# 4 th Order LC-Based Sigma Delta Modulators
## 2 Materials and Methods
It is possible to reduce the coupling factor by putting the coils apart [17], however, this solution has several disadvantages: (1) it is very hard to obtain very low coupling factors with a good precision [17]; (2) it is difficult to have reliable simulation results; (3) significant increase in the surface area of the filter. A configuration of magnetically coupled LC filter is using non-identical LC tanks. In this case, we consider the same inductor value for both tanks, but two different capacitors \(C_{1}\) and \(C_{2}\), resulting in this transfer function:  
\[H_{G_{m}-LC(s)}=\frac{G_{m}w_{0}^{4}L^{2}s^{2}}{s^{2}+w_{0}^{2}}, \tag{1}\]1  
and \(w_{01},w_{02},w_{1},w_{2}\), the resonance frequencies of the first and the second tanks and the poles of the 4 th order magnetically coupled LC filter are respectively:  
\[H_{G_{m}-LC(s)}=\frac{G_{m}w_{0}^{4}L^{2}s^{2}}{s^{2}+w_{0}^{2}}, \tag{1}\]2  
\[H_{G_{m}-LC(s)}=\frac{G_{m}w_{0}^{4}L^{2}s^{2}}{s^{2}+w_{0}^{2}}, \tag{1}\]3  
\[H_{G_{m}-LC(s)}=\frac{G_{m}w_{0}^{4}L^{2}s^{2}}{s^{2}+w_{0}^{2}}, \tag{1}\]4  
\[H_{G_{m}-LC(s)}=\frac{G_{m}w_{0}^{4}L^{2}s^{2}}{s^{2}+w_{0}^{2}}, \tag{1}\]5  
\[H_{G_{m}-LC(s)}=\frac{G_{m}w_{0}^{4}L^{2}s^{2}}{s^{2}+w_{0}^{2}}, \tag{1}\]6 Frequency response of 4 th order magnetically coupled LC filter: (**a**) With different values of the coupling factor k for \(w_{0}\) = 432 MHz. (**b**) With a fixed k = 0.8, \(C_{2}\) = 10.8 pF, L = 12.5 nH and different \(C_{1}\). Through the equations below, it is possible to define a \"forbidden zone\", where the poles cannot be placed. This zone is located between \(w_{1_{max}}\) and \(w_{2_{min}}\):  
\[H_{G_{m}-LC(s)}=\frac{G_{m}w_{0}^{4}L^{2}s^{2}}{s^{2}+w_{0}^{2}}, \tag{1}\]7  
\[H_{G_{m}-LC(s)}=\frac{G_{m}w_{0}^{4}L^{2}s^{2}}{s^{2}+w_{0}^{2}}, \tag{1}\]8  
In Figure 4b, we plotted the frequency response of a 4 th order magnetically coupled LC filter with a fixed \(k,C_{2}\) and L and different \(C_{1}\). Capacitively coupled LC filters, as in Figure 2c, can be described by the transfer function in Equation (16) when identical resonators are used. Thus, we have that:  
\[H_{G_{m}-LC(s)}=\frac{G_{m}w_{0}^{4}L^{2}s^{2}}{s^{2}+w_{0}^{2}}, \tag{1}\]9  
Assuming that C and L are respectively capacitance and inductance of the tank circuit, and \(C_{c}\) is the coupling capacitance, from Equation (16), \(w_{1}\) and \(w_{2}\) can be defined as:  
Figure 2:0  
Figure 2:1  
The \(w_{1}\) and \(w_{2}\) represent respectively the frequency of oscillation of the two tanks and the oscillation frequency due to the coupling capacitor. One can observe that, as \(C_{c}\) tends to infinity, one pole is fixed, \(w_{1}\), and the other pole tends to zero, \(w_{2}\). In the case of two different tanks, the transfer function used to describe the capacitively coupled LC resonators filter is:  
Figure 2:2  
and the poles are positioned at:  
Figure 2:3  
Figure 2:4  
The minimum distance between the poles is achieved when the capacitance values are equal, being that minimum distance defined by the coupling capacitor. In general, a big coupling capacitor increases the minimum distance. It is illustrated in Figure 5b the poles positioning of 4 th order capacitively coupled filter with different tanks with a \(C_{c}\) = 65 fF, L = 12.5 nH, \(C_{2}\) = 10.8 pF and different values for \(C_{1}\) (1 fF, 1 pF, 10 pF, 200 pF, and 800 pF). One can see that the distance between the poles varies with the values of \(C_{1}\) and that the minimum distance is reached when \(C_{1}\) equals \(C_{2}\)DT to CT Equivalence_  
The 4 th order LC filters presented above can be used as a loop filter of a 4 th order bandpass \(\Sigma\Delta\)D Modulator. It is possible to design an \(\Sigma\Delta\)M with a 4 th order CT based on the equivalence between the Continuous Time loop gain and the Discrete Time loop gain [18; 19]. To this approach, the design of a CT \(\Sigma\Delta\)M start with the calculation of the DT equivalent model, as shown in Figure 6[13].