Connecting RL Combination to Output Time Derivative of Input Voltage

[e_sovalye]

1-How could an RL combination be connected to produce an output voltage which is the time derivative of the input voltage?

2-Show that taking the time derivative of a sinusoidal function [such as cos(wt+a)] always has the effecton increasing its phase pi/2.

3-İf the internal series resistance of an inductor is not negligible , how will this change the relative phase of voltage and current fort he inductor?

 

[fargoth]

1. parallel connecet R to the input and parallel to the put L, the voltage on L is proportional to the derivative of the current which is proportional to the input voltage...
2. you know what the derivative of cos(wt+a)? what's the phase change between sin and cos?

 

[piercas]

1- An RL combination can be connected in series to produce an output voltage that is the time derivative of the input voltage. This can be achieved by connecting the input voltage to the resistor and the output voltage is taken across the inductor. The inductor will act as a low-pass filter, allowing only the high frequency components of the input voltage to pass through. As a result, the output voltage will be proportional to the rate of change of the input voltage, which is the time derivative.

2- Taking the time derivative of a sinusoidal function, such as cos(wt+a), will always result in an increase in phase by pi/2. This can be seen by differentiating the function, which results in -w sin(wt+a). The negative sign indicates a phase shift of pi/2 or 90 degrees. This is due to the fact that the derivative of a sinusoidal function is a cosine function, which is shifted by pi/2 from the original function.

3- If the internal series resistance of an inductor is not negligible, it will change the relative phase of voltage and current for the inductor. This is because the internal resistance will cause a voltage drop across the inductor, leading to a phase shift between the input voltage and the voltage across the inductor. This phase shift will depend on the magnitude of the internal resistance and the frequency of the input voltage. At low frequencies, the phase shift will be small, but at high frequencies, it can be significant. This is an important consideration in circuit analysis and design, as it can affect the overall performance of the circuit.