Series and Parallel Circuits and Resonant Circuits

 Electronic components like resistors, capacitors and inductors can be connected together in series or parallel. Components connected in series are connected along a single electrical path and same electric current passes through each of the components, which will be equal to the current through the network. Voltage across the series network will be equal to the sum of voltages across each component. On the other hand, when components are connected in parallel, they have multiple current paths and each component has the same voltage across it, which will be equal to the voltage across the network. Current through the network is the sum of currents through each component. Circuit with only components connected in series can be called a series circuit and a circuit with only components connected in parallel is called a parallel circuit. Most electronic circuits will have a combination of series and parallel circuits.

As we have seen earlier, when resistors are connected in series, the total resistance will be the sum of the resistances. Similarly, when inductors are connected in series, total inductance will be the sum of the inductances. But for capacitors, we have to take the inverse of the capacitances, add them and finally get the inverse of that sum. When inductors or resistors are connected in parallel, it is the same method to get the total inductance or resistance. For capacitors connected in parallel, the total capacitance is the sum of their capacitances. But things become much more complicated if you combine resistors, inductors and capacitors in series or parallel. Reactance and impedance of inductors and capacitors come into play and they are dependent on the frequency of the alternating current as well. We have already discussed the calculation of reactance of individual components.

When an inductor and a capacitor are connected in series, the reactance of the inductance tends to cancel the capacitive reactance. So the total reactance will be the difference between the inductive and capacitive reactances. That is the ideal case when there is no resistance in the circuit which is practically impossible. In real circuits there will be some resistance of the coil forming the inductance, which may be negligible so as to be ignored in calculations. Reactance of the inductor is directly proportional to the frequency while that of a capacitor is inversely proportional. For every combination of inductor and capacitor, there will be a frequency at which inductive reactance will fully cancel the capacitive reactance. That will be the resonant frequency of the combination.

If the resistance in the circuit is negligibly low, then current can go very high at resonant frequency. Resonance for a particular frequency can be achieved by varying the capacitance of a variable capacitor to match the reactive inductance of a coil for the frequency to be received. In a series alternating current circuit inductive reactance leads the zero reference angle of resistance by 90 degrees while the capacitive reactance lags by 90 degrees in phase. Therefore the inductive and capacitive reactances are 180 degrees out of phase, meaning that they are opposite to each other and tend to cancel each other at resonance. At resonance 2πfL = 1/(2πfC). In a series resonant circuit, the impedance is lowest at the resonant frequency. Frequencies quite away from the resonant frequency have a high impedance and will not pass through a series resonant circuit.

It is the other way round for a parallel resonant circuit. Parallel resonant circuits are also called 'tank' circuits as they offer high impedance to the resonant frequency and 'captures' the resonant frequency within it. Frequencies away from the resonant frequency are shunted down to the ground as the other end of the tank circuit is usually grounded.

Parallel resonant circuit is often used for tuning of radios. The antenna coil couples inductively to the tank circuit and the tank circuit resonates at the tuned frequency. Tuning is achieved by turning the rotor of a variable capacitor. Signals greatly amplified by tuning is then taken to other circuits for further processing. Though L₁ captures all the frequencies from the antenna, only the resonant frequency will be practically transferred to other circuits through L₂ for further processing. Other frequencies are grounded by the tuned circuit. In case of amateur radio antennas, another level of tuning is occurring in resonant antennas like a dipole antenna cut to the resonant frequency.

Resonant circuits are also used in frequency filter. Series resonant circuit allows the resonant frequency to pass through while a parallel resonant circuit blocks the resonant frequency. High pass filter allows frequencies above a certain limit to pass through while a low pass filter allows frequencies below a certain limit to pass through. They also find application in traps used in multi-band antennas. Traps are essentially resonant circuits which do not allow the resonant frequency to pass beyond them so that the effective length of the antenna becomes shorter for that frequency. Using multiple traps allow the same antenna, typically a vertical antenna, to be used for multiple bands.

An inductor with less ohmic resistance can be considered as a better one and the quality or Q factor is used to express this. Q can be calculated by dividing the inductive reactance with the resistance of the coil. Substituting the formula for inductive reactance, Q = 2πfL/R, where f is the frequency and L the inductance of the coil. It can be understood from the formula that the same coil or inductor has a higher Q at higher frequency. 

At higher frequencies, skin effect also causes loss of efficiency of a coil or inductor. At higher frequencies, electrons flow nearer to the surface of the conducting wire. This is because of opposing eddy currents in the conductor induced by the changing magnetic field resulting from the alternating current. As the current flows only through a small portion of the cross section of the wire, ohmic resistance is more and the Q becomes lower. The depth up to which current flows in the conductor is known as the skin depth.