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Series RLC Circuit

Current is the same in any series circuit. This major characteristic must always be remembered as you solve for series circuit values. Opposition to this current is impedance. In a series RLC circuit, impedance consists of resistance and reactance. Coil or inductive reactance is X_L = 6.28 fL, and capacitive reactance is X_C = 0.159/fC. Series current then develops voltage across the circuit components. Note the inductor voltage leads, while the capacitor voltage lags the referenced resistor voltage.

The resistor ohmic value remains the reference as X_L and X_C are plotted to show that each reactance is, indeed, 180 degrees apart. Since this is true, subtraction of the values will yield total reactance. Total reactance is now vectored with resistance to obtain the reactance angle for the total circuit. Next the impedance triangle is used and the Pythagorean theorem reveals the total "Z" in ohms. Verify this with your calculator (a check is Z = V/I). When reactances are subtracted, a negative-sign result tells us that the circuit is capacitive.

IR voltage drops must follow Kirchhoff's voltage law, but remember that the total must be a vector result with reactive components installed. Note that reactor voltages subtract as well. The phase difference between V_S and I may be calculated with the arctan formula for either ratio of total X (X_L – X_C) or total V (V_L – V_C) divided by R.

Power in a series RLC circuit is similar to what you have learned about series RL and RC power. Since both reactive powers are imaginary, true power still remains across the resistor, therefore, P_R = I²R. Apparent power (P_A) is V_T x I² with a unit value of volt-amperes. The power factor remains the same. When the ratio of true to apparent power is 1, or towards 1, the circuit is resistive, and when towards 0, or 0, the circuit is reactive.

Parallel RLC Circuit

When resistors, inductors, and capacitors are connected in parallel, voltage remains as the reference, is equal, and is in phase across all components. However, it should be noted that it is current that changes. With each component it is in phase, or leads, or lags.

Once resistance and individual reactances determined by value and frequency are known, branch currents are calculated using Ohm's law. Currents are calculated using ac ohmic values. After reactive currents are subtracted, the Pythagorean theorem is used to determine (vector) total current. Again the phase angle may be obtained with the arctan formula. The ratio is the subtracted reactance current divided by the resistive current. A negative angle indicates that current is lagging the source voltage.

Parallel impedance may be calculated with Ohm's law (Z), after total current has been determined. Total current is also used for apparent power calculation. True power is still resistive, and the cosine of the angle remains the power factor.

Resonance

Resonance is a circuit condition that occurs when the inductive reactance is equal to the capacitive reactance or X_L = X_L. When a dc voltage is applied to an LC circuit, the inductor will act as a short, while the capacitor will act as an open. With ac, capacitive reactance will continue to fall as frequency increases, while inductive reactance will climb as frequency increases. When the two reactances are equal in ohmic value, resonance has been reached.

For a given value inductor with a given value capacitor, there can only be one resonant frequency. The formula 0.159/square root of LC determines a given set of components' resonant frequency. As it increases, frequency determines reactance; its plot will be first below, at, and then above resonance. Notice with the vector arrows that reactances are equal at the resonant frequency, and that the circuit is considered capacitive below resonance, and more inductive above resonance.

LC circuits act differently when wired in series than they do when wired in parallel. Here, for the sake of discussion, the ohmic values of all series components are the same. If they are different, the voltage across the resistor will still be in phase with the series current, and the voltage across the capacitor remains 180 degrees out of phase with the voltage across the inductor. But here, their equal voltages cancel when measured in series. Remember, this only occurs at resonance. Other things occur too. Total impedance at resonance is only equal to R, because the reactances cancel. Because only R affects the circuit, voltage and current are now in phase, power is only I²R, and the power factor is 1. When R is removed, or is a small value, individual LC voltage drops can far exceed the source.

Another characteristic of a series resonant circuit, when the circuit is at resonance, is coil quality or selectivity. As mentioned in the inductance chapter, the Q factor is the ratio of reactance to resistance, or X_L/R.

