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# December 2017

## Low-pass Filtered Signals

All periodic signals can be represented through *Fourier analysis* as a sum of sinusoidal oscillations whose frequencies are integral multiples of the fundamental signal frequency. For example, a *square-wave signal **s*(*t*) of a frequency *f*_{0} is represented as follows:

## Cut-off Frequency of an RC Element

A RC element's frequency response depends on the ohmic resistance *R* and the capacitance *C*. However, the basic amplitude and phase characteristics (curve shapes) are always the same. A change in resistance merely results in a corresponding *phase shift* of the curves on the frequency axis. An RC element's characteristic parameter is its *cut-off frequency* *f*_{G}, given by the following equation:

## RC Element as a Frequency-Dependent Voltage Divider

The diagram below shows the previously considered RC element once again in a slightly different configuration. Here it is supplied with an alternating voltage *U*_{0} (input voltage to the element) and itself supplies an alternating voltage *U*_{C} to the capacitor (output voltage to the element). This configuration makes it clear that an RC element acts as a *frequency-dependent *voltage divider.

## Phase Shift in an RL Element

An ideal coil comprises a pure reactance *X*_{L} where the voltage leads the current by a phase angle j = 90°, the two vectors thus forming a right angle (illustrations on the left below). Connecting an ohmic resistance and inductance in series to form an *RL element* produces, as in the case of the RC element considered earlier, an overall resistance comprising an active and a reactive component.

## Reactance of a Coil

As in the case of a capacitor, the current and voltage of an ideal inductor coil (i.e. one with no active resistance) are displaced in phase by 90°, resulting in purely reactive power. The coil's higher AC resistance compared with its DC resistance is due to the *inductive reactance X* _{L}. This reactance arises from an opposing, self-induced voltage.

## Sinusoidal Alternating Voltage for a Coil

As observed during energising and de-energising of a coil, current starts to flow through an inductor after a certain delay. Connecting a pure inductance (i.e. a coil with an ohmic resistance of zero) to a sinusoidal alternating voltage of the form

## Energizing and De-energizing a Coil

The processes of energising and de-energising a coil are already familiar from the treatment of *DC circuits*. If a coil is connected to a direct voltage source as shown in the diagram below (switch closed), a voltage *U*_{ind} is self-induced at the same time that a current starts to flow. According to Lenz's law, this voltage is in the opposite direction to the externally applied DC voltage *U*_{0}.

## Series and Parallel Connection of Coils

Inductors connected in series or parallel are governed by the same formulae as those applicable to resistive loads. The following diagram shows an array of inductors connected in series.

## Inductance of a Coil

Inductance of a coil In addition to the electric fields that occur, for instance, between the plates of a charged capacitor, electrical engineering also relies on a second kind of field, magnetic fields. While electric fields are generated by static charge, magnetic fields are generated by moving charge carriers, i.e. electric current. If an electric current passes through a coil comprising an array of conductive windings, magnetic field lines run through the coil as a result.

## Phase Shift in a RC Element

If a sinusoidal voltage is applied to an ohmic resistance *R*, the resulting current is in phase with the voltage, i.e. the corresponding vectors *I* and *U* of the rms values point in the same direction (illustrations on the left below). A capacitor acts as a pure reactance *X*_{C}, where the voltage trails the current by a phase angle of j = 90° so that the two corresponding vectors form a right angle (illustrations in the centre below).

## Reactance of a Capacitor

The instantaneous value *p*(*t*) of the power consumed by a capacitor is the product of the instantaneous current and voltage. Because these two variables are separated in phase by 90° in a capacitor, however, the power consumption characteristic has a frequency that is double that of the voltage and current themselves, as shown by the green curve in the diagram below.

## Sinusoidal Alternating Voltage for a Capacitor

If a capacitor is subjected to a *sinusoidal* alternating voltage of the form: