Resonant Circuits and Passive Filters, Physics tutorial

Introduction:

The LC circuit is resonant circuit or tuned circuit which comprises of an inductor and a capacitor. Whenever we join them altogether, an electric current arrives at maximum at the circuit's resonant frequency. We can make use of LC circuits to either produce signals at a particular frequency, or for picking out a signal at a specific frequency from a more complex mix of signals.

They are the main components in some applications like oscillators, filters, tuners and frequency mixers and LC circuits are an idealization as it is supposed that there is no dissipation of energy due to resistance. The more practical supposition is that we approve a model incorporating resistance that dissipates a little energy as no practical circuit exists devoid of losses, In order to achieve a good understanding of the process, we shall study LC circuits in the pure form that supposes lossless elements.

LC Circuit Operation:

1239_L-C circuit operation.jpg

The LC circuit can store electrical energy vibrating at its resonant frequency. The capacitor stores energy in the electric field between its plates, based on the voltage across it, and inductor stores energy in its magnetic field, based on the current via it.

Whenever a charged capacitor is joined across an inductor, charge will began to flow via the inductor, building up a magnetic field around it and decreasing the voltage on the capacitor. Finally all the charge on the capacitor will be gone and the voltage across it will arrive at zero. Though, the current will continue, as inductors resist changes in current and energy to keep it flowing is extracted from the magnetic field that will start to decline. The current will start to charge the capacitor by a voltage of opposite polarity to its original charge. If the magnetic field is fully dissipated, then the current will stop and the charge will again be stored in the capacitor, by the opposite polarity as before. Then the cycle will start again, by the current flowing in the opposite direction via the inductor.

The charge flows backward and forward between the plates of the capacitor, via the inductor. The energy oscillates backward and forward between the capacitor and the inductor till (if not refilled by power from the external circuit) internal resistance makes the oscillations die out. Its action, termed mathematically as a harmonic oscillator, is alike to a pendulum swinging backward and forward, or water moving back and forth in a tank and for this reason, the LC circuit is as well termed as a tank circuit. The oscillation frequency is found out by the capacitance and inductance values employed. In typical tuned circuits in electronic equipment the oscillations are extremely fast, thousands to millions of times per second.

Time domain solution:

By means of Kirchhoff's voltage law, the voltage across the capacitor, VC, should equivalent the voltage across the inductor, VL:

VC = VL

Similarly, through Kirchhoff's current law, the current via the capacitor plus the current via the inductor should equivalent to zero:

iC + iL = 0

From the constitutive relations for the circuit elements, we as well familiar that:

VL(t) = L diL/dt

And iC(t) = C dVC/dt

Rearranging and replacement provides the second order differential equation:

(d2i(t)/dt2) + (1/LC) i(t) = 0

The parameter 'ω', the radian frequency, can be stated as: ω = (LC)-1/2. By employing this, we can simplify the differential equation as:

(d2i(t)/dt2) + ω2i(t) = 0

The related polynomial is s22 = 0, therefore

s = +jω

Or s = -jω

Here, 'j' is the imaginary unit. 

Therefore, the complete solution to the differential equation is:

i(t) = Ae+jωt + Be-jωt

And can be resolved for A and B by assuming the initial conditions. As the exponential is complex, the solution stands for a sinusoidal alternating current.

When the initial conditions are such that A = B, then we can make use of Euler's formula to get a real sinusoid having amplitude 2A and angular frequency:

ω = (LC)-1/2

Therefore, the resultant solut ion becomes:

i(t) = 2Acos(ωt)

The primary conditions which would persuade this result are:

i(t = 0) = 2A

Resonance effect:

The resonance effect takes place if inductive and capacitive reactances are equivalent in absolute value. (It will be noted that the LC circuit doesn't, by itself, resonate.) The term resonance signifies to a class of phenomena in which a small driving perturbation gives mount to a big effect in the system. The LC circuit should be driven, for illustration through an AC power supply, for resonance to take place. The frequency at which this equality holds for the specific circuit is termed as the resonant frequency. The resonant frequency of the LC circuit is:

ω = √(1/LC)

Here, 'L' is the inductance in Henries and 'C' is the capacitance in Farads. The angular frequency 'ω' has units of radians per second.

The equivalent frequency in units of hertz is:

f = ω/2π = 1/2π √(LC)

LC circuits are frequently employed as filters; the L/C ratio is one of the factors which find out their 'Q' and as a result selectivity. For a series resonant circuit having a given resistance, the higher the inductance and the lower the capacitance, the narrower the filter bandwidth. For a parallel resonant circuit the opposite applies. The positive feedback around the tuned circuit (regeneration) can as well raise selectivity.

Stagger tuning can give an acceptably wide audio bandwidth, yet good selectivity.

Series LC circuit Resonance:

In this, 'L' and 'C' are joined in series to an AC power supply. Inductive reactance magnitude (XL) rises as frequency rises whereas capacitive reactance magnitude (XC) reduces with the rise in frequency. At a specific frequency such two reactances are equivalent in magnitude however opposite in sign. The frequency at which this occurs is the resonant frequency (fr) for the given circuit.

Therefore, at fr:

XL = - XC

ωL = 1/ωC

Transforming angular frequency into hertz we obtain:

2πfL = 1/2πfC

In this, 'f' is the resonant frequency. Then on reorganize we get,

f = 1/2π √(LC)

In a series AC circuit, XC leads via 90 degrees whereas XL lags by 90. Thus, they cancel one other out. The only opposition to a current is the coil resistance. Therefore in series resonance the current is maximum at resonant frequency.

At fr, current is maximum. The circuit impedance is minimum. In this state a circuit is termed as an acceptor circuit.

Below fr, XL << (-XC) Therefore the circuit is capacitive. 

