9.2:
Frequency Response of a Circuit
Consider an inductive circuit in the time domain. The equivalent frequency domain circuit is obtained by replacing the resistance and inductance with their impedances.
The circuit is connected to an input with variable frequency, and the output is taken across the inductance.
The output-to-input voltage division determines the transfer function.
The expression of the transfer function gives the magnitude and phase derived from it.
The ratio of inductance to resistance is equal to the time constant of the inductive circuit.
The magnitude and phase can be written in terms of the time constant.
As the frequency approaches zero, the transfer function approaches zero, and the phase shift approximates pi over two.
When the frequency equals the inverse of the time constant, the amplitude reaches about 0.707 times its maximum value, and the phase shift is pi over four.
Similarly, at higher frequencies, the transfer function approaches unity, and the phase shift tends to zero.
These points, among a few additional ones, depict the frequency response through amplitude or magnitude and phase response graphs.
9.2:
Frequency Response of a Circuit
Inductive circuits present intriguing challenges in electrical engineering, particularly during the transition from the time domain to the frequency domain. This transformation involves converting inductors into impedances and utilizing phasor representation.
The transfer function is pivotal in characterizing how these circuits react to various frequencies, facilitating a profound understanding of their behavior. An essential parameter is the time constant, signifying the inductance-to-resistance ratio in the circuit.
The magnitude of the transfer function is derived by integrating the inverse of the time constant into the transfer function equation. Furthermore, this approach allows for the calculation of phase shifts, illuminating differences in phase angles between output and input sinusoids.
Graphing amplitude and phase responses visually represent the circuit's frequency characteristics, effectively conveying fundamental aspects of its performance. As the frequency spectrum is explored, distinctive trends emerge. The transfer function approaches zero at lower frequencies, accompanied by an approximate π/2 phase shift. Conversely, the transfer function converges towards unity at higher frequencies while the phase shift approaches zero.
Of paramount importance is the identification of the half-power frequency, signifying the inverse of the time constant. At this specific frequency, an accurate assessment of the transfer function's magnitude and phase is attainable, offering a comprehensive understanding of the circuit's response characteristics.