## Basics of tuned circuits

The tuned circuit is a fundamental building block among electrical and analog electronic circuits. Understanding how it functions provides insight to power quality, communications, controls, electronic equipment, and many other systems found in industrial manufacturing plants. This article provides a brief tutorial on tuned circuit theory and operation.

## RCL circuits

Recent articles in PLANT ENGINEERING magazine have discussed capacitance and inductance as separate entities (see January and May 2004 issues of PLANT ENGINEERING magazine, Plant Electronics). They explain how voltage and current behave in circuits containing only one or the other. However, many practical applications of these devices make use of the interactions between them. Although capacitance and reactance are presented as “pure” in theory, it is virtually impossible to have only one in an actual circuit.

Voltage and current in a pure inductive or a pure capacitive circuit is 90 degrees out-of-phase with voltage and current through a resistive load. Because current in a capacitive load *leads* the current through a resistive load, and the current in an inductive load *lags* the current in a resistive load, the current in a pure inductance will be 180 degrees out-of-phase with the current in a pure capacitance (Fig. 1).

When used together in a circuit, a capacitor stores the charge while the magnetic field around an inductor is collapsing. The inductor’s magnetic field expands while the capacitor discharges.

In addition, the effect of frequency on the opposition to the flow of current through each device is exactly opposite. Higher frequency currents flow more easily through capacitors than do lower frequency currents. Lower frequency currents flow more easily through inductors than do the higher frequency currents. This opposition is similar to resistance in a dc circuit, but it behaves differently when ac is present. This resistance to ac is called *reactance* . It can be thought of as ac resistance.

Opposition to current flow that is caused by inductance is called *inductive reactance* (X _{L} ); opposition to current flow that is caused by capacitance is called *capacitive reactance* (X _{C} ). Combining the effects of capacitive reactance, inductive reactance, and resistance, the total opposition to the flow of ac through a circuit is obtained. This opposition is called impedance (Z) of a circuit. X _{L,} X _{C} , Z, and resistance are measured in Ohms.

## Tank circuits

We can take advantage of this phenomenon by placing a capacitor in parallel with an inductor. In a purely resistive parallel circuit, the total resistance is always less than the value of the resistor with the lowest resistance (Fig. 2). As the frequency of the voltage applied to the parallel LC circuit increases from zero (dc), the X _{C} decreases while the X _{L} increases.

At some point, the X _{L} equals the X _{C} . At this particular frequency there is maximum opposition to the flow of current through the parallel circuit. This is because the inductive reactance decreases as the frequency decreases. As the total opposition is always less than the lowest value, the total opposition decreases (see “DC resistance”). As the frequency increases, from this point the capacitive reactance decreases, and again the net opposition decreases. This frequency is termed the resonant frequency (f _{O} ) of the circuit (Fig. 3).

If there were no dc resistance in this parallel circuit, the discharging capacitor would build up a magnetic field in the inductor. During the next cycle the collapsing magnetic field would charge up the capacitor, which would then discharge and this cyclic process continues. At this resonant frequency, the circuit would continue to cycle forever with no further need for external excitation, as the capacitance and the inductance would exactly balance each other — in theory. This condition never exists. There are always resistance losses, which dampen out the oscillations.

At the resonant frequency, the impedance of the parallel tank circuit is at its maximum, the current flow is at its minimum, and the voltage drop across the circuit is at its peak (see “Understanding Ohm’s Law”). The circuit is “tuned” to that frequency, and the voltage drop across it is the maximum that can be obtained.

There are a couple of things occurring within the tuned circuit at the resonance frequency (Fig. 4). First, the total current flowing through the tank to the external circuit is at a minimum because the Z is at the maximum value. At the same time, the current flowing *inside* the tank circuit is at a peak. The capacitor is charging and discharging while the magnetic field is expanding and collapsing. They are resonating at their natural frequency. The energy stored in the capacitor creates the magnetic field around the coil, and the coil’s collapsing field charges the capacitor. Depending on the values involved, this internal current can be extremely high.