Above fr, XL >> (-XC) Therefore the circuit is inductive.

=> Impedance:

First assume that the impedance of the series LC circuit. The net impedance is given by the sum of the inductive and capacitive impedances:

Z = ZL + ZC 

By representing the inductive impedance as ZL = jωL and capacitive impedance as ZC = (jωC)-1 and replacing we have:

Z = jωL + (1/jωC)

By writing this expression beneath a common denominator gives:

Z = [(ω2LC - 1)j]/ωC

The numerator means that if ω2LC = 1 then the net impedance 'Z' will be zero and otherwise non-zero. Thus the series LC circuit, if joined in series by a load, will act as the band-pass filter containing zero impedance at the resonant frequency of the LC circuit.

Parallel LC circuit Resonance:

In this, a coil (L) and capacitor (C) are joined in parallel by an AC power supply. Assume that 'R' be the internal resistance of the coil. If XL equivalents XC, the reactive branch currents are equivalent and opposite. Therefore they cancel out one other to give minimum current in the main line. As the total current is minimum, therefore in this state the total impedance is maximum.

Resonant frequency given by:

f = 1/2π √(LC)

=> Impedance:

The similar analysis might be applied to the parallel LC circuit. The net impedance is then given by:

Z = ZLZC/(ZL + ZC)

And after replacing ZL and ZC and simplification, gives:

Z = (-jωL)/(ω2LC - 1)

It will be noted that:

lim ω2LC→1 Z = ∞

However for all other values of ω2LC, the impedance is finite (and as a result less than infinity). Therefore the parallel LC circuit joined in series by a load will act as band-stop filter having infinite impedance at the resonant frequency of the LC circuit.

Applications of Resonance Effect:

Most of the common application is tuning. For illustration, if we tune a radio to a particular station, the LC circuits are set at resonance for that specific carrier frequency. 

a) A series resonant circuit gives the voltage magnification. 

b) A parallel resonant circuit gives the current magnification. 

c) A parallel resonant circuit can be employed as load impedance in output circuits of RF amplifiers. Because of the high impedance, the gain of amplifier is maximum at resonant frequency. 

The LC circuits act as electronic resonators, which is a main component in such applications as Filters, Oscillators, Tuners, Mixers, Foster-Seeley discriminator, Contactless cards, Graphics tablets and  Electronic Article Surveillance (that is, Security Tags)

Electronic filters:

The electronic filters are electronic circuits that perform signal processing functions, particularly whenever we wish for to remove unwanted frequency components from  signals, to improve wanted signals, or when we wish for to do  both. We will find out that the family of electronic filters is quite a big one; including passive, active, analog, digital, High-pass, low-pass, band pass, band-reject, all-pass, discrete-time, continuous-time,  linear, non-linear, infinite impulse response and finite impulse response.

Most of the general kinds of electronic filters are linear filters in spite of other features of their design. They are passive analog linear filters, constructed by using merely resistors and capacitors or resistors and inductors. These are termed as RC and RL single-pole filters correspondingly. More complex multiple LC filters have as well existed for numerous years, and their operation is well comprehended.

Passive filters:

The Passive implementations of linear filters are mainly dependent on the combinations of resistors (R), inductors (L) and capacitors (C). Such kinds are collectively termed as passive filters, as they don't depend on an external power supply and they don't have active components like transistors.

The inductors block high-frequency signals and conduct low-frequency signals, whereas capacitors do the reverse. A filter in which the signal passes via an inductor, or in which a capacitor gives a path to ground, presents less attenuation to low-frequency signals than high-frequency signals and is a low-pass filter. When the signal passes via a capacitor, or has a path to ground via an inductor, then the filter presents less attenuation to high-frequency signals than low-frequency signals and is a high-pass filter. Resistors on their own encompass no frequency-selective properties, however are added to capacitors and inductors to find out the time-constants of the circuit and thus the frequencies to which it responds.

The capacitors and inductors are the reactive elements of the filter. The number of elements finds out the order of the filter. In this framework, an LC tuned circuit being employed in a band-pass or band-stop filter is considered as a single element even although it comprises of two components.

At high frequencies (higher than around 100 megahertz), at times the inductors comprise of single loops or strips of sheet metal and the capacitors comprise of adjacent strips of metal. Such inductive or capacitive pieces of metal are termed as stubs.

Single element types:

1011_single element types.jpg

It is a low-pass electronic filter that we can realize through an RC circuit. The simplest passive filters, RC and RL filters, comprise only one reactive element, apart from hybrid LC filter that is characterized through inductance and capacitance integrated in one element. 

L filter:

An L filter comprises of two reactive elements, one in series and one in the parallel.

T and π filters:

679_T-pi filters.jpg

This Low-pass π filter consists of the topology of the symbol π.

We can observe that this High-pass T filter in reality consists of the topology of the letter 'T'. Three-element filters can encompass a 'T' or 'π' topology and in geometries, a low-pass, high-pass, band-pass or band-stop characteristic is possible. The components can be selected symmetric or not, based on the required frequency characteristics. The high-pass 'T' filter in the above diagram consists of extremely low impedance at high frequencies and very high impedance at low frequencies. That signifies that it can be inserted in a transmission line, resultant in the high frequencies being passed and low frequencies being reflected. Similarly, for the described low-pass π filter, the circuit can be joined to a transmission line, transmitting low frequencies and reflecting high frequencies. By employing m-derived filter sections having correct termination impedances, the input impedance can be reasonably constant in the pass band.

Multiple element types:

The multiple element filters are generally constructed as a ladder network. These can be observed as a continuation of the L, T and π designs of filters. More elements are required if it is desired to enhance some of the parameter of the filter like stop-band reject ion or slope of transit ion from pass-band to stop-band.

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