# Types of tuned circuits

### Types of tuned circuits

### Tuned Circuits

Types of tuned circuits- virtually all communication equipment contains tuned circuits, circuits made up of inductors and capacitors that resonate at specific frequencies. In this section, you will review how to calculate the reactance, resonant frequency, impedance, Q, and bandwidth of series and parallel resonance circuits.

### Reactive Components

All tuned circuits and many filters are made up of inductive and capacitive elements, including discrete components such as coils and capacitors and the stray and distributed inductance and capacitance that appear in all electronic circuits.

Both coils and capacitors offer an opposition to alternating current flow known as reactance, which is expressed in ohms (abbreviated Ω). Like resistance, reactance is an opposition that directly affects the amount of current in a circuit.

In addition, reactive effects produce a phase shift between the currents and voltages in a circuit. Capacitance causes the current to lead the applied voltage, whereas inductance causes the current to lag the applied voltage. Coils and capacitors used together form tuned, or resonant, circuits.

### Capacitors:

A capacitor used in an ac circuit continually charges and discharges. A capacitor tends to oppose voltage changes across it. This translates to an opposition to alternating current known as capacitive reactance X

_{C}.

The reactance of a capacitor is inversely proportional to the value of capacitance C and operating frequency f. It is given by the familiar expression

#### Some formula:

The formula can also be used to calculate either frequency or capacitance depending on the application. These formulas are

#### Note:

Stray and distributed capacitances and inductances can greatly alter the operation and performance of a circuit.

The wire leads of a capacitor have resistance and inductance, and the dielectric has leakage that appears as a resistance value in parallel with the capacitor.

##### These characteristics, which are illustrated in Fig. 2-8, are sometimes referred to as residuals or parasitics.

The series resistance and inductance are very small, and the leakage resistance is very high, so these factors can be ignored at low frequencies. At radio frequencies, however, these residuals become noticeable, and the capacitor functions as a complex RLC circuit. Most of these effects can be greatly minimized by keeping the capacitor leads very short.

This problem is mostly eliminated by using the newer chip capacitors, which have no leads as such.

##### Figure 2-8 What a capacitor looks like at high frequencies.

#### Capacitance:

Capacitance is generally added to a circuit by a capacitor of a specific value, but capacitance can occur between any two conductors separated by an insulator. For example, there is capacitance between the parallel wires in a cable, between a wire and a metal chassis, and between parallel adjacent copper patterns on a printed-circuit board.

##### These are known as stray, or distributed, capacitances.

Stray capacitances are typically small, but they cannot be ignored, especially at the high frequencies used in communication. Stray and distributed capacitances can significantly affect the performance of a circuit.

**I**nductors:

An inductor, also called a coil or choke, is simply a winding of multiple turns of wire. When current is passed through a coil, a magnetic field is produced around the coil. If the applied voltage and current are varying, the magnetic field alternately expands and collapses.

This causes a voltage to be self-induced into the coil winding, which has the effect of opposing current changes in the coil. This effect is known as inductance.

#### The basic unit of inductance:

The basic unit of inductance is the henry (H).

Inductance is directly affect by the physical characteristics of the coil, including the number of turns of wire in the inductor, the spacing of the turns, the length of the coil, the diameter of the coil, and the type of magnetic core material. Practical inductance values are in the millihenry (mH =10

^{-3}H), microhenry (μH= 10

^{-6}H), and nanohenry (nH= 10

^{-9}H) regions.

#### Fig. 2-9 shows several different types of inductor coils.

**Fig. 2-9(a)**is an inductor made of a heavy, self-supporting wire coil.**In Fig. 2-9(b)**the inductor is formed of a copper spiral that is etched right onto the board itself.**In Fig. 2-9(c)**the coil is wound on an insulating form containing a powdered iron or ferrite core in the center, to increase its inductance.**Fig. 2-9(d)**shows another common type of inductor, one using turns of wire on a toroidal or doughnut-shaped form.**Fig. 2-9(e)**shows an inductor made by placing a small ferrite bead over a wire; the bead effectively increases the wire’s small inductance.**Fig. 2-9( f )**shows a chip inductor. It is typically no more than 1⁄8 to 1⁄4 in long. A coil is contained within the body, and the unit is soldered to the circuit board with the end connections. These devices look exactly like chip resistors and capacitors.

#### In a dc circuit:

Types of tuned circuits an inductor will have little or no effect. Only the ohmic resistance of the wire affects current fl ow. However, when the current changes, such as during the time the power is turned off or on, the coil will oppose these changes in current.

