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Filter circuits
From our work with Laplace transforms (and the AC analysis work from EE 201), it is clear that the behavior of a circuit depends on the frequency used. The impedance of the reactive elements varies with frequency.
As s → 0 , ZC → ∞. An open circuit.
As s → ∞ , ZC → 0. A short circuit.
As s → 0 , ZL → 0. A short circuit.
As s → ∞ , ZL → ∞. An open circuit.
We can use this frequency dependence to build circuits that discriminate between frequencies. The circuit would allow signals at some frequencies to pass through while signals at other frequencies are attenuated – filtered out.
V&
=/ = V/
M&
( =/ = M/ )
As s → 0, ZC → ∞ and ZL → 0.
As s → ∞ , ZC → 0 and ZL → ∞.
Consider
This would be a low-pass filter. Low frequency signals are passed through to the output. High-frequency
signals are cut off — Vo = 0).
Vi ( s ) Vo ( s ) = Vi ( s )
Vi ( s ) Vo ( s ) = 0
L
C
Vi ( s ) Vo ( s )
ZL = sL
V&
Basic filter types
Low Pass: Frequencies below fc pass through, and those above are cut off.
fc cut-off frequency
High Pass: Frequencies above fc pass through, and those below are cut off.
fc cut-off frequency
Band reject: Frequencies between fc1 and fc2 are blocked. Everything else is
allowed to pass through.
fc1 fc
bandwidth: BW = fc2 – fc1 = ∆f
fnull : frequency at
which minimum
occurs
quality factor: QP = fnull / ∆f
Transfer function
The properties of a filter circuit can be specified in detail by the transfer function, defined in terms of either voltages or currents. (Using voltages is most common.)
Soon, we will learn how to recognize the basic properties of most filters
by examining T ( s ). However, when we need to compute some specific
property of a filter, we will move to AC analysis, where s = jω.
The transfer function can be plotted as a function of frequency. The graph is called a frequency response plot. A complete frequency
response is actually two plots, since T ( jω ) is a complex number and complex numbers have two components, described by rectangular (real and imaginary) or polar (magnitude and phase) forms. In working with filters, we almost always use magnitude and phase.
The general shape of the curve is characteristic of the type of filter.
7 ( V ) =
9 R ( V )
9 L ( V )
7 ( M ) =
9 R ( M )
9 L ( M )
Cut-off frequency
For the ideal filters, it is fairly obvious where to locate the dividing line between the passband and the cut-off region. For realistic filters, the transition is gradual and it is less obvious where the dividing line between pass and cut-off should be.
The usual convention is to choose the frequency where the magnitude of
the transfer function is down by (= 0.707) from the peak value.
- Find the peak of the magnitude of transfer function, | T ( jω )| max — low frequencies for low pass, high frequencies for high-pass, in the middle of the pass band for bandpass.
- Find the frequency where the magnitude is
This is also the frequency at which the power being transmitted through a passive circuit is one-half of the power provided by the source. So it is also know as the half-power frequency.
PD[
7 ( MF ) =
PD[
7 PD[
7 PD[
fc
7 PD[
fc
f or ω?
7 PD[
To some degree, it doesn’t matter which you use, just make sure that you
keep them straight. f is the real frequency and is the number of
oscillations per second (Hz or hertz). This is used in the lab and is
usually the number that is reported. The angular frequency ω is the
number of radians per second. This is what goes into our equations.
Important: ω = 2 π f !!!!
fc
Bode plot
In the magnitude plot, using straight magnitude on the vertical axis is OK, but most practitioners make a Bode plot, which uses decibels on the vertical axis.
fo = 1000 Hz GHFLEHOV · log (| 7 |)
· log
at fo : = G%
corner frequency cut-off frequency 3-dB frequency
