# Nonlinear Dynamics of an Electromagnetically Actuated
## 6 Analysis of the Stability for the Primary Resonance
### Circuit Model Verification  
The term operational amplifier, or op-amp for short, denotes a special type of amplifier that, via the proper selection of its external components, can be configured for a variety of operations, such as amplification, addition, subtraction, differentiation, and integration. The ability to perform mathematical operations was the result of combining high gain with negative feedback. Most operational amplifiers are powered by a double voltage source with opposite polarities and the usual values being \(+15\) V and \(-15\) V. A double source is obtained by connecting two simple DC sources, \(V_{cc}\) and \(V_{ee}\), in series (Figure 16). The inverting amplifier represents one of the most frequently used configurations and has the structure from Figure 17a. Figure 16: Op-amp power supply with a dual (differential) voltage source. Figure 14: The error \(e_{1}\) on the domain [0, 2300]. The circuit is in a closed loop because the resistor \(R_{f}\) is connected between the output terminal and the one corresponding to the inverting input. Assuming linear and stable operation, the differential input voltage is forced to zero, and thus, the two input potentials are equal. But the non-inverting input is connected to the ground, so its potential is 0, and the inverting input will also have zero potential of the ground. The fact that the inverting input has a potential equal to that of the mass leads to the conclusion that the input voltage is found entirely at the terminals of resistor \(R_{1}\). Since the inverting input is a virtual ground point, the output voltage \(\mathrm{V_{out}}\) is equal to the voltage drop across the resistor \(R_{f}\) but is opposite to the feedback voltage and can be written as follows: \(\mathrm{V_{out}}=-\frac{R_{t}}{R_{1}^{*}}\mathrm{V_{in}}\), where \(V_{in}\) is the input voltage. As shown in Figure 17b, the summation of several input signals can be achieved if they are connected to the op-amp inputs, depending on the sign of the input voltage coefficients. By making all resistors equal, the sum (or difference) of the input voltages is obtained. The derivation and integration circuits are circuits that perform the mathematical operations of integration and derivation. Both circuits change the shape of the processed signal in accordance with the associated mathematical operation. Depending on the placement of the capacitor at the inverting input (c) or in the reaction network (d), the derivative or integrator circuits can be obtained. The derivation circuit from Figure 17c is the circuit where the input voltage \(V_{in}\) and the output voltage \(V_{out}\) establishes the following relationship: \(\mathrm{V_{out}}=-\mathrm{RC}\frac{dV\mathrm{in}(t)}{\mathrm{dt}}\)  
The integrator configuration from Figure 17d is the circuit that implements the integration function, and the dependence of the output voltage on the input voltage is the following: \(\mathrm{V_{out}}=-\frac{1}{\mathrm{RC}}\int_{0}^{t}\mathrm{V_{in}}(t)\mathrm{ dt}+\mathrm{V_{out}}(0)\), where \(V_{out}(0)\) represents the initial value of the output voltage (calculated at time \(t=0\)). In our approach, for the implementation of the ALPHA function, we designed an electronic circuit composed of AD741 op-amps and passive components, including resistors and capacitors, and an AD633 multiplier with output according to the following catalog specifications: \(\mathrm{W}=\frac{(X_{1}-X_{2})(Y_{1}-Y_{2})}{10}-Z\), which is used to generate nonlinear items (where \(W\) is the output of the integrated circuit, \(X_{1}\), \(X_{2}\), \(Y_{1}\), and \(Y_{2}\) are the inputs of the integrated circuit, and the amplifier summing node \(Z\) allows the user to add two or more multiplier outputs, convert the output voltage to a current, and configure various analog computational functions; in our case, pin 6 is always connected to the ground).