#### When an inductor is used in an ac circuit:

When an inductor is use in an ac circuit, this opposition becomes continuous and constant and is known as inductive reactance. Inductive reactance X

_{L}is expressed in ohms and is calculated by using the expression

X

_{L}= 2πfL

#### Example:The inductive reactance of a 40-μH coil at 18 MHz is

#### X_{L}= 6.28(18 ×10^{6})(40×10^{-6}) = 4522 Ω

In addition to the resistance of the wire in an inductor, there is stray capacitance between the turns of the coil. See Fig. 2-10(a).

#### The overall effect:

The overall effect is as if a small capacitor were connected in parallel with the coil, as shown in Fig. 2-10(b). This is the equivalent circuit of an inductor at high frequencies.

At low frequencies, capacitance may be ignored, but at radio frequencies, it is sufficiently large to affect circuit operation. The coil then functions not as a pure inductor, but as a complex RLC circuit with a self-resonating frequency.

##### Figure 2-9: Types of inductors. (a) Heavy self-supporting wire coil. (b) Inductor made as copper pattern. (c) Insulating form. (d) Toroidal inductor. (e) Ferrite bead inductor. (f) Chip inductor.

Types of tuned circuits any wire or conductor exhibits a characteristic inductance. The longer the wire, the greater the inductance. Although the inductance of a straight wire is only a fraction of a micro henry, at very high frequencies the reactance can be significant.

#### For this reason:

It is important to keep all lead lengths short in interconnecting components in RF circuits.

This is especially true of capacitor and transistor leads, since stray or distributed inductance can significantly affect the performance and characteristics of a circuit.

##### Figure 2-10**: **Equivalent circuit of an inductor at high frequencies. (a) Stray capacitance between turns. (b) Equivalent circuit of an inductor at high frequencies.

### Another important characteristic of an inductor is its quality factor Q, the ratio of inductive power to resistive power:

This is the ratio of the power returned to the circuit to the power actually dissipated by the coil resistance.

#### For example:

The Q of a 3-μH inductor with a total resistance of 45Ω at 90 MHz is calculated as follows:

#### Resistors:

At types of tuned circuits low frequencies, a standard low-wattage color-coded resistor offers nearly pure resistance, but at high frequencies its leads have considerable inductance, and stray capacitance between the leads causes the resistor to act as a complex RLC circuit, as shown in Fig. 2-11. To minimize the inductive and capacitive effects, the leads are keep very short in radio applications.

#### Use of tiny resistor chip:

The tiny resistor chips used in surface-mount construction of the electronic circuits preferred for radio equipment have practically no leads except for the metallic end pieces soldered to the printed-circuit board. They have virtually no lead inductance and little stray capacitance.

Many resistors are made from a carbon-composition material in powder form sealed inside a tiny housing to which leads are attach.

#### Cause of noise:

The type and amount of carbon material determine the value of these resistors. They contribute noise to the circuit in which they are use. The noise is cause by thermal effects and the granular nature of the resistance material.

The noise contribute by such resistors in an amplifier used to amplify very low level radio signals may be so high as to obliterate the desired signal.

#### Solution:

To overcome this problem, fi lm resistors were develope. They are make by depositing a carbon or metal film in spiral form on a ceramic form.

The size of the spiral and the kind of metal film determine the resistance value. Carbon fi lm resistors are quieter than carbon-composition resistors, and metal fi lm resistors are quieter than carbon film resistors.

Metal film resistors should be use in amplifier circuits that must deal with very low level RF signals. Most surface-mount resistors are of the metallic film type.

##### Figure 2-11 Equivalent circuit of a resistor at high (radio) frequencies.

#### Skin Effect:

The resistance of any wire conductor, whether it is a resistor or capacitor lead or the wire in an inductor, is primarily determine by the ohmic resistance of the wire itself.

However, other factors influence it. The most significant one is skin effect, the tendency of electrons flowing in a conductor to flow near and on the outer surface of the conductor frequencies in the VHF, UHF, and microwave regions (Fig. 2-12).

This has the effect of greatly decreasing the total cross-sectional area of the conductor, thus increasing its resistance and significantly affecting the performance of the circuit in which the conductor is used.

#### For example:

Skin effect lowers the Q of an inductor at the higher frequencies, causing unexpected and undesirable effects.

Thus many high-frequency coils, particularly those in high-powered transmitters, are made with copper tubing. Since current does not flow in the center of the conductor, but only on the surface, tubing provides the most efficient conductor.

##### Very thin conductors, such as a copper pattern on a printed-circuit board, are also used. Often these conductors are silver or gold-plated to further reduce their resistance.

#### Figure 2-12 Skin effect increases wire and inductor resistance at high frequencies.

#### Tuned Circuits and Resonance:

A tuned circuit is make up of inductance and capacitance and resonates at a specific frequency, the resonant frequency. In general, the terms tune circuit and resonant circuit are use interchangeably.

