Electrical Engineering Concepts

The average power consumed by a resistor is $I_{rms}^2 R$, by a capacitor is $I_{rms}^2 X_c$, and by an inductor is $I_{rms}^2 X_l$.

For a complex number $z = x + yi$, the magnitude |z| is given by $|z| = \/sqrt{x^2 + y^2}$, and the phase angle θ is given by $θ = \/arctan(y/x)$.

In the context of impedance, the complex generalization of resistance is impedance, denoted by Z [Ω].

The first order differential equation describing the time evolution of UR(t) in a series sinusoidally forced RC circuit is $R \frac{dUR}{dt} + UR = V_m \sin(ωt)$.

Sinusoidal voltages and currents can be expressed as $V(t) = V_m \cos(ωt + φ)$ and $I(t) = I_m \cos(ωt + φ)$, where $V_m$ and $I_m$ are the maximum voltages and currents, ω is the angular frequency, and φ is the phase angle.

For an inductor, the phase shift is +90 degrees and the impedance is jωL.

UR(t) = VR * cos(ωt - φ)

A phasor is a complex number representing the amplitude and phase of a sinusoidal voltage or current.

The instantaneous phase θ of z is the angle measured anti-clockwise from the x-axis in the Argand diagram.

In parallel, the complex impedances add reciprocally: 1/Z = 1/Z1 + 1/Z2 + 1/Z3 + ...

The instantaneous phase of a complex number $z$ is represented by the angle $\theta$ measured anti-clockwise from the x-axis in the Argand diagram.

The expression for the potential difference across the resistor $UR(t)$ in a series sinusoidally forced RC circuit is obtained by applying Kirchhoff's voltage law (KVL) to establish the 1st order differential equation describing the time evolution of $UR(t)$.

When resistors, capacitors, and inductors are connected in parallel, the complex impedances are combined using the reciprocal of the sum of the reciprocals, for resistors and inductors, and directly for capacitors.

The magnitude $|z|$ of a complex number $z$ in the Argand diagram can be expressed as $\sqrt{x^2 + y^2}$, and the angle $\theta$ is given by $\arctan\left(\frac{y}{x}\right)$.

The average power consumed by each component per cycle can be calculated using the formula $P_{\text{avg}} = \frac{V_{\text{rms}}^2}{|Z|}$, where $V_{\text{rms}}$ is the root mean square voltage and $|Z|$ is the magnitude of the complex impedance.