THIS PAGE IS UNDER CONSTRUCTION.

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.

- 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.

- 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}

- 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.

- 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**

- For a series
*RLC*circuit, the resonant frequency*f*_{r}is given by the formula:*f*_{r}= 1 ÷ (2p √(*LC*))

- 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.

- 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.

- 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.

- 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.

- 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 .)

- 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*))

- 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.

- 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.

- 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

- 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.

- 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.

- 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.

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

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

- 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.

- 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*)

- 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*)

- 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.

- 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.

- 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.