#### Reason:

Because tuned circuits are frequency-selective, they respond best at their resonant frequency and at a narrow range of frequencies around the resonant frequency.

##### Figure 2-13 Series RLC circuit.

##### Figure 2-14 Variation of reactance with frequency.

#### Series Resonant Circuits:

A series resonant circuit is make up of inductance, capacitance, and resistance, as show in Fig. 2-13. Such circuits are often refer to as LCR circuits or RLC circuits. The inductive and capacitive reactances depend upon the frequency of the applied voltage.

#### When Resonance occurs :

Resonance occurs when the inductive and capacitive reactances are equal. A plot of reactance versus frequency is show in Fig. 2-14, where f

_{r}is the resonant frequency.

The total impedance of the circuit is give by the expression

When X

_{L}equals X

_{C}, they cancel each other, leaving only the resistance of the circuit to oppose the current.

##### At resonance:

The total circuit impedance is simply the value of all series resistances in the circuit. This includes the resistance of the coil and the resistance of the component leads, as well as any physical resistor in the circuit.

The resonant frequency can be express in terms of inductance and capacitance. A formula for resonant frequency can be easily derive. First, express X

_{L}and X

_{C}as an equivalence: XL 5 XC. Since

#### In this formula:

The frequency is in hertz, the inductance is in henrys, and the capacitance is in farads.

As indicated earlier, the basic definition of resonance in a series tune circuit is the point at which X

_{L}equals X

_{C}. With this condition, only the resistance of the circuit impedes the current.

The total circuit impedance at resonance is Z= R. For this reason, resonance in a series tuned circuit can also be defined as the point at which the circuit impedance is lowest and the circuit current is highest.

Types of tuned circuits Since the circuit is resistive at resonance, the current is in phase with the applied voltage. Above the resonant frequency, the inductive reactance is higher than the capacitive reactance, and the inductor voltage drop is greater than the capacitor voltage drop.Types of tuned circuits.

Therefore, the circuit is inductive, and the current will lag the applied voltage. Below resonance, the capacitive reactance is higher than the inductive reactance; the net reactance is capacitive, thereby producing a leading current in the circuit. The capacitor voltage drop is higher than the inductor voltage drop.

##### Figure 2-15 Frequency and phase response curves of a series resonant circuit.

The response of a series resonant circuit is illustrate in Fig. 2-15, which is a plot of the frequency and phase shift of the current in the circuit with respect to frequency. At very low frequencies, the capacitive reactance is much greater than the inductive reactance;therefore, the current in the circuit is very low because of the high impedance. In addition, because the circuit is predominantly capacitive, the current leads the voltage by nearly 90°.

As the frequency increases, X

_{C}goes down and X

_{L}goes up. The amount of leading phase shift decreases. As the values of reactances approach one another, the current begins to rise.

When Types of tuned circuits X

_{L}equals X

_{C}, their effects cancel and the impedance in the circuit is just that of the resistance. This produces a current peak, where the current is in phase with the voltage (0°). As the frequency continues to rise, X

_{L}becomes greater than X

_{C}. The impedance of the circuit increases and the current decreases.

Types of tuned circuits:With the circuit predominantly inductive, the current lags the applied voltage. If the output voltage were being take from across the resistor in Fig. 2-13, the response curve and phase angle of the voltage would correspond to those in Fig. 2-15.

As Fig. 2-15 shows, the current is highest in a region center on the resonant frequency. The narrow frequency range over which the current is highest is call the bandwidth. This area is illustrate in Fig. 2-16.

**Figure 2-16 Bandwidth of a series resonant circuit.**

The upper and lower boundaries of the bandwidth are define by two cutoff frequencies designate f1 and f

_{2}. These cutoff frequencies occur where the current amplitude is 70.7 percent of the peak current. In the figure, the peak circuit current is 2 mA, and the current at both the lower (f

_{1}) and upper (f

_{2}) cutoff frequency is 0.707 of 2 mA, or 1.414 mA.

Current levels at which the response is down 70.7 percent are call the half-power points because the power at the cutoff frequencies is one-half the power peak of the curve.

P= I

^{2}R= (0.707 I

_{peak})

^{2}R =0.5I

_{peak}

^{2}R

Types of tuned circuits the bandwidth BW of the tuned circuit is define as the difference between the upper and lower cutoff frequencies:

BW= f

_{2}- f

_{1}

For example, assuming a resonant frequency of 75 kHz and upper and lower cutoff frequencies of 76.5 and 73.5 kHz, respectively, the bandwidth is BW= 76.5μ73.5 = 3 kHz.

