**7.1. Mathematical model of the FM-AFM**

60 Nonlinearity, Bifurcation and Chaos – Theory and Applications

The AGC loop also depends on the amplitude detector output, shown in Figure 17. The amplitude detector is composed of diode followed by a first-order low-pass filter. The circuit holds the output A(t) for a while, allowing the AGC to determine the control signal.

**Figure 15.** Block Diagram of the FM-AFM control system. Source: (Bueno et al., 2011).

**Figure 16.** PLL block diagram

The mathematical model of the FM-AFM considers the microcantiler dynamics, the tipsample interaction, the amplitude detector circuit and the PLL. The microcantilever is assumed to be a second order system with natural frequency *c* and damping coefficient . Concerning to tip-sample interaction, there are short, medium and long range forces. Since the FM-AFM operates in long-range distance, the predominant force is the Van Der Waals

 <sup>2</sup> 6 () () *AH dt zt* , where AH is the Hamaker constant and d(t) is the tip-sample distance, Figure

7.1. Besides, the microcatilever is excited by an external forcing signal with a previously determined amplitude. The signal is a sinusoid with phase ( ) *<sup>c</sup> t t* and amplitude ( ) *<sup>o</sup> rtv* , where *r t*( ) is the AGC signal, and *<sup>o</sup> v* is constant. The tip-sample interaction forces cause modulations both in the amplitude and in the frequency of oscillation of the AFM microcantilever. The modulations are detected by the PLL and used by the AGC and by the ADC, in order to control the microcantilever, drivin it to oscillate according to *z t A t sen t t c c* . The microcantilever mathetamical model is given by equation 37.

**Figure 17.** Amplitude detector. Source (Bueno et al., 2011)

The mathematical model of the amplitude detector is given by equation 38, where <sup>1</sup> *<sup>d</sup> RC* and ( ), ( ) 0 0, ( ) 0 *<sup>d</sup> zt zt z z t* . The PLL model can be seen in many works in the literature. The mathematical model in equation 39 follows Bueno et al., 2010 and 2011. Considering the filter transfer function <sup>0</sup> 2 1 0 *f t s s* and the gain <sup>1</sup> 2 *G k kvA moo c* , where *mk* is the phase detector gain, *<sup>o</sup> k* is the VCO gain, *<sup>o</sup> v* is the VCO output amplitude and *Ac* is the nominal microcantilever amplitude of oscillation, Figure 14. Equations 40 and 41 represent the AGC and ADC, respectively. Equations 37 to 41 are the model of the FM-AFM.

$$\dot{\varepsilon}\dot{z} + \gamma \dot{z}(t) + \alpha\_c^2 z(t) = r(t)\upsilon\_o \text{sen}(\alpha\_c t + \varphi\_o(t)) + \frac{A\_H}{6\left(d(t) + z(t)\right)^2} \tag{37}$$

$$
\dot{A}(t) + \tau\_d A(t) = \tau\_d \tau\_d(t) \tag{38}
$$

On an Overview of Nonlinear and Chaotic Behavior and

Their Controls of an Atomic Force Microscopy (AFM) Vibrating Problem 63

Despite the good transient response and high frequency noise rejection - such as the double frequency jiter -, provided by the all pole filter the steady state response may need improvement. The PLL must demodulate an FSK (Frequency Shift Keying) signal, that actually is a frequency step. In order to track a frequency shift the loop filter must have at least a pure integration, i.e., the PLL must be at least a type 2 system (Bueno et al., 2010 and Bueno et al., 2011; Bueno, 2009, Ogata, 1993). According to that, and considering the loop

Figure 18 illustrates the PLL response to an FSK signal, showing the PLL FM output (figure 16). After the transient the mean value of the FM output is the same value of the FSK signal. The oscillation is due to the Double frequency jitter (Bueno et al., 2010 and Bueno, 2009). This shows that the PLL design must provide strong damping to noise and to the double frequency jitter. Besides, if the PLL is not at least of type 2 the PLL presents steady state error in the FSK demodulation, impairing the AFM imaging process. The PLL perfomance was analysed and presented under the FM-AFM perspective. Besides, a PLL design method was shown and illustrated by simulation, making clear the PLL performance importantance

This chapter deals with emergent problems in the Engineering Science research, presenting study and research related to NEMS systems, specially microcantilevers with many modes of vibration, for which the tip-sample interaction forces are highly nonlinear, impairing the stability of the latent image, while the others modes of vibration can be explored in order to

the stability of the PLL is assured only if *a b* .

filter <sup>1</sup> ( ) ( 1) *as F s s bs*

in the AFM control system.

