltages in even-mode at nodes 1, 2, 3 and 4, respectively; \(V_{1}^{o}\), \(V_{2}^{o}\), \(V_{3}^{o}\) and \(V_{4}^{o}\) are the node voltages in odd-mode at nodes 1, 2, 3 and 4, respectively. Therefore, the input impedance in even-mode and odd-mode can be calculated as:  
\[Z_{in}^{e}=Z_{0e}\frac{Z_{0}+jZ_{0e}\tan\beta l}{Z_{0e}+jZ_{0}\tan\beta l} \tag{8}\]  
\[V(z)=V^{+}\cdot(e^{-j\beta l}+\Gamma e^{j\beta l}) \tag{1}\]0  
where \(Z_{0}=\sqrt{Z_{0e}\cdot Z_{00}}\). With the voltage relationship, we have:  
\[V(z)=V^{+}\cdot(e^{-j\beta l}+\Gamma e^{j\beta l}) \tag{1}\]1  
\[V(z)=V^{+}\cdot(e^{-j\beta l}+\Gamma e^{j\beta l}) \tag{1}\]2  
\[V(z)=V^{+}\cdot(e^{-j\beta l}+\Gamma e^{j\beta l}) \tag{1}\]3  
\[V(z)=V^{+}\cdot(e^{-j\beta l}+\Gamma e^{j\beta l}) \tag{1}\]4  
where \(I_{1}^{e}\) and \(I_{1}^{o}\) are the current flows into node 1 in even-mode and odd-mode, respectively. By substituting (8)-(9) into (10)-(13) and taking the superstition method into consideration, the voltage at each port (\(V_{1}\), \(V_{2}\), \(V_{3}\) and \(V_{4}\)) can be expressed as:  
\[V(z)=V^{+}\cdot(e^{-j\beta l}+\Gamma e^{j\beta l}) \tag{1}\]5  
\[V(z)=V^{+}\cdot(e^{-j\beta l}+\Gamma e^{j\beta l}) \tag{1}\]6  
\[V(z)=V^{+}\cdot(e^{-j\beta l}+\Gamma e^{j\beta l}) \tag{1}\]7  
\[V(z)=V^{+}\cdot(e^{-j\beta l}+\Gamma e^{j\beta l}) \tag{1}\]8  
where \(C=(Z_{0e}-Z_{0e})/(Z_{0e}+Z_{00})\) is the voltage coupling coefficient. When the length of the transmission line is equal to \(\lambda/4\), namely \(\beta l=\pi/2\), (14)-(17) are simplified as:  
\[V(z)=V^{+}\cdot(e^{-j\beta l}+\Gamma e^{j\beta l}) \tag{1}\]9  
\[I(z)=\frac{V^{+}}{Z_{c}}^{+}\cdot(e^{-j\beta l}-\Gamma e^{j\beta l}) \tag{2}\]0  
When \(C=\sqrt{2}/2\), perfect quadrature signals with same magnitude and \(90^{\circ}\) phase difference are generated at the coupled port and through port. The transmission line quadrature hybrid can achieve a matched I/Q signal at the output ports, but the chip area occupied by the \(\lambda/4\) transmission line is considerably large,especially in the low-RF frequency band. However, a transmission line hybrid could be realized by transformers with a compact chip area and similar I/Q balance performance.