## 3 Transformer-Based Quadrature Hybrid
Figure 4 shows the equivalent lumped-element circuit of a fully differential transformer-based quadrature hybrid. Leakage and series losses of inductors are represented by an inductor in parallel with the resistor. Similar to the transmission line quadrature hybrid, a transformer-based quadrature hybrid has four ports: input, coupled (CPL), through (THRU) and isolate (ISO). It also utilizes inductive coupling (provided by transformers 1 and 2) and capacitive coupling (provided by the parasitic capacitor Cm) to generate I/Q signal at the through port and coupled port. The isolate port is connected to Z0 (100\(\Omega\)), which is the matched impedance of the port. Figure 5 shows the 3D EM model of the proposed quadrature hybrid, wherein two transformers are folded and stacked so that the quadrature hybrid occupies only one inductor footprint. The inductors on the routes from input port to through port are designed in the OB layer (purple spiral polylines), inductors on the routes from coupled port to isolate port are designed in the OA layer (orange spiral polylines) and crossovers are designed in the LD and UA layers. Ground walls composed of M1-LD layers are placed around the quadrature hybrid to serve as a signal return path and to improve magnetic coupling isolation. With a proper design, it will produce a 90\({}^{\circ}\) phase difference between coupled port and through port at a given frequency. A 100 \(\Omega\) resistor is placed at the isolate port to avoid signal reflection and improve impedance matching. The whole layout is symmetric in order to reduce magnitude and phase mismatch, except crossovers. The width of all metal polylines is 6\(\upmu\)m, and the space between them is 10 \(\upmu\)m. The layout size of the proposed quadrature hybrid is only 220 \(\upmu\)m \(\times\) 160 \(\upmu\)m.  
The quadrature hybrid proposed in this article has the following advantages over the hybrids presented in [18; 19]: (1) a vertically stacked, multiturn transformer topology is adopted, enhancing the magnetic coupling coefficient (0.89) between transformers 1 and 2 so that a more compact layout, lower passive loss and a wider quadrature balance bandwidth can be achieved; (2) signal routes from input port to through port and from coupled port to isolate port are properly designed in OA and OB layers with same thickness and sheet resistance, which could reduce intrinsic magnitude and phase imbalance; (3) the coupling capacitor Cm shown in Figure 5 is absorbed in parasitic capacitance between metal spiral polylines. The absence of coupling capacitor results in a smaller chip area and removes modeling uncertainty. The advantages are justified by the following simulation results. \[I(z)=\frac{V^{+}}{Z_{c}}^{+}\cdot(e^{-j\beta l}-\Gamma e^{j\beta l}) \tag{2}\]1 Equivalent model of a fully differential transformer-based quadrature hybrid. Figure 6 presents the simulation results of the proposed quadrature hybrid, the port definition of which is shown in \[I(z)=\frac{V^{+}}{Z_{c}}^{+}\cdot(e^{-j\beta l}-\Gamma e^{j\beta l}) \tag{2}\]2 Figure 5(a) shows that the return loss of input, through and coupled ports are all below \(-10\) dB from \(10\) GHz to \(60\) GHz; the insertion loss of coupled and through ports are better than \(-5\) dB (including \(3\) dB from \(1\):\(2\) power dividing) from \(20\) GHz to \(60\) GHz; and the insertion loss is \(3.6\) dB at \(29\) GHz (also including \(3\) dB from power splitting), which is the perfect magnitude matched point. Figure 5(b) shows the magnitude and phase imbalance of the quadrature hybrid. The magnitude mismatch is below \(2\) dB from \(20.5\) GHz to \(60\) GHz, and the phase mismatch is below \(5^{\circ}\) from \(10\) GHz to \(45\) GHz, which demonstrates the wideband operation capability of the proposed quadrature hybrid.