Resonance & Filter Circuits (RLC)

https://drstienecker.com/tech-261-material/16-resistorinductorcapacitor-circuits-chapter-16/

TOPIC 1: Resonance

When an inductor and a capacitor are used in a series circuit their associated reactance works against each other and can operate at a certain frequency as if there were no inductor or capacitor in the circuit at all.

This phenomenon is commonly used to function as a filter either only letting through a certain range of frequencies or only blocking a certain range of frequencies.

In this situation, the frequencies near resonance are not affected by the reactance in the circuit but when the frequencies are far away from resonant frequency the impedance is much higher and they are affected greater.  These component values could be used in a circuit to filter out all frequencies below 3500Hz and above 3600Hz.

TOPIC 2: Low Pass Filters

However, many filters are even simpler than the RLC circuit.  A simple low pass filter consists of a resistor and a capacitor and all frequencies below a specified frequency are allowed to “pass” through the filter, while all frequencies greater than the specified frequency are “blocked”.

TOPIC 3: High Pass Filters

A High Pass Filter is the opposite of the Low Pass Filter because it allows all frequencies above a specified frequency to “pass” through and blocks all frequencies lower.

TOPIC 4: Band Pass and Band Reject Filters

The Band Pass Filter allows a small window of frequencies to pass through and blocks all frequencies above and below this window.  The Band Pass Filter is just a simple combination of the Low Pass and High Pass filters.  The corner frequency of the low pass filter must be chosen high than the frequency of the high pass filter.  The same equation applies to determine each corner frequency.

A Band Reject Filter will reject all frequencies in a window and allow all frequencies greater or lesser than the window to pass.  The center of the reject window is determined by the corner frequency equation from the low pass or high pass filter.

You should now be prepared to answer the following questions.

1. If an inductor of 100mH and a capacitor of 100nF where used in a series circuit together give the resonant frequency.
2. If a resistor (1000 Ohms), an inductor (250mH), and a capacitor (180nF) where placed in series with one another what is the change in impedance that would occur if an AC power source applied to the circuit was changed from 1000Hz to 2000Hz?
3. A low pass filter is needed to filter a signal coming from a microphone to eliminate high frequency electrical noise.  You decide to filter out everything above 18kHz and you are stuck using a 10nF capacitor but have an array of different resistors to choose from.  Give the resistance you need to complete the circuit.
4. A certain vibration sensor produces a frequency as its output based on the frequency of the vibration.  This sensor is then sent to a computer for processing but along the way it picks up high frequency electrical noise through crosstalk with other signals in the computer.  Additionally the sensor picks up some low frequency vibrations that you want to ignore in your results.  Both the high frequency noise and the low frequency vibrations need to be filtered out with a Band Pass Filter.  The area of interest to your test ranges between 10kHz and 25kHz.  Once again you are stuck with using a 10nF capacitor.  Give the two values of resistors needed for the circuit.

Q factor final Math

The Q, quality factor, of a resonant circuit is a measure of the “goodness” or quality of a resonant circuit.

        Q = P_stored/P_dissipated = I²X/I²R

        Q = X/R

        where:         X = Capacitive or Inductive reactance at resonance

                       R = Series resistance.

A high Q resonant circuit has a narrow bandwidth as compared to a low Q

   BW = fc/Q

        Where fc = resonant frequency 

        Q = quality factor 

Bandwidth, Δf is measured between the 70.7% amplitude points of series resonant circuit.

        BW = Δf = fh-fl = fc/Q

        Where: fh = high band edge fl = low band edge

        fl = fc - Δf/2

        fh = fc + Δf/2

        Where fc = center frequency (resonant frequency) 

In Figure above, the 100% current point is 50 mA. The 70.7% level is .707(50 mA)=35.4 mA. The upper and lower band edges read from the curve are 291 Hz for f_l and 355 Hz for f_h. The bandwidth is 64 Hz, and the half power points are ± 32 Hz of the center resonant frequency:

        BW = Δf = fh-fl = 355-291 = 64

        fl = fc - Δf/2 = 323-32 = 291

        fh = fc + Δf/2 = 323+32 = 355

Since BW = f_c/Q:

        Q = fc/BW = (323 Hz)/(64 Hz) = 5

What is Q-factor or Quality factor ?

# Q- factor measures the quality of a circuit or the sharpness of resonance means how the sharp circuit is . The more sharpness , the circuit has more tendency to receive a particular signal.

# Defination of Q factor .

It is defined as the ratio of the resonant frequency to the the band width frequency at which the current is reduced to 70% of the resonant current .

To understand it clearly .

Renonant frequency - in this frequency current is maximum in the circuit .

Bandwidth frequency - it is the frequency at which current is reduced to 70% of the resonant current (You may find w1 and w2 in the book , the w1 and w2 may has different frequency but at this frequency current is reduced to 70% of the resonant frequency ) . So, more the value of Q , the band width tries to become small vallue and sharpness of resonance increases .

Also , there are many definations .

Here , i do not claim what i said , its true . Good luck .

#

Q means Quality Factor, a factor which determines the efficiency of an LCR circuit.

A circuit which contains only L (inductor) and C (capacitor) works as a Harmonic Oscillator (or an undamped harmonic oscillator). Adding an ohmic resistance R makes it a Damped Harmonic Oscillator since it is responsible for energy dissipation.

Ohmic resistance R in LCR circuit also causes damping and thus dissipates energy in the oscillator)

The oscillations in this LCR circuit involves transfer of energy from capacitor ( in the form of electric field) to inductor ( in the form of magnetic field) and back to it again and again. 

But don't forget there is R also present there, when it comes to play, and as oscillations proceed, the energy remaining in the fields gradually diminishes or dissipates ( due to damping nature of ohmic resistance in this LCR circuit)

q factor simply means how efficient a system is , higher q factor means low energy loss. for example q factor of inductor is the ratio between the inductive reactance and ohmic resistance, in which higher inductive reactance means less power loss.

Q factor is a dimensionless parameter that describes how underdamped an oscillator or resonator is, and characterizes a resonator's bandwidth relative to its center frequency.