The bandwidth of a resonant circuit is determin by the Q of the circuit. Recall that the Q of an inductor is the ratio of the inductive reactance to the circuit resistance. This holds true for a series resonant circuit, where Q is the ratio of the inductive reactance to the total circuit resistance, which includes the resistance of the inductor plus any additional series resistance:

Recall that bandwidth is then compute as

If the Q of a circuit resonant at 18 MHz is 50, then the bandwidth is BW= 18/50=0.36 MHz = 360 kHz.

#### Parallel Resonant Circuits:

A parallel resonant circuit is form when the inductor and capacitor are connect in parallel with the applied voltage, as show in Fig. 2-19(a).

##### Figure 2-19 Parallel resonant circuit currents. (a) Parallel resonant circuit. (b) Current relationships in parallel resonant circuit.

In general, resonance in a parallel tuned circuit can also be defined as the point at which the inductive and capacitive reactances are equal. The resonant frequency is therefore calculate by the resonant frequency formula give earlier. If we assume lossless components in the circuit (no resistance), then the current in the inductor equals the current in the capacitor:

I

_{L}=I

_{C}

Although the currents are equal, they are 180° out of phase, as the phasor diagram in Fig. 2-19(b) shows. The current in the inductor lags the applied voltage by 90°, and the current in the capacitor leads the applied voltage by 90°, for a total of 180°.

Now, by applying Kirchhoff’s current law to the circuit, the sum of the individual branch currents equals the total current drawn from the source. With the inductive and capacitive currents equal and out of phase, their sum is 0.

Thus, at resonance types of tuned circuits, a parallel tuned circuit appears to have infinite resistance, draws no current from the source and thus has infinite impedance, and acts as an open circuit. However, there is a high circulating current between the inductor and capacitor.

Energy is being store and transferred between the inductor and capacitor. Because such a circuit acts as a kind of storage vessel for electric energy, it is often refer to as a tank circuit and the circulating current is refer to as the tank current.

In a practical resonant circuit where the components do have losses (resistance), the circuit still behaves as described above. Typically, we can assume that the capacitor has practically zero losses and the inductor contains a resistance, as illustrated in Fig. 2-20(a).

#### Figure 2-20 A practical parallel resonant circuit. (a) Practical parallel resonant circuit with coil resistance RW. (b) Phase relationships.

At resonance types of tuned circuits, where X

_{L}= X

_{C}, the impedance of the inductive branch of the circuit is higher than the impedance of the capacitive branch because of the coil resistance.

The capacitive current is slightly higher than the inductive current. Even if the reactances are equal, the branch currents will be unequal and therefore there will be some net current flow in the supply line.

The source current will lead the supply voltage, as shown in Fig. 2-20(b). Nevertheless, the inductive and capacitive currents in most cases will cancel because they are approximately equal and of opposite phase, and consequently the line or source current will be significantly lower than the individual branch currents. The result is a very high resistive impedance, approximately equal to

The circuit in Fig. 2-20(a) is not easy to analyze. One way to simplify the mathematics involved is to convert the circuit to an equivalent circuit in which the coil resistance is translate to a parallel resistance that gives the same overall results, as shown in Fig. 2-21.

##### The equivalent inductance L_{eq} and resistance R_{eq} are calculate with the formulas:

where R

_{W}is the coil winding resistance.

If Q is high, usually more than 10, L

_{eq}is approximately equal to the actual inductance value L. The total impedance of the circuit at resonance is equal to the equivalent parallel resistance:

Z= R

_{eq}

#### Figure 2-21 An equivalent circuit makes parallel resonant circuits easier to analyze.

A frequency and phase response curve of a parallel resonant circuit is show in Fig. 2-22. Below the resonant frequency, XL is less than XC; thus the inductive current is greater than the capacitive current, and the circuit appears inductive.

The line current lags the applied voltage. Above the resonant frequency, XC is less than XL; thus the capacitive current is more than the inductive current, and the circuit appears capacitive.

Therefore, the line current leads the applied voltage. The phase angle of the impedance will be leading below resonance and lagging above resonance.

**Note: The bandwidth of a circuit is inversely proportional to the circuit Q. The higher the Q, the smaller the bandwidth. Low Q values produce wide bandwidths or less selectivity.**

At the resonant frequency, the impedance of the circuit peaks. This means that the line current at that time is at its minimum. At resonance, the circuit appears to have a very high resistance, and the small line current is in phase with the applied voltage.

**Figure 2-22**Response of a parallel resonant circuit.

#### Note that:

the Q of a parallel circuit, which was previously express as Q 5 X

_{L}/R

_{W}, can also be compute with the expression

where R

_{P}is the equivalent parallel resistance, R

_{eq}in parallel with any other parallel resistance, and X

_{L}is the inductive reactance of the equivalent inductance L

_{eq}.

You can set the bandwidth of a parallel tune circuit by controlling Q. The Q can be determine by connecting an external resistor across the circuit. This has the effect of lowering RP and increasing the bandwidth.