Question

Z-Transform and Frequency Response. The transfer function of the LTI system is given by Ω H(z)=1−0.8eʲΩ . This transfer function indicates that the system exhibits a high gain for low frequencies. Hence, it functions as a low-pass filter, allowing low-frequency components to pass while attenuating high-frequency components.

Answer 1

The subject addresses the use of LTI system transfer functions, capacitive and inductive impedances, and the impact of these components at different frequencies within the field of electrical engineering, illustrating the principle of frequency filtering in electronic circuits.

Step-by-step explanation:

The question involves the concepts of Z-Transform and frequency response in the realm of electrical engineering, specifically analyzing an LTI (Linear Time-Invariant) system. The transfer function provided suggests that the system acts as a low-pass filter. Inductors and capacitors play key roles at different frequencies; at 60 Hz, a capacitor's impedance (Z) is significantly higher compared to having no capacitor, indicating a substantial effect at low frequencies. Conversely, at 10 kHz, the capacitor does not alter the impedance as drastically, illustrating a diminished impact at high frequencies. The impedance formula for a circuit with resistance (R), inductive reactance (XL), and capacitive reactance (Xc) is given by Z = √(R²+(XL - Xc)²), where the resonant frequency (fo) is specified when XL equals Xc. This explains the concept that inductors impede high-frequency currents more than low-frequency ones due to their higher reactance at increased frequencies, which is leveraged in applications such as audio systems or computers to filter out undesirable high-frequency signals.