## 2 Analysis and Design of the Active Phase Shifter Circuit
### Quadrature Signal Generation Circuit Design  
The quadrature signal generation circuit in this paper employs an RC-CR frequency-selective network. The traditional quadrature signal generator based on an RC-CR network is shown in Figure 2. This circuit consists of an RC low-pass filter and a CR high-pass filter, which produces output signals \(I_{\text{out}}\) and \(Q_{\text{out}}\) with a \(90^{\circ}\) phase difference. When the angular frequency \(\omega=1/RC\), the magnitudes of the output signals \(I_{\text{out}}\) and \(Q_{\text{out}}\) are equal. However, as the signal bandwidth increases, the orthogonality of the output signals degrades. Therefore, a single-stage RC-CR quadrature signal generator is typically used in narrowband systems [20].  
To expand the bandwidth, this paper adopts a cascaded configuration of the single-stage RC-CR network. By introducing multiple poles, this approach extends the bandwidth while making the structure less sensitive to the load capacitance at the output. The more stages added to the RC-CR network, the wider the covered bandwidth becomes, but at the cost of increased insertion loss [21]. Figure 3 shows the insertion loss and amplitude error of the RC-CR network vary with the number of its stages. Figure 4 shows the simulation results of phase error and amplitude error. Considering the design requirements for bandwidth, insertion loss, and phase shifting error, a two-stage cascaded polyphase filter structure was ultimately selected.  
Figure 3: Insertion loss and amplitude error of RC-CR network vary with number of stages.  
Figure 2: RC-CR quadrature signal generation circuit.  
Figure 5 illustrates the two-stage polyphase filter obtained by cascading RC-CR networks. This polyphase filter has two orthogonal frequency points: \(\omega_{1}=1/R_{1}C_{1}\) and \(\omega_{2}=1/R_{2}C_{2}\). When the resistor and capacitor values are appropriately chosen, the orthogonality of the output signals is optimized. To cover the 20 GHz to 30 GHz bandwidth, the two poles of the cascaded polyphase filter are positioned at 20 GHz and 25 GHz, respectively, thereby covering the entire frequency range.  
Figure 6 shows the layout of the proposed highly symmetrical miniaturized two-stage RC-CR polyphase filter. Compared with traditional quadrature signal generation modules, RC-CR networks have a small natural area and can expand the bandwidth through multi-stage cascade to realize broadband application. The second-order RC-CR network in this paper further reduces the layout area through highly symmetrical miniaturized layout and makes the length of each signal path equal as much as possible to reduce the amplitude and phase errors caused by the layout.  
Figure 4: (a) Phase error and (b) amplitude error for 1-/2-/3-stage PPF from 20 to 30 GHz.  
In the design and pre-simulation of the polyphase filter, resistors and capacitors in series were used as the equivalent load impedance for the subsequent variable-gain amplifier (VGA). The simulation results demonstrate that the two-stage polyphase filter exhibits excellent orthogonality, with a phase difference of 90.13\({}^{\circ}\) between the two output signals at 25 GHz, which is very close to the ideal value. The filter also resonates at approximately 20 GHz and 25 GHz, with the maximum amplitude error within the bandwidth being only 0.3 dB.