# 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
### Bandgap Reference with a Start-Up Circuit  
The bandgap reference circuit proposed in this paper is shown in Figure 3, which mainly consists of a start-up circuit, a low-voltage common-source common-gate current mirror, diodes and resistors. For a diode, the relationship between current and voltage is  
\[I=I_{S}\bigg{(}e^{\frac{V}{V_{T}}}-1\bigg{)} \tag{1}\]  
Figure 3: Transistor-level block diagram of a bandgap reference circuit. Figure 2: The overall structure of low phase noise crystal oscillator. in which \(V\) is the voltage across the diode, \(V_{T}=kT/q\), and \(I_{S}\) is the reverse saturation current. At room temperature, that is, when \(T=300\) K, \(V_{T}\approx 26\) mV, so that \(e^{\frac{V}{V_{T}}}\gg 1\). Thus, Equation (1) can be simplified to  
\[I=I_{S}e^{\frac{V}{V_{T}}} \tag{2}\]  
From Equation (2), we can ascertain the relationship between \(V\) and \(I\) as  
\[V=V_{T}\ln(I/I_{S}) \tag{3}\]  
In Figure 3, the current flowing through \(Q_{1}\) (\(Q_{1}\) is regarded as the set of n diodes) is  
\[I_{1}=nI_{S}e^{\frac{V_{1}}{V_{T}}} \tag{4}\]  
If the voltage of the \(Q_{1}\) anode is \(V_{d1}\) and the voltage of the \(Q_{2}\) anode is \(V_{d2}\), the relationship between \(V_{d1}\) and \(I_{1}\) can be obtained as  
\[V_{d1}=V_{T}\ln\left(\frac{I_{1}}{nI_{S}}\right) \tag{5}\]  
The current flowing through \(Q_{2}\) is  
\[I_{2}=I_{S}e^{\frac{V_{2}}{V_{T}}} \tag{6}\]  
The relationship between \(V_{2}\) and \(I_{2}\) can be obtained as  
\[V_{d2}=V_{T}\ln\left(\frac{I_{2}}{I_{S}}\right) \tag{7}\]  
In the circuit we designed, \(I_{1}=I_{3}=I_{4}\), \(I_{2}=I_{3}+I_{4}\). So one can ascertain  
\[ \begin{array}{ll}I_{1}\cdot R_{1}&=V_{d2}-V_{d1}\\ &=V_{T}\ln\left(\frac{I_{2}}{S_{2}}\right)-V_{T}\ln\left(\frac{I_{1}}{nI_{S}} \right)\\ &=V_{T}\ln\left(n\cdot\frac{I_{2}}{I_{1}}\right)\\ &=V_{T}\ln(2n)\end{array}  \tag{8}\]  
Therefore, the current \(I_{1}\) can be obtained as  
\[I_{1}=\frac{V_{T}\ln(2n)}{R_{1}} \tag{9}\]  
We can express the output voltage \(V_{REF}\) with the following expression::  
\[V_{REF}=\frac{V_{T}\ln(2n)}{R_{1}}\cdot R_{2}+V_{d2} \tag{10}\]  
At room temperature, \(\partial_{V2}/\partial_{T}\approx-1.5\) mV/K. However, \(\partial_{VT}/\partial_{T}\approx 0.087\) mV/K, so we chose \((R_{2}/R_{1})\cdot\ln(2n)\cdot(0.087\) mV/K) = 1.5 mV/K, that is, \((R_{2}/R_{1})\cdot\ln(2n)\approx 17.2\). In the design of this article, we chose n = 160. Figure 4 shows the change in \(V_{REF}\) with temperature obtained by simulation, from which it can be seen that the \(V_{REF}\) has changed by a total of 1.9 mV in the range of temperature from \(-40\)\({}^{\circ}\)C to 125 \({}^{\circ}\)C at the ss process corner, in which the voltage changes are most obvious. In addition, temperature drift coefficient can be calculated to be about 10 ppm in the range of \(-40\)\({}^{\circ}\)C to \(+125\)\({}^{\circ}\)C. By the way, the ss process corners mentioned above denotes the NEFT-Slow and PEET-Slow corner. An important issue in the power-independent bias circuits described above is the existence of a degenerate bias point, which necessitates a start-up circuit to ensure that the current mirror works properly at power-up. The circuit in the green box in Figure 3 is the start-up circuit designed in this article. Upon powering up, transistor PM\({}_{0}\) turns on due to the gate voltage \(V_{3}\) of transistor PM\({}_{0}\) being 0 V. The turning on of transistor PM\({}_{0}\) causes the gate voltage \(V_{2}\) of transistor NM\({}_{1}\) to become high. When \(V_{2}\) becomes high, transistor NM\({}_{1}\) is turned on, and the conduction of transistor NM\({}_{1}\) causes \(V_{1}\) voltage to drop. The drop in \(V_{1}\) voltage causes transistors PM\({}_{3}\) and PM\({}_{4}\) to turn on, which in turn turns on the current mirror. After the current mirror is turned on, the \(V_{3}\) voltage rises, and the increase in \(V_{3}\) voltage causes the transistor NM\({}_{2}\) to turn on.