Selectivity determines the value of the bandwidth that will cause a larger circuit current than that from frequencies well above or below resonance. Bandwidth is more than just a high current frequency. It consists of additional side frequencies that allow at least 70.7% of this maximum current to flow. If you recall, this percentage is half-power. The two half-power points make up the total bandwidth, or one power point indicates half a bandwidth. Bandwidth is also proportional to the resonant frequency and inversely proportional to its resonant quality.

BW therefore equals f/Q. As R is increased, BW flattens or increases, but selectivity (Q) decreases. As Q decreases, bandwidth increases, but current becomes lower. A high value of Q will cause a high value of current, producing a narrow or very selective bandwidth. As Q decreases, the bandwidth of a series resonant circuit increases, causing unwanted frequencies to be included.

In a parallel resonant circuit, component currents are of different values and phase relationships. Still, they will offer a high impedance at the frequency of resonance. For a given frequency inductor, current and capacitor current are in 180 degree opposition. As valued, the current will flow first in one direction through the inductor and then in the opposite direction through the capacitor. In the ideal case, current due to capacitor discharge, coil buildup and collapse, and capacitor recharge simply flows back and forth. This is known as the "flywheel" effect.

If there were no losses in the circuit, the flywheel effect would never stop. However, this is the real world, and so occasionally circuit degeneration or loss will occur; therefore, the circuit must be allowed to have source current occasionally flow to keep the tank circuit resonating. With a parallel tank circuit at resonance, external current is minimum, while inside the tank current is at maximum. Noted another way, tank impedance to the circuit is maximum, and minimum internally. With series resonance, it was just the opposite.

With parallel resonance, since current is the biggest factor, quality (Q) is determined by I_TANK/I_SOURCE, or Z_TANK/X_L, as well as X_L/R.

In response curves for a parallel resonant circuit, when impedance is maximum, current is minimum. With frequencies below resonance, X_L is low and X_C is high, with the opposite being true above resonance. This is just the opposite of series resonance. Bandwidth, however, uses the same formula, but high Q components with narrow selectivity are a concern in parallel situations. External "loading" of a tank circuit changes selectivity. This alternate path for line current decreases Q, resulting in a wider bandwidth to compensate for those devices that have very high, unwanted Q.

Applications of RLC Circuits

Bandpass filter circuits pass a group of frequencies between a lower and an upper cutoff frequency while heavily attenuating frequencies outside this band. The series circuit, due to low impedance at resonance, simply passes the required frequencies to the output load resistor. With a parallel tank, impedance is high at resonance, so very little current is shunted away from the output load resistor. A parallel resonant tank coil may be replaced by a transformer.

A band-stop filter attenuates those frequencies that are between the upper and lower cutoff values, while allowing the passage of frequencies above and below the resonant band. Impedance is low near resonance, therefore center frequencies are shunted away from the output. The parallel tank impedance is high, blocking center frequencies from passage. In both cases, those frequencies above and below the resonant band will be passed to the output.

QUALITY CONTROL

• Resonance occurs when inductive reactance and capacitive reactance values within a tuned circuit are balanced.
• A given resonant frequency is inversely proportional to the inductance and capacitance values in the circuit.
• In a series RLC circuit, impedance is resistive and current is maximum at resonance. In a parallel RLC circuit, impedance and circulating current are maximum, and source current is minimum. In a series resonant circuit, reactive voltages cancel one another due to their opposite, but equal, values.
• The Q of a circuit is a ratio of reactance to resistance or reactive power (stored) to resistive (lost) power. It indicates the quality of the circuit. Selectivity is better at a moderately high Q, which results in a narrower bandwidth.
• A low-pass filter provides an output of frequencies below the cutoff frequency, while a high-pass filter provides an output of frequencies above its cutoff frequency.
• A bandpass filter passes a selected band of frequencies, while a band-stop or band-reject filter prevents a band of frequencies from being a useable output.

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