# Electronics/Expanded Edition Resonance

## ResonanceEdit

Simple resonant circuit, description.

## Amplification, Q, form factorsEdit

## FiltersEdit

### Ideal FiltersEdit

(Discussion of how ideal filters provide the signal until the break frequency/cies and then provide total attenuation. That is they look like a for low pass filter, where u(w) is the unit step function or Heaviside function. That is they have infinite drop off at the cutoff frequency. How this is not possible. Pretty diagrams of all the filters Low Pass, High Pass, Band Pass, Band Stop.)

This section introduces first order butterworth low pass and high pass filters. An understanding of Laplace Transforms or at least Laplace Transforms of capacitors, inductors and resistors.

### Low PassEdit

Transforming the Resistor and Capacitor to the Laplace domain we get:

- R and .

Expressing using terms of .

The transfer function is

So

For the Frequency Domain we put

The magnitude is

and the angle is

As increases decreases so this circuit must represent low pass filter.

Using the -3 dB definition of band width.

Therefore

Which gives the general form of a low pass butterworth filter as:

, where k is the order of the filter and is the cut-off frequency.

### High PassEdit

- (Image of a first order RL high pass filter)

If all the component of the circuit are transformed into the Laplace Domain. The resistor becomes and the inductor becomes . Using voltage divider rule below is reached.

If is transformed into the frequency domain by putting .

Which has a magnitude of

and an angle of

- (cut-off frequency is w (R/L)^0.5)