# A Low Phase Noise Crystal Oscillator with a Fast Start-Up Bandgap Reference for WLAN Applications
## 2 Circuit Composition of Low Phase Noise Crystal Oscillator
### Design of Low Phase Noise Crystal Oscillator
#### 2.2.3 Design of Broadband LDO  
Figure 12 depicts the LDO circuit designed with a closed-loop configuration comprising a conventional power tube, error amplifier, and sampling feedback. The LDO employs NMOS transistors as the power tube, leading to improved response time and reduced on-chip capacitance area. This configuration is also referred to as a broadband regulator. The amplifier input adopts a folded common-source and common-gate structure, which enhances common-mode rejection and output impedance. The op-amp gain (\(A_{\frac{p}{p}}\)) is obtained by multiplying the cross-conductor of the input tube by the output impedance. The power tube is designed with a source-follower configuration and has a gain almost equal to 1, hence, the gain of the complete loop is the product of the op-amp gain and the feedback coefficient \(A_{\frac{p}{p}}\beta\). By eliminating feedback resistors, the feedback coefficient \(\beta\) is 1 and, therefore, the loop gain is equal to the op-amp gain. Regarding stability, the primary pole of the LDO is located at the op-amp's output, while the secondary pole is at the power tube's output, which restricts the load current at the output and complicates in-band compensation for capacitance. Even if there is only a small amount of load capacitance at the output, it can either provide or absorb the load current during transient response. As a result, a capacitor can be added at the output of the op-amp to create a low-frequency main pole \(p_{0}\), using high resistance at this location. \[p_{0}=\frac{1}{2\pi R_{oc}C_{c}} \tag{14}\]  
In the aforementioned equation, the equivalent impedance of the LDO output node is represented by \(R_{0}\), while the compensation capacitor is represented by \(C_{c}\). It is worth noting that the secondary pole \(p_{1}\) is located at the LDO output node. \[p_{1}=\frac{g_{m,power}}{2\pi C_{load}} \tag{15}\]  
The \(g_{m,\;power}\) in the above equation is the trans-conductance value of the power tube, and \(C_{load}\) is the load capacitance value. The structure without feedback resistors of the LDO causes the secondary pole to move too close to the main pole at the time of no-load, making stability a problem. Therefore, a zero point needs to be introduced to offset the effect of the secondary pole. In this paper, a resistor is connected in series with the compensation  
Figure 12: Transistor-level block diagram of the broadband LDO. capacitor to form a current-limiting zero-point \(z_{0}\), which is placed at a higher frequency than the secondary pole. \[z_{0}=\frac{1}{2\pi R_{0}C_{c}} \tag{16}\]  
\(R_{0}\) in the above equation is the zero-point compensation resistor value. The design of a crystal oscillator with low phase noise is crucial for a phase-locked loop with low phase noise, as the phase noise of the oscillator has a significant impact on the in-band noise [26]. Among various sources influencing the phase noise of the crystal oscillator, power supply noise has a substantial effect due to the high Q value of the crystal. Neglecting power supply noise can result in a low phase noise of the crystal oscillator, even if it is not optimized. Therefore, it is crucial to optimize power supply noise to design a low phase noise crystal oscillator. This paper proposes the use of an RC low-pass filter to suppress bandgap reference noise, and its structure is depicted in Figure 13a. The transfer function of the proposed RC low-pass filter is  
\[H(s)=\frac{V_{\mathrm{LDO}}}{V_{EAMP}}=\frac{1}{1+R_{p}C_{p}} \tag{17}\]  
Let \(RC=T\), and the logarithmic amplitude-frequency characteristic of Equation (17) is  
\[ \begin{array}{rl}20\mathrm{lg}|G(j\omega)|&=20\mathrm{lg}\frac{1}{\sqrt{T^{ 2}\omega^{2}+1}}\\ &=-20\mathrm{lg}\sqrt{T^{2}\omega^{2}+1}\end{array}  \tag{18}\]  
The Bode plot of Equation (17) is shown in Figure 13b, from which it can be seen that the cutoff frequency is \(1/T\). To minimize the effect of bandgap reference noise on the crystal oscillator phase noise while taking area into account, we chose a cutoff frequency of 10 kHz.