Low-pass filters

By definition, a low-pass filter is a circuit offering easy passage to low-frequency signals and difficult passage to high-frequency signals. There are two basic kinds of circuits capable of accomplishing this objective, and many variations of each one: The inductive low-pass filter in Figure below and the capacitive low-pass filter in Figure below

Inductive low-pass filter

The inductor's impedance increases with increasing frequency. This high impedance in series tends to block high-frequency signals from getting to the load. This can be demonstrated with a SPICE analysis: (Figure below)

inductive lowpass filter          
v1 1 0 ac 1 sin
l1 1 2 3
rload 2 0 1k
.ac lin 20 1 200
.plot ac v(2)
.end

The response of an inductive low-pass filter falls off with increasing frequency.

Capacitive low-pass filter.

The capacitor's impedance decreases with increasing frequency. This low impedance in parallel with the load resistance tends to short out high-frequency signals, dropping most of the voltage across series resistor R₁. (Figure below)

capacitive lowpass filter                             
v1 1 0 ac 1 sin
r1 1 2 500 
c1 2 0 7u  
rload 2 0 1k
.ac lin 20 30 150  
.plot ac v(2)
.end

For the capacitive low-pass filter with R = 500 Ω and C = 7 µF, the Output should be 70.7% at 45.473 Hz.

           fcutoff = 1/(2πRC) = 1/(2π(500 Ω)(7 µF)) = 45.473 Hz

When dealing with filter circuits, it is always important to note that the response of the filter depends on the filter's component values and the impedance of the load. If a cutoff frequency equation fails to give consideration to load impedance, it assumes no load and will fail to give accurate results for a real-life filter conducting power to a load.

One frequent application of the capacitive low-pass filter principle is in the design of circuits having components or sections sensitive to electrical “noise.” As mentioned at the beginning of the last chapter, sometimes AC signals can “couple” from one circuit to another via capacitance (C_stray) and/or mutual inductance (M_stray) between the two sets of conductors. A prime example of this is unwanted AC signals (“noise”) becoming impressed on DC power lines supplying sensitive circuits: (Figure below)

Noise is coupled by stray capacitance and mutual inductance into “clean” DC power.

The oscilloscope-meter on the left shows the “clean” power from the DC voltage source. After coupling with the AC noise source via stray mutual inductance and stray capacitance, though, the voltage as measured at the load terminals is now a mix of AC and DC, the AC being unwanted. Normally, one would expect E_load to be precisely identical to E_source, because the uninterrupted conductors connecting them should make the two sets of points electrically common. However, power conductor impedance allows the two voltages to differ, which means the noise magnitude can vary at different points in the DC system.

If we wish to prevent such “noise” from reaching the DC load, all we need to do is connect a low-pass filter near the load to block any coupled signals. In its simplest form, this is nothing more than a capacitor connected directly across the power terminals of the load, the capacitor behaving as a very low impedance to any AC noise, and shorting it out. Such a capacitor is called a decoupling capacitor: (Figure below)

Decoupling capacitor, applied to load, filters noise from DC power supply.

A cursory glance at a crowded printed-circuit board (PCB) will typically reveal decoupling capacitors scattered throughout, usually located as close as possible to the sensitive DC loads. Capacitor size is usually 0.1 µF or more, a minimum amount of capacitance needed to produce a low enough impedance to short out any noise. Greater capacitance will do a better job at filtering noise, but size and economics limit decoupling capacitors to meager values.

• REVIEW:
• A low-pass filter allows for easy passage of low-frequency signals from source to load, and difficult passage of high-frequency signals.
• Inductive low-pass filters insert an inductor in series with the load; capacitive low-pass filters insert a resistor in series and a capacitor in parallel with the load. The former filter design tries to “block” the unwanted frequency signal while the latter tries to short it out.
• The cutoff frequency for a low-pass filter is that frequency at which the output (load) voltage equals 70.7% of the input (source) voltage. Above the cutoff frequency, the output voltage is lower than 70.7% of the input, and vice versa.

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