# A 12 dBm B1 dB N-Path Notch Filter for Transmitter Leakage Suppression in Wideband Receiver
## 3 Proposed N-Path Notch Filter Architecture
Stage I of the filter exhibits a matched impedance within the passband and a low impedance within the stopband, for a single-ended eight-path bandpass filter \(\gamma=\frac{8}{\pi^{2}}\sin^{2}\frac{\pi}{g}\approx 0.1187\). Figure 4:0  
The N-path BPF undergoes impedance transformation via a quarter-wavelength transmission line with an appropriately selected impedance, resulting in an impedance flip [18]. This transformation converts the input impedance from a matched impedance to a low impedance at the center frequency and from low impedance to a matched impedance as the frequency deviates from the center, as described in Equation (9):  
Figure 4:1  
The transformed input impedance Equation (9) is substituted into Equation (6), obtaining a notch filter with high blocking power. Quarter-wavelength transmission lines are characterized by specific frequencies and impedance values. In the RF range of this design, the size of the on-chip quarter-wavelength transmission line is too large to be implemented, and it can be equated by the structure of CLC, as shown in Figure 7a, and the structure of LCL, as shown in Figure 7b. The CLC structure equivalent to a quarter-wavelength transmission line used the least amount of inductor. The CLC structure quarter-wavelength transmission lines' specific frequencies were set to \(f=450\) MHz, and the transmission lines' specific impedances were set to \(Z_{line}=25\)\(\Omega\). The inductance and capacitance parameters in their CLC equivalent structure are calculated by Equations (10) and (11), \(L=8.84\) nH, \(C=14.15\) pF. Figure 4:2  
Figure 4:3  
At this point the impedance characteristics of the notch filter can be rewritten as Equation (12), where \(Q_{L}\) is the Q value of the inductor in the CLC structure. \[ \begin{array}{rl}Z_{in,sh}(\omega)&=\left[\left(R_{SW}+\left[\left(R_{SW}+R _{S}\right)\frac{8\gamma}{1-8\gamma}\|\gamma Z_{BB}(\omega-\omega_{0})\right. \right]\right)\right]\left\|\frac{1}{\mu\omega\text{C}_{11}}+j\omega L\\ &\qquad\qquad+\frac{\omega L}{Q_{L}}\right]\left\|\frac{1}{\mu\omega\text{C}_ {12}}\end{array}  \tag{12}\]  
The quality factor of the inductor affects the impedance characteristics of the filter after it has been transformed by the CLC structure. First, for the reject band impedance, the base-band capacitor impedance \(Z_{BB}(\omega-\omega_{0})\) is approximately equal to the load impedance \(Z_{B}\)  
Figure 7: **(a)** CLC equivalent structure; **(b)** LCL equivalent structure. and the reject impedance can be expressed as Equation (14). Next, the analyzed passband impedance can be expressed as Equation (15). From here, the resistance introduced by the finite inductor Q factor can actually be corrected by an increase in \(C_{T1}\) or a decrease in \(C_{T2}\). Therefore, the quality factor does not limit the impedance of the filter's reject band or the linearity. However, this adjustment leads to a shift in the transmission line's characteristic frequency toward higher frequency. \[Z_{B}\approx R_{S} \tag{13}\]  
\[Z_{in,reject}(\omega)=\left[\left(R_{SW}+(R_{SW}+R_{S})\frac{8\gamma}{1-8\gamma} \|\gamma Z_{B}\right)\right]\left\|\frac{1}{j\omega C_{T1}}+j\omega L+\frac{ \omega L}{Q_{L}}\right]\left\|\frac{1}{j\omega C_{T2}}\right. \tag{14}\]  
\[Z_{in,pass}(\omega)\approx\left(R_{SW}\|\frac{1}{j\omega C_{T1}}+j\omega L+ \frac{\omega L}{Q_{L}}\right)\left\|\frac{1}{j\omega C_{T2}}\right. \tag{15}\]  
The simulation of the CLC uses an ideal inductor versus finite quality factor inductor equivalent structure with an ideal quarter-wavelength transmission line; the adjusted CLC structure is shown in Figure 8a, where it can be concluded that the CLC structure is approximately equivalent to a quarter-wavelength transmission line in 0 to 600 MHz, and equivalent deterioration of the impedance transformation at the band edges leads to an additional differential loss shortfall and poor equivalence at high frequencies due to its low-pass filtering characteristics.