Unit 9: Resonance and Passive Filters

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Resonant circuits and filters are two types of circuits that rely on the fact that capacitive and inductive reactances change as frequency changes.

Filters are circuits that pass signals of certain frequencies but block signals of other frequencies. Filters have many uses: one of the most obvious is in a radio tuner, which must select radio signals of one frequency out of the many frequencies present in the air around us. A series RC circuit can serve as a simple filter; its operation relies on the fact that capacitors have high reactance at low frequencies, and low reactance at high frequencies. A series RL circuit can also serve as a simple filter; its operation relies on the fact that inductors have low reactance at low frequencies, and high reactance at high frequencies.

Resonant circuits make use of the fact that, in a circuit containing capacitors and inductors, there must be a frequency at which the circuit's capacitive reactance is equal to its inductive reactance. At this frequency, called the circuit's resonant frequency, the circuit's currents and voltages typically reach extreme values (maximums or minimums). This phenomoneon is useful in building a variety of circuits, including certain types of filters.

Unit 8 Review

• Check all self-tests from here regarding usage of "impedance".
• This unit will build on material that you studied in Unit 8. So let's begin by taking this self-test to review what you learned in that unit.
• 

Impedance of Series RLC Circuits

• A series RLC circuit contains both inductive reactance (X_L) and capacitive reactance (X_C).
• Since X_L and X_C have opposite phase angles, they tend to cancel each other out, and the circuit's total reactance is smaller than either individual reactance:
  X_T < X_L and X_T < X_C

The Effect of Changing Frequency

• A series RLC circuit's reactances change as you change the voltage source's frequency. Its total impedance also changes.
• At low frequencies, X_C > X_L and the circuit is primarily capacitive.
• On the other hand, at high frequencies, X_L> X_C and the circuit is primarily inductive.

Series Resonance

• For every series RLC circuit there is one frequency, called the resonant frequency, at which X_C = X_L and the reactances perfectly cancel each other out.
• At this resonant frequency, the circuit is resistive: its total impedance is equal to its resistance:

  Z_T = R

Formula for Resonant Frequency

• For a series RLC circuit, the resonant frequency f_r is given by the formula:

  f_r = 1 ÷ (2p √(LC))

Current at Series Resonance

• The current in a series RLC circuit changes as you change the frequency.
• The current reaches its maximum value at the resonant frequency, when it's equal to

  I = V_S ÷ R

• The current is a maximum at this frequency because impedance is a minimum at this frequency.
• At lower or higher frequencies, current decreases because the circuit's total impedance increases.

Resistor Voltage at Series Resonance

• In a series RLC circuit, the resistor's voltage drop also changes as you change the frequency.
• The resistor voltage reaches its maximum value at the resonant frequency, when the resistor's voltage is equal to the source voltage.

Capacitor and Inductor Voltages at Series Resonance

• In a series RLC circuit, the capacitor's voltage drop and inductor's voltage drop also change as you change the frequency.
• These voltages reach their maximum value at the resonant frequency, but they cancel each other out so that the combined capacitor-plus-inductor voltage is zero.

Impedance of Parallel LC Circuits

• Up to this point we've been discussing resonance in series RLC circuits. As we'll see now, similar comments apply to parallel LC circuits.
• A parallel LC circuit contains both inductive reactance (X_L) and capacitive reactance (X_C), which tend to cancel each other.
• But in a parallel circuit the smaller reactance dominates, since a smaller reactance results in a larger branch current.

The Effect of Changing Frequency

• A parallel LC circuit's reactances change as you change the voltage source's frequency.
• At low frequencies, X_L < X_C and the circuit is primarily inductive.
• At high frequencies, X_C < X_L and the circuit is primarily capacitive.
• (Note: this is the opposite of what we said earlier about series RLC circuits, which are primarily capacitive at low frequencies and primarily inductive at high frequencies .)

Parallel Resonance

• For every parallel LC circuit there is one frequency, called the resonant frequency, at which X_C = X_L and the reactances cancel each other out.
• For the ideal case (in which the inductor has zero winding resistance), this resonant frequency f_r is given by the same formula as for series-resonant circuits:

  f_r = 1 ÷ (2p √(LC))

Current at Parallel Resonance

• In a parallel LC circuit, total impedance and total current change as you change the frequency.
• Total impedance reaches its maximum value at the resonant frequency, when (ideally) it's infinitely large.
• Therefore, total current reaches its minimum value at the resonant frequency, when (ideally) it's equal to zero.

The Non-Ideal Case

• Our discussion of parallel resonance up to now has assumed ideal inductors with zero winding resistance.
• For a real inductor (with non-zero winding resistance):
  • The formula for computing parallel resonant frequency is more complicated.
  • The circuit's total impedance is not infinite.
  • The circuit's total current is not zero.

Filters

• A filter is a circuit that passes signals of certain frequencies but blocks signals of all other frequencies.
• The four general categories of filters are:
  • Low-pass filters
  • High-pass filters
  • Band-pass filters
  • Band-stop filters

Frequency-response Curve

• A filter's frequency-response curve is a graph that shows how the filter's output voltage changes as the frequency of the input signal is changed.
• Such a graph has frequency on the horizontal axis and output voltage on the vertical axis.

Bode Plots

• Frequency-response curves are often drawn with the vertical (voltage) axis expressed in units of decibels, and with a logarithmic scale on the horizontal (frequency) axis.
• Such a graph is called a Bode plot , named after its creator.
• Multisim has a Bode plotter that can graph the response curve of filters and other circuits.

Low-pass Filters

• A low-pass filter passes signals whose frequencies are less than a certain frequency (called the cutoff frequency or critical frequency), but blocks higher-frequency signals.

High-pass Filters

• A high-pass filter passes signals whose frequencies are greater than the cutoff frequency, but blocks lower-frequency signals.

Band-pass Filters

• A band-pass filter passes signals whose frequencies are in a certain range, but blocks lower-frequency and higher-frequency signals.

Band-stop Filters

• A band-stop filter blocks signals whose frequencies are in a certain range, but passes lower-frequency and higher-frequency signals.
• Now that we've discussed some types of filters in terms of how they behave, let's now look at how you can build circuits that behave in these ways.

RC Filters

• A series RC circuit serves as a simple low-pass filter if the output is taken across the capacitor.
• A series RC circuit serves as a simple high-pass filter if the output is taken across the resistor.
• In either case, the cutoff frequency is given by the formula:

  f_c = 1 ÷ (2p RC)

RL Filters

• A series RL circuit serves as a simple low-pass filter if the output is taken across the resistor.
• A series RL circuit serves as a simple high-pass filter if the output is taken across the inductor.
• In either case, the cutoff frequency is given by the formula:

  f_c = 1 ÷ (2p L ÷ R)

Series-Resonant RLC Filters

• A series RLC circuit serves as a simple band-pass filter if the output is taken across the resistor.
• A series RLC circuit serves as a simple band-stop filter if the output is taken across the combined inductor-plus-capacitor.
• In either case, the filter's center frequency is equal to the circuit's resonant frequency.

Parallel-Resonant RLC Filters

• A series-parallel RLC circuit serves as a simple band-pass filter if the output is taken across the parallel inductor-plus-capacitor.
• A series-parallel RLC circuit serves as a simple band-stop filter if the output is taken across the resistor.
• In either case, the filter's center frequency is equal to the circuit's resonant frequency.

Unit 9 Review

• This e-Lesson has covered several important topics, including:
• To finish the e-Lesson, take this self-test to check your understanding of these topics.
• 

Congratulations! You've completed the e-Lesson for this unit.

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