# 4 th Order LC-Based Sigma Delta Modulators
## 2 Materials and Methods  
A high-order filter LC can be obtained using three main techniques of coupling that are shown in Figure 2.  
The Gm coupling technique, for example, is based on converting the output voltage of the first tank into the input current of the second. This technique is generally applied to improve Gm linearity by the reason of it has an associated higher power consumption [14; 15]. The Equation (1) defines a transfer function of two LC tanks that have the same inductors and the same capacitors where a transconductance Gm is placed between the two LC tanks, as shown in Figure 2a. \[H_{G_{m}-LC(s)}=\frac{G_{m}w_{0}^{4}L^{2}s^{2}}{s^{2}+w_{0}^{2}}, \tag{1}\]  
Figure 2: 4 th order LC filters using different coupling techniques: (**a**) Transconductance (Gm) coupling. (**b**) Magnetic coupling. (**c**) Capacitive coupling. where,  
\[w_{0}=\frac{1}{\sqrt{LC}}. \tag{2}\]  
Figure 3a shows the peak positioning for different center frequencies. The Equation (3) is the transfer function that defines two LC tanks with the same inductor L and different capacitance. Where Gm is the transconductance placed between the two tanks and \(w_{1}\) and \(w_{2}\) are the resonance frequencies of the two LC. \[H_{G_{m}-LC(s)}=\frac{G_{m}w_{1}^{2}w_{2}^{2}L^{2}s^{2}}{(s^{2}+w_{1}^{2})(s^{ 2}+w_{2}^{2})}. \tag{3}\]  
Figure 3b shows that the poles \(w_{1}\) and \(w_{2}\) are independent, and their position is directly defined. Another technique of coupling is magnetic coupling, presented in Figure 2b. The working principle of this technique is the induction of the current in the second tank by a magnetic field due to current variation in the first one. From the occupied area point of view, this approach could be attractive, taking that a transformer can be easily implemented with two inductors occupying the same area on different metal layers. Some expertise concerning the used process technology is needed because the technical parameters are not always available and the translation to a specific tool is not a trivial task. The transfer function of a magnetically coupled LC filter with identical LC tanks is defined by Equation (4):  
\[H_{\text{MC}}(s)=\frac{\frac{k\kappa L}{(1-k)\text{C}^{2}L^{2}}}{s^{4}+s^{2} \frac{2}{(1-k^{2})\text{C}L}+\frac{1}{(1-k^{2})\text{C}^{2}L^{2}}}, \tag{4}\]  
where \(k=M/L\) is the coupling factor and M is the mutual-inductance, \(w_{0}\), the resonance frequency of the LC tanks and \(w_{1}\) and \(w_{2}\). The resonance frequencies of the 4 th order magnetically coupled filter are defined like:  
\[w_{0}=\frac{1}{\sqrt{LC}}, \tag{5}\]  
\[w_{1}=\frac{1}{\sqrt{(1+k)LC}}, \tag{6}\]  
\[w_{2}=\frac{1}{\sqrt{(1+k)LC}}. \tag{7}\]  
Figure 3: Frequency response of 4 th order Gm coupled filters having: (**a**) \(w_{1}=w_{2}=w_{0}\) (each line represents a center frequency (\(w_{0}\)) plotted for values of \(C_{1}=C_{2}\) and \(L_{1}=L_{2}\). With L values varying in the range 2 nH to 11 nH 2 and C values varying in the range 4 pF to 12 pF). (**b**) \(w_{1}\
eq w_{2}\) (each line represents \(w_{1}\) and \(w_{2}\) to \(L_{1}=L_{2}\) values varying in the range 2 nH to 11 nH, \(C_{1}\) values varying in the range 4 pF to 12 pF, and \(C_{2}\) values varying in the range 6 pF to 14 pF). \(k\) in terms of resonance frequency, in terms of \(w_{1}\) and \(w_{2}\) is given by :  
\[H_{G_{m}-LC(s)}=\frac{G_{m}w_{0}^{4}L^{2}s^{2}}{s^{2}+w_{0}^{2}}, \tag{1}\]0  
It is important to note that the main issue of this topology, which is the high dependence on the coupling factor \(k\): the resonance frequencies, \(w_{1}\) and \(w_{2}\), are a part from \(w_{0}\) by a factor of \(1/(1+k)\). Figure 4a shows poles spreading with respect to the coupling factor \(k\), for \(w_{0}\) = 432 MHz. To achieve a narrowband (few MHz), the coupling factor should be very small (\(\approx\)0.01). The coupling factor of integrated transformers normally varies between 0.5 and 0.8 [16].