Introduction to Frequency

Selective Circuits

� A graph of |H(jω)| versus frequency ω, called the

magnitude plot.

� A graph of θ(jω) versus frequency ω, called the phase

angle plot.

� The signals passed from the input to the output fall within

a band of frequencies called the passband.

� Frequencies not in a circuit’s passband are in its stopband.

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� A series RL circuit is shown in Fig. The circuit’s input is a sinusoidal voltage source with varying frequency.

� Recall that the impedance of an inductor is jωL. At low frequencies, the inductor’s impedance is very small compared with the resistor’s impedance, and the inductor effectively functions as a short circuit. The term low frequencies thus refers to any frequencies for which ωL << R.

� At high frequencies, the inductor’s impedance is very large compared with the resistor’s impedance, and the inductor thus functions as an open circuit, effectively blocking the flow of current in the circuit. The term high frequencies thus refers to any frequencies for which ωL >> R.

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� Based on the behavior of the output voltage

magnitude, this series RL circuit selectively

passes low-frequency inputs to the output,

and it blocks high-frequency inputs from

reaching the output.

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� The cutoff frequency is defined as when the voltage across the resistor and the voltage across the inductor or capacitor are equal.

� Compute the power delivered to the load at the cutoff frequency:

� The cutoff frequency ωc, the average power delivered by the circuit is one half the maximum average power. Thus, ωc is also called the half-power frequency. In the filter’s passband, the average power delivered to a load is at least 50% of the maximum average power.

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� The series RC circuit shown in Fig. also behaves as a low-pass filter.

� Zero frequency (ω = 0): The impedance of the capacitor is infinite, and the capacitor acts as an open circuit. The input and output voltages are thus the same.

� Frequencies increasing from zero: The impedance of the capacitor decreases relative to the impedance of the resistor, and the source voltage divides between the resistive impedance and the capacitive impedance. The output voltage is thus smaller than the source voltage.

� Infinite frequency (ω = ∞): The impedance of the capacitor is zero, and the capacitor acts as a short circuit. The output voltage is thus zero.

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� A series RC circuit is shown in

Fig. The output voltage here is

defined across the resistor, not

the capacitor.

� At ω = 0, the capacitor behaves

like an open circuit, so there is

no current in the resistor.

� When the frequency of the

source is infinite (ω = ∞), the

capacitor behaves as a short

circuit, so the capacitor voltage

is zero.

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� The ideal bandpass filters have two cutoff frequencies, ωc1 and ωc2, which identify the passband.

� The center frequency, ωo, defined as the frequency for which a circuit’s transfer function is purely real. Another name for the center frequency is the resonant frequency. The center frequency is the geometric center of the passband; that is, ωo = ωc1ωc2

� .

� The bandwidth, β, which is the width of the passband.

� The final parameter is the quality factor Q, which is the ratio of the center frequency to the bandwidth.

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� Figure depicts a series RLC circuit.

� At ω = 0, the capacitor behaves like an open circuit, and the inductor behaves like a short circuit. The open circuit representing the capacitor impedance prevents current from reaching the resistor, and the resulting output voltage is zero.

� At ω = ∞, the capacitor behaves like a short circuit, and the inductor behaves like an open circuit. The inductor now prevents current from reaching the resistor, and again the output voltage is zero.

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� But what happens in the frequency region between ω= 0 and ω = ∞? Between these two extremes, both the capacitor and the inductor have finite impedances.

� Remember that the capacitor impedance is negative, whereas the inductor impedance is positive. Thus, at some frequency, the capacitor impedance and the inductor impedance have equal magnitudes and opposite signs; the two impedances cancel out, so the output voltage equals the source voltage.

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� At the center frequency, ωo, the phase angles of the input and output voltages are equal, so the phase angle of the transfer function is zero.

� As the frequency decreases, the capacitor phase angle is larger than the inductor phase angle. Because the capacitor contributes positive phase shift, the transfer function phase angle is positive. At very low frequencies, the transfer function phase angle is +90°.

� Conversely, if the frequency increases from the center frequency, the inductor phase angle is larger than the capacitor phase angle. The inductor contributes negative phase shift, so the transfer function phase angle is negative. At very high frequencies, the transfer function phase angle is –90°.

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� Recall that the center frequency, ωo, is

defined as the frequency for which the

circuit’s transfer function is purely real. The

will be real when the capacitor and inductor

impedances sum to zero:

Solving for ωo, we get

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� A graphic equalizer is an audio amplifier that

allows you to select different levels of

amplification within different frequency

regions. Using the series RLC circuit, choose

values for R, L, and C that yield a bandpass

circuit able to select inputs within the 1 kHz–

10 kHz frequency band. Such a circuit might

be used in a graphic equalizer to select this

frequency band from the larger audio band

(generally 0–20 kHz) prior to amplification.

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Solution

� We need to select a value for either R, L, or C

and use the two equations we’ve chosen to

calculate the remaining component values. Here,

we arbitrarily choose 1 µF as the capacitor value.

We compute the center frequency as the

geometric mean of the cutoff frequencies:

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� Next, find L using C and the center frequency,

which must be converted to radians/sec:

The bandwidth is the difference between the two

cutoff frequency values, so

Now convert the bandwidth to calculate R:

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� To check whether these component values

produce the bandpass filter we want. We find

that

which are the cutoff frequencies specified for

the filter.

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� Bandreject filters and bandpass filters have

the same characteristic parameters: the two

cutoff frequencies, the center frequency, the

bandwidth, and the quality factor. Again,

only two of these five parameters can be

specified independently.

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� This is because in the bandreject filter, the center

frequency is in the stopband, not in the passband. Because

the sum of the capacitor and inductor impedances is zero

at the center frequency,

� The cutoff frequencies, the bandwidth, and the quality

factor are defined and calculated for the bandreject filter

and the bandpass filter in exactly the same way.

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� Use the cutoff frequencies to generate an

expression for the bandwidth, β:

� The center frequency and the bandwidth

produce an equation for the quality factor, Q:

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� The bandwidth and center frequency, as we

did for the bandpass filter:

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