## 3 Overview of Modelling Issues for Energy Recovery Systems
In addition to assuming a linear relationship in the constitutive equations and assuming the homogeneity of the MFC transducers, one simplification is to assume an ideal attachment of the transducer to the vibrating substrate. As shown in [6], the adoption of this assumption slightly affects the results obtained when using a continuous as well as a discrete model of the piezoelectric transducer, however, to a lesser extent than the adoption of an incorrect attachment model. It has also been shown that the use of an overly complex attachment model results in a significant increase in the difficulty of calculations with little change in the results obtained [10]. The considerations carried out in this way allow for a more accurate understanding of the performance of piezoelectric energy recovery devices. Once the performance characteristics of the piezoelectric elements have been taken into account, a diagram can be created, as shown in Figure 1.  
In this case, the origin of the sources does not matter, it can be force, displacement, or acceleration. The coupling mechanism can be very simple, as it is sufficient to attach a piezoelectric transducer to an oscillating element; alternatively, a structure containing an additional oscillating (seismic) mass can be used to convert accelerations into loads in the chosen transducer axis, allowing the most favourable piezoelectric constant to be selected [7]. The role of the transducer here is played by the piezo element, while a DC-DC converter rectifier can, for example, be used as the conditioning and collecting circuit. The role of the receiver can be played by a battery, capacitor, or directly powered element. Rectifiers with a DC-DC converter should therefore operate in such a way that the consumers can make optimum use of the energy generated, which is why additional measuring circuits are used to fine-tune the parameters of the conditioning circuit [7]. It is also worth noting that the resulting energy recovery device can operate under a variety of conditions. The vibrations powering the device do not have to be present for the entire time it is operating; therefore, the consumers can be powered intermittently or even with short, individual vibration spikes occurring regularly or even randomly [12]. When designing an energy recovery device, it is therefore necessary to take into account the mode of operation of the vibration source with which it is to interact. Continuous operation of the device can be useful, for example, in sensors mounted on continuously running machinery; intermittent vibration occurrence can be used, for example, in sensors monitoring the condition of bridges or viaducts; and pulsed, sudden vibrations can be used in push-button ignition systems (due to the very high voltage that can be achieved) or in projectiles detecting their launch [12; 22]. A number of piezoelectric energy recovery devices or devices that directly rely on piezoelectric sensors are currently on the market. Among the former are transducers that, according to the manufacturer, are capable of producing power of up to 60 mW at resonance, operating simultaneously over a wide frequency band, or devices capable of producing 27 mW at two output voltages of 5 V and 8 V, as well as a circuit capable of producing electricity at 1.8 V, 2.5 V, 3.3 V, and 5 V with a current of up to 100 mA [12]. Piezoelectric energy recovery devices are still being researched and developed and are being applied in many systems under development. For example, they have been applied to a system for recovering energy from the movements of the human jaw, which achieves possible output powers of up to 7 \(\mu\)W, which are nevertheless sufficient to power some devices such as hearing aids [7]. Piezoelectric transducers are also being developed to power pressure sensors inside car types. These transducers are intended to replace the batteries used in available solutions, which would significantly extend the possible operating time of the sensor without replacement [12].