LC circuit

An LC circuit consists of an inductor, represented by the letter L, and a capacitor, represented by the letter C. When connected together, an electrical current can alternate between them at an angular frequency of

<math>\omega = \sqrt{1 \over LC}</math>

where L is the inductance in henries, and C is the capacitance in farads. The angular frequency has units of radians per second. LC circuits are key components in many applications such as oscillators, filters, tuners and frequency mixers. An LC circuit is an idealized model since it assumes there is no dissipation of energy due to resistance. For a model incorporating resistance see RLC circuit.

Contents 1 Resonance effect 1.1 Series resonance 1.2 Parallel resonance 1.3 Applications of resonance effect 2 Circuit analysis 3 Impedance of LC circuits 3.1 Series LC 3.2 Parallel LC 4 Selectivity 5 Applications 6 See also 7 External links

Resonance effect

The resonance effect occurs when inductive and capacitive reactances are equal. See: Reactance. [Notice that the LC circuit does not, by itself, resonate. The word resonance refers to a class of phenomena in which a small driving perturbation gives rise to a large effect in the system. The LC circuit must be driven, for example by an AC power supply, for resonance to occur (below).] The frequency at which this equality holds for the particular circuit is called the resonant frequency. The resonant frequency of the LC circuit (in radians per second) is

<math>\omega = \sqrt{1 \over LC}</math>

The equivalent frequency in units of hertz is

<math>f = { \omega \over 2 \pi } = {1 \over {2 \pi \sqrt{LC}}} </math>

Series resonance

Here R, L, and C are in series in an ac circuit. Inductive reactance (X_L) increases as frequency increases while capacitive reactance (X_C) decreases with increase in frequency. At a particular frequency these two reactances are equal in magnitude but opposite in phase. The frequency at which this happens is the resonant frequency (f_r) for the given circuit. Hence, at f_r : X_L = X_C <math>{\omega {L}} = {{1} \over {\omega} {C}}</math> Converting angular frequency into hertz we get <math>{2 \pi fL} = {1 \over {2 \pi fC}}</math> Here f is the resonant frequency. Then rearranging, <math>f = {1 \over {2 \pi \sqrt{LC}}} </math> In a series ac circuit, X_C leads by 90 degrees while X_L lags by 90. Therefore, they both cancel each other out. The only opposition to a current is coil resistance. Hence in series resonance the current is maximum at resonant frequency.

• At f_r, current is maximum. Circuit impedance is minimum. In this state a circuit is called an acceptor circuit.
• Below f_r, X_L < X_C. Hence cct is capacitive.
• Above f_r, X_L > X_C. Hence cct is inductive.

Parallel resonance

Here a coil (L) and capacitor (C) are connected in parallel with an ac power supply. Let R be the internal resistance of the coil. When X_L equals X_C, the reactive branch currents are equal and opposite. Hence they cancel out each other to give minimum current in the main line. Since total current is minimum, in this state the total impedance is maximum. Resonant frequency given by: <math>f = {1 \over {2 \pi \sqrt{LC}}} </math>. Note that any reactive branch current is not minimum at resonance, but each is given separately by dividing source voltage (V) by reactance (Z). Hence I=V/Z, as per Ohm's law.

• At f_r,line current is minimum. Total impedance is maximum. In this state cct is called rejector circuit.
• Below f_r, cct is inductive.
• Above f_r,cct is capacitive.

Applications of resonance effect

1. Most common application is tuning. For example, when we tune a radio to a particular station, the LC circuits are set at resonance for that particular carrier frequency.
2. A series resonant circuit provides voltage magnification.
3. A parallel resonant circuit provides current magnification.
4. A parallel resonant circuit can be used as load impedance in output circuits of RF amplifiers. Due to high impedance, the gain of amplifier is maximum at resonant frequency.
5. A parallel resonant circuit can be used in induction heating.