## Series LC circuits

The ideal series LC circuit would have no resistance and consequently no losses (Fig. 5). The ideal is never achieved, but it’s useful for descriptive purposes. Just like the parallel circuit, as the frequency is varied, there is a frequency at which the X _{L} is equal to the X _{C} .

At that frequency, the coil uses the same amount of energy to build up its magnetic field as the capacitor can store in its electrical charge. The current delay through the coil is exactly canceled by the voltage delay through the capacitor. Effectively, it appears that neither the capacitor nor the inductor are in the circuit *electrically* , and the total reactance is zero, *at that frequency* (f _{O} ). Realistically, the coil, capacitor and the conductors connecting them have some resistance, which is the total circuit Z *at the resonant frequency* .

As the applied frequency varies above and below the resonant frequency, X _{L} and X _{C} are no longer equal and the circuit Z increases. With frequencies higher than resonance, the circuit behaves like an inductor with the current lagging the voltage — but not by as much as with a pure inductance. With frequencies below resonance, the circuit behaves like a capacitor with the current peak leading the voltage peak — but not by as much as with a pure capacitive circuit.

Because the Z of the series LC circuit is lowest at resonance, then the voltage drop is lowest, and more electrons get through (higher current). The only opposition to the current flow is pure resistance. The lower the resistance, the more difference there is between current flow at the resonant frequency and current flow at all other frequencies. This is the “Q” of the circuit (Fig. 6). Q has no units; it’s a measure of how much of the total impedance of the circuit is resistance as opposed to reactance. Higher Q means the circuit can be tuned more sharply to reject (or pass, depending on how it’s used) the desired frequency.

Author Information |

Wendell Rice, Instrument and Controls Engineer, Parsons Infrastructure & Technology, Pasadena, CA, has been a controls engineer for more than 25 years. He can be reached at 765-245-5357 or wendell.rice@parsons.com . Mr. Rice currently is assigned to a project in Newport, IN that provides support for the U. S. Army’s chemical weapon neutralization program. |

## DC resistance

In order to fully understand the combined effects of inductance and capacitance, it is important to have an intuitive feel for how pure resistance affects dc circuits. In series resistive circuits, the total resistance is simply the total of all the individual resistances. Each successive resistance restricts the flow of current to the next, and so on. However, in parallel circuits, the total resistance is always less than the lowest resistance.

Less resistance means more current flow; more resistance means lower current flow. If there are two or more resistances in parallel, the smallest resistance allows the highest current flow through it. Regardless of the values of the other resistors, their current flow, however small, will be in addition to the current through the smallest resistor. The total current flow will be larger and, therefore the total resistance must be less. No matter how many resistors you add in parallel, and regardless of how high their individual resistance might be, they all add a finite amount of current to the total circuit current — including the current flowing through the smallest resistor.

For equal value resistances, the total resistance of two in series equals twice the resistance of each one alone. Therefore, the resulting current is half of what would flow through just one. For two equal resistances in parallel, twice the current will flow in the resulting circuit, so the total resistance must be half of what each resistor is alone.

## Understanding Ohm’s Law

Obviously, higher resistance reduces current flow — resistance goes up, current goes down. Current is literally electrons flowing through a wire; amperage is a measure of how many electrons per second pass a given point. Resistance is like friction; it slows down the electrons.

Electrical pressure, called voltage, causes these electrons to move. A higher voltage pushing the electrons through the wire increases the number of electrons per second — more amperage. One V of electrical pressure pushes 1 A of current through 1 Ohm of resistance. Another way of saying this is that 1 A of current flowing through 1 Ohm of resistance will cause an electrical pressure drop (voltage drop) of 1 V. If the voltage stays the same, the current through a circuit will decrease when the resistance to that current increases.

When the resistance of a circuit increases there is a higher voltage drop across that circuit, just like the water pressure in your house drops as the pipes build up a layer of minerals that restrict the flow of water. More electrical pressure (voltage) is required to push a given number of electrons through a higher resistance.