**Figure 18.** PLL response to a FSK signal.

improve the AFM performance.

**8. Conclusions** 

$$
\ddot{\varphi}\_o(t) + \beta\_1 \ddot{\phi}\_o(t) + \beta\_0 \dot{\phi}\_o(t) + \alpha\_0 \text{Gen}\left(\varphi\_o(t) - \varphi\_c(t)\right) = 0\tag{39}
$$

$$r\left(t\right) = \Phi\_{A\mathcal{GC}}\left(A\_c - A\left(t\right)\right) \tag{40}$$

$$d\left(t\right) = \Phi\_{ADC} \left(\Delta o\_c - \phi\_o\left(t\right)\right) \tag{41}$$

## **7.2. Local stability and PLL design for FM-AFM**

From equation 39, and considering the phase error *o c* , results that:

$$
\ddot{\mathcal{H}} + \beta\_1 \ddot{\mathcal{G}} + \beta\_0 \dot{\mathcal{G}} + \alpha\_0 \text{Gsen}(\mathcal{G}) = \ddot{\phi}\_c + \beta\_1 \ddot{\phi}\_c + \beta\_0 \dot{\phi}\_c \tag{42}
$$

that represents the phase erros between the microcantilever oscillation and the PLL. The PLL behavior analysis is conveniently perfomed considering the cylindric state space, considering , . In that case, the synchronous state, corresponding to an asymptocally stable equilibrium point of equation 42 (See Bueno et al., 2010 and Bueno et al., 2011), corresponds to a Constant phase error and to null frequency and acceleration errors, i.e., 0 . For small phase erros it can be considered that *sen* in (42). In addition, considering *<sup>c</sup> t* , (42) can be rewritten as:

$$
\ddot{\mathcal{G}} + \mathcal{J}\_1 \ddot{\mathcal{G}} + \mathcal{J}\_0 \dot{\mathcal{G}} + \alpha\_0 G \mathcal{G} = \beta\_0 \Omega \tag{43}
$$

that represents the PLL linear response to a frequency shift (step) of amplitude . The local stability of equation 42 can be determined by the position of the poles of equation 43, or by the Routh-Hurwitz criterion (See Bueno et al., 2010 and Bueno et al., 2011; Ogata, 1993). Therefore, considering that the coefficients of the filter are all positive and real, the poles of equation 43 have negative real parts if:

$$G < \frac{\beta\_0 \beta\_1}{a\_0} \,. \tag{44}$$

Considering the filter coefficients <sup>2</sup> 0 0 *<sup>n</sup>* and 1 2 *<sup>n</sup>* , where is the damping factor and *<sup>n</sup>* the natural frequency, then, from equation 44, results:

$$\mathcal{G} < 2\xi a\_n \,. \tag{45}$$

Equation 45 establishes a design criterion that assure the local stability of the PLL, i.e., for small phase and frequency steps the PLL synchronizes to the microcantilever oscillation. Additionally, from the design parameters and *<sup>n</sup>* the loop gain *G* has a superior bound, and can be determined in order to satisfy the requirements of performance and stability.

Despite the good transient response and high frequency noise rejection - such as the double frequency jiter -, provided by the all pole filter the steady state response may need improvement. The PLL must demodulate an FSK (Frequency Shift Keying) signal, that actually is a frequency step. In order to track a frequency shift the loop filter must have at least a pure integration, i.e., the PLL must be at least a type 2 system (Bueno et al., 2010 and Bueno et al., 2011; Bueno, 2009, Ogata, 1993). According to that, and considering the loop filter <sup>1</sup> ( ) ( 1) *as F s s bs* the stability of the PLL is assured only if *a b* .

Figure 18 illustrates the PLL response to an FSK signal, showing the PLL FM output (figure 16). After the transient the mean value of the FM output is the same value of the FSK signal. The oscillation is due to the Double frequency jitter (Bueno et al., 2010 and Bueno, 2009). This shows that the PLL design must provide strong damping to noise and to the double frequency jitter. Besides, if the PLL is not at least of type 2 the PLL presents steady state error in the FSK demodulation, impairing the AFM imaging process. The PLL perfomance was analysed and presented under the FM-AFM perspective. Besides, a PLL design method was shown and illustrated by simulation, making clear the PLL performance importantance in the AFM control system.

**Figure 18.** PLL response to a FSK signal.
