Fundamental Physics/Electricity/Frequency Response

Frequency response is the measure of a system's response at the output to a signal of varying frequency (but constant amplitude) at its input. It is often referred to in connection with electronic audio amplifiers, loudspeakers and similar systems. The frequency response is typically characterized by the magnitude of the system's response, measured in dB, and the phase, measured in radians, versus frequency. The frequency response of a system can be measured by:

• applying an impulse to the system and measuring its response (see impulse response)
• sweeping a constant-amplitude pure frequency sinusoidal signal through the frequency bandwidth of interest and measuring the output level and phase shift relative to the input
• applying a signal with a wide frequency spectrum (e.g., maximum length sequence, white noise, or pink noise), and calculating the impulse response by deconvolution of this input signal and the output signal of the system.

Once a frequency response has been measured, providing the system is linear and time-invariant, its characteristic can be approximated with arbitrary accuracy by a digital filter. Similarly, if a system is demonstrated to have a poor frequency response, a digital or analog filter can be applied to the signals prior to their reproduction to compensate for these deficiencies.

Frequency response curves are often used to indicate the accuracy of amplifiers and speakers for reproducing audio. As an example, a high fidelity audio amplifier may be said to have a frequency response of 20 Hz - 20,000 Hz ±1 dB. This means that the system amplifies all frequencies within that range with precision of 1dB. 'Good frequency response' therefore does not guarantee a specific fidelity, but only indicates that a piece of equipment meets the basic frequency response requirements.

Resistor

$Z_{R}=R$

Resitor does not depend on frequency

Capacitor

AC response of a capacitor connected with AC voltage source

Lossless capacitor

For lossless capacitor , its reactance is calculated by

$X_{C}={\frac {v_{c}}{i_{c}}}={\frac {{\frac {1}{C}}\int idt}{c{\frac {dv}{dt}}}}$

take Fourier transform

$X_{C}={\frac {1}{j\omega C}}$

Capacitor is an electronics component depends on frequency

At $\omega =0$ , $X_{C}={\frac {1}{0}}=00$  . Capacitor opens circuit at low frequency
At $\omega =00$ , $X_{C}={\frac {1}{00}}=0$  . Capacitor shorts circuit at high frequency

For lossy capacitor

$Z_{C}=X_{C}+R_{C}$
$X_{C}={\frac {v_{c}}{i_{c}}}+R_{C}={\frac {{\frac {1}{C}}\int idt}{c{\frac {dv}{dt}}}}+R_{C}$

take Fourier transform

$Z_{C}={\frac {1}{j\omega C}}+R_{C}$

Inductor

AC response of a inductor connected with AC voltage source

Lossless inductor

For lossless capacitor , its reactance is calculated by

$X_{C}={\frac {v_{c}}{i_{c}}}={\frac {{\frac {1}{C}}\int idt}{c{\frac {dv}{dt}}}}$

take Fourier transform

$X_{L}=j\omega L$

Capacitor is an electronics component depends on frequency

At $\omega =0$ , $X_{L}={\frac {1}{00}}=0$  . Inductor shorts circuit at low frequency
At $\omega =00$ , $X_{L}={\frac {1}{0}}=00$  . Inductor opens circuit at high frequency

For lossy capacitor

$Z_{L}=X_{L}+R_{L}$
$X_{L}={\frac {v_{L}}{i_{L}}}$

take Fourier transform

$Z_{L}=R_{L}+j\omega L$