Circuit analysis

By Kirchhoff's voltage law, we know that the voltage across the capacitor, <math>V _{C}</math> must equal the voltage across the inductor, <math>V _{L}</math>:

<math>V _{C} + V_{L}</math> = 0

Likewise, by Kirchhoff's current law, the current through the capacitor plus the current through the inductor must equal zero:

<math>i_{C} = i_{L}</math>

From the constitutive relations for the circuit elements, we also know that

<math>V _{L}(t) = L \frac{di_{L}}{dt}</math>

and

<math>i_{C}(t) = C \frac{dV_{C}}{dt}</math>

After rearranging and substituting, we obtain the second order differential equation

<math>\frac{d ^{2}i(t)}{dt^{2}} + \frac{1}{LC} i(t) = 0</math>

We now define the parameter ω as follows:

<math>\omega = \sqrt{\frac{1}{LC}}</math>

With this definition, we can simplify the differential equation:

<math>\frac{d ^{2}i(t)}{dt^{2}} + \omega^ {2} i(t) = 0</math>

The associated polynomial is <math>s ^{2} + \omega^ {2} = 0</math>, thus

<math>s = +j \omega</math>

or

<math>s = -j \omega</math>

where j is the imaginary unit.

Thus, the complete solution to the differential equation is

<math>i(t) = Ae ^{+j \omega t} + Be ^{-j \omega t}</math>

and can be solved for <math>A</math> and <math>B</math> by considering the initial conditions. Since the exponential is complex, the solution represents a sinusoidal alternating current. If the initial conditions are such that <math>A = B</math>, then we can use Euler's formula to obtain a real sinusoid with amplitude <math>2A</math> and angular frequency <math>\omega = \sqrt{\frac{1}{LC}}</math>. Thus, the resulting solution becomes:

<math>i(t) = 2 A cos(\omega t) </math>

The initial conditions that would satisfy this result are:

<math>i(t=0) = 2 A</math>

and

<math>\frac{di}{dt}(t=0) = 0</math>

Impedance of LC circuits

Series LC

First consider the impedance of the series LC circuit. The total impedance is given by the sum of the inductive and capacitive impedances:

<math>Z = Z_{L} + Z_{C}</math>

By writing the inductive impedance as <math>Z_{L} = j \omega L</math> and capacitive impedance as <math>Z_{C} = \frac{1}{j{\omega C}}</math> and substituting we have

<math>Z = j \omega L + \frac{1}{j{\omega C}}</math> .

Writing this expression under a common denominator gives

<math>Z = \frac{(\omega^{2} L C - 1)j}{\omega C}</math> .

Note that the numerator implies if <math>\omega^{2} L C =1</math> the total impedance Z will be zero and otherwise non-zero. Therefore the series connected circuit, when connected to a circuit in parallel, will act as a band-pass filter having zero impedance at the resonant frequency of the LC circuit.

Parallel LC

The same analysis may be applied to the parallel LC circuit. The total impedance is then given by:

<math>Z=\frac{Z_{L}Z_{C}}{Z_{L}+Z_{C}}</math>

and after substitution of <math>Z_{L}</math> and <math>Z_{C}</math> we have

<math>Z=\frac{\frac{L}{C}}{\frac{(\omega^{2}LC-1)j}{\omega C}}</math>

which simplifies to

<math>Z=\frac{-L\omega j}{\omega^{2}LC-1}</math> .

Note that <math> \lim_{\omega^{2}LC \to 1}Z = \infty </math> but for all other values of <math>\omega^{2} L C</math> the impedance is finite (and therefore less than infinity). Hence the parallel connected circuit will act as band-stop filter having infinite impedance at the resonant frequency of the LC circuit.

Selectivity

LC circuits are often used as filters; the L/C ratio determines their selectivity. For a series resonant circuit, the higher the inductance and the lower the capacitance, the narrower the filter bandwidth. For a parallel resonant circuit the opposite applies.

Applications

LC circuits behave as electronic resonators, which are a key component in many applications:

• Oscillators
• Filters
• Tuners
• Mixers
• Foster-Seeley discriminator
• Contactless_cards
• Graphics tablets

See also

• Resonant frequency

External links

• Lessons In Electric Circuits