Show That This Filter Can Be Described As a Low-Pass Filter

Show That This Filter Can Be Described As a Low-Pass Filter

3.1

Filters:

Since capacitors and inductors result in frequency dependent impedance elements, it is possible to create circuits that filter out (or pass) signal energy over a specified frequency region.

Example:

Find the transfer function for the circuit below and sketch magnitude of the transfer function. Describe the filtering properties of this circuit with input vi(t) and output vo(t):

Show that this filter can be described as a low-pass filter.

(i.e. as , then , and as 0, then Gain at DC > 0).

Filtering Properties of Capacitive and Inductive Elements:

Consider the impedance magnitudes of the capacitor and inductor as a function of :

Capacitor: Inductor:

  • For higher frequencies the capacitor impedance acts more like a short circuit while the inductor impedance acts more like an open circuit.
  • For lower frequencies the inductor impedance acts more like a short circuit while the capacitor impedance acts more like an open circuit.

Describe the capacitor impedance at DC.

Describe the inductor impedance at DC.

What kind of filtering goes on for current passing through a capacitor?

What kind of filtering goes on for a voltage drop across a capacitor?

What kind of filtering goes on for current passing through an inductor?

What kind of filtering goes on for a voltage drop across an inductor?

Four Basic Filter Types:

The most common types of filters are listed below:

  • Low-pass
  • High-pass
  • Band-pass
  • Band-reject (or notch)

Filter Terms:

The passband of the filter refers to a range of frequencies that pass through the filter relatively unimpeded. The stopband of the filter refers to a range of frequencies that pass through the filter with relatively strong attenuation.

The cut-off frequency refers to a frequency that determines the end of the passband region. For the band-pass and band-reject filters there are 2 cut-off frequencies. The 3 dB cut-off (or half power) frequency defines a cut-off frequency as the point where the magnitude to of the transfer function is 3 dB down from the maximum value of the transfer function.

Construction and Critical Design Parameters for the Four Basic Filter Types:

For all examples below assume voltage-input and voltage-output circuits:

Use a capacitor and resistor to construct a low-pass filter. Demonstrate it is indeed a low-pass filter. Determine its 3dB cut-off frequency (fc) and gain at DC (G0).

Exercise: Do the same as above with an inductor and a resistor.

Use an inductor and resistor to construct a high-pass filter. Demonstrate it is indeed a high-pass filter. Determine its 3dB cut-off frequency (fc) and gain as  (G0).

Exercise: Do the same as above with a capacitor and a resistor.

Use 2 capacitors and 2 resistors to construct a band-pass filter. Demonstrate it is indeed a band-pass filter. Determine its 3 dB bandwidth (B), center (resonant) frequency (fo ), and gain at resonance (G0).

Exercise: Do the same as above with an inductor, capacitor, and resistor.

Use an inductor, capacitor, and resistor to construct a band-reject filter. Demonstrate it is indeed a band-reject filter. Determine its bandwidth (B), center (anti-resonant) frequency (fo ), ratio of gain at anti-resonance (G0) to gain at DC.

Exercise: Do the same as above with 2 inductors and 2 resistors.

Filter Design Example:

For the high-pass filter shown below, determine the circuit parameters so that the 3 dB cutoff is at 1000 Hz and the gain as  approaches 40 dB.

Show that fc = 1/(2R1C) and G = Rf / R1 . (Don’t forget to convert dB to a ratio of voltage-amplitudes gain.)

Filter Design Example:

For the band-reject filter given below, determine the circuit parameters so that the center (anti-resonant) frequency is 15 kHz, and the bandwidth 1.5 kHz. Determine the resulting gain at anti resonance, DC, and as .

Show that

In general, if a second order band-reject filter can be put in canonical form:

where G = gain at DC (GDC ) = gain as  (G ),

B = 3 dB bandwidth, 0 = center frequency. The gain at anti-resonance is 0.

Active Filters:

Filters that contain active elements (amplifiers - elements that add energy to the output) are called active filters. Filters that contain only passive elements (elements that do not add energy to the system) are called passive filters.

Active filters have the following advantages over passive filters:

  • The gain of the filter is not limited between 0 and 1, and in most cases the gain can be easily set to a desired value.
  • The input and output impedance properties can be configured to eliminate loading effects. Therefore, the filter will have the same properties independent of the load.

Most active filter designs are limited to resistors, capacitors, and op amps (transistors). Inductors are not usually used because they are harder to reduce in size for implementing the circuit on a small IC.

Loading Effects:

If the output of the filter is not properly buffered, then for different loads, the frequency characteristics of the filter will change.

Example: Both low-pass filters below have fc = 1 kHz and GDC = 1 (or 0 dB).

Find fc and GDC when a 100  load is placed across the output vo(t).

Show: fc = 11 kHz and GDC = 1/11 (or -21 dB) for the passive circuit, and values remain unchanged for the active circuit.

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