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.

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 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.

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 XC. 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

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 complex RLC circuits. 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. Capacitance is generally added to circuits by a capacitor of a specific value, but capacitance can occur between any two conductors separated by an insulator. For example, there is the 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.

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 is the henry (H). Inductance is directly affected 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 5 1023 H), microhenry (μH 5 1026 H), and nanohenry (nH 5 1029 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 boar 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, an inductor will have little or no effect. Only the ohmic resistance of the wire affects the current flow. 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, this opposition becomes continuous and constant and is known as inductive reactance. Inductive reactance XL is expressed in ohms and is calculated by using the expression

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

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 microhenry, at very high frequencies the reactance can be signifi cant.

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. 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 45V at 90 MHz is calculated as follows:

At 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 kept very short in radio applications. 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 powdered form sealed inside a tiny housing to which leads are attached. The type and amount of carbon material determine the value of these resistors.

They contribute noise to the circuit in which they are used. The noise is caused by thermal effects and the granular nature of the resistance material. The noise contributed by such resistors in an amplifi er used to amplify very low level radio signals may be so high as to obliterate the desired signal. To overcome this problem, fi lm resistors were developed. They are made by depositing a carbon or metal fi lm in spiral form on a ceramic form. The size of the spiral and the kind of metal fi lm determine the resistance value. Carbon fi lm resistors are quieter than carbon-composition resistors, and metal fi lm resistors are quieter than carbon fi lm resistors. Metal fi lm resistors should be used in amplifi er circuits that must deal with very low level RF signals. Most surface-mount resistors are of the metallic fi lm type.

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

Tuned circuits are made up of inductance and capacitance and resonate at a specific frequency, the resonant frequency. In general, the terms tuned circuits and resonant circuit are used interchangeably. Because tuned circuits are frequency-selective, they respond best at their resonant frequency and at a narrow range of frequencies around the resonant frequency.

A series resonant circuit is made up of inductance, capacitance, and resistance, as shown in Fig. 2-13. Such circuits are often referred to as LCR circuits or RLC circuits. The inductive and capacitive reactances depend upon the frequency of the applied voltage. Resonance occurs when the inductive and capacitive reactances are equal. A plot of reactance versus frequency is shown in Fig. 2-14, where fr is the resonant frequency.

When XL equals XC, 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 expressed in terms of inductance and capacitance. A formula for resonant frequency can be easily derived. First, express XL and XC as an equivalence: XL = XC. Since

In this formula, the frequency is in hertz, the inductance is in henrys, and the capacitance is in farads.

**Example 2-14** What is the resonant frequency of a 2.7-pF capacitor and a 33-nH inductor?

It is often necessary to calculate capacitance or inductance, given one of those values and the resonant frequency. The basic resonant frequency formula can be rearranged to solve for either inductance and capacitance as follows:

**Example 2-15** What value of inductance will resonate with a 12-pF capacitor at 49 MHz?

As indicated earlier, the basic definition of resonance in a series tuned circuits is the point at which XL equals XC. 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 circuits can also be defined as the point at which the circuit impedance is lowest and the circuit current is highest. 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. 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.

The response of a series resonant circuit is illustrated 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, XC goes down and XL goes up. The amount of leading phase shift decreases. As the values of the reactances approach one another, the current begins to rise. When XL equals XC, 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, XL becomes greater than XC. The impedance of the circuit increases and the current decreases. With the circuit predominantly inductive, the current lags the applied voltage. If the output voltage were being taken 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 centered on the resonant frequency. The narrow frequency range over which the current is highest is called the bandwidth. This area is illustrated in Fig. 2-16. The upper and lower boundaries of the bandwidth are defined by two cutoff frequencies designated f1 and f2. 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 ( f1) and upper ( f2) cutoff frequency is 0.707 of 2 mA or 1.414 mA. Current levels at which the response is down 70.7 percent are called the half-power points because the power at the cutoff frequencies is one-half the power peak of the curve.

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 = 76.5 – 73.5 = 3 kHz. The bandwidth of a resonant circuit is determined 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:

**Example 2-16** What is the bandwidth of a resonant circuit with a frequency of 28 MHz and a Q of 70?

The formula can be rearranged to compute Q, given the frequency and the bandwidth:

**Q = fr/BW**

Thus the Q of the circuit whose bandwidth was computed previously is Q = 75 kHz /3kHz = 25. Since the bandwidth is approximately centered on the resonant frequency, f1 is the same distance from fr as f2 is from fr. This fact allows you to calculate the resonant frequency by knowing only the cutoff frequencies:

If the circuit Q is very high (>100), then the response curve is approximately symmetric around the resonant frequency. The cutoff frequencies will then be roughly equidistant from the resonant frequency by the amount of BW/2. Thus the cutoff frequencies can be calculated if the bandwidth and the resonant frequency are known:

Keep in mind that although this procedure is an approximation, it is useful in many applications. The bandwidth of a resonant circuit defines its selectivity, i.e., how the circuit responds to varying frequencies. If the response is to produce a high current only over a narrow range of frequencies, a narrow bandwidth, the circuit is said to be highly selective. If the current is high over a broader range of frequencies, i.e., the bandwidth is wider, the circuit is less selective. In general, circuits with high selectivity and narrow bandwidths are more desirable. However, the actual selectivity and bandwidth of a circuit must be optimized for each application.

The relationship between circuit resistance Q and bandwidth is extremely important. The bandwidth of a circuit is inversely proportional to Q. The higher Q is, the smaller the bandwidth. Low Qs produce wide bandwidths or less selectivity. In turn, Q is a function of the circuit resistance. A low resistance produces a high Q, a narrow bandwidth, and a highly selective circuit. A high circuit resistance produces a low Q, wide bandwidth, and poor selectivity. In most communication circuits, circuit Qs are at least 10 and typically higher. In most cases, Q is controlled directly by the resistance of the inductor. Fig. 2-17 shows the effect of different values of Q on bandwidth.

**Example 2-17** The upper and lower cutoff frequencies of a resonant circuit are found to be 8.07 and 7.93 MHz. Calculate (a) the bandwidth, (b) the approximate resonant frequency, and (c) Q.

**Example 2-18** What are the approximate 3-dB down frequencies of a resonant circuit with a Q of

200 at 16 MHz?

Resonance produces an interesting but useful phenomenon in a series RLC circuit. Consider the circuit in Fig. 2-18(a). At resonance, assume XL 5 XC 5 500 V. The total circuit resistance is 10 V. The Q of the circuit is then

When the reactances, the resistances, and the current are known, the voltage drops across each component can be computed:

As you can see, the voltage drops across the inductor and the capacitor are significantly higher than the applied voltage. This is known as the resonant step-up voltage. Although the sum of the voltage drops around the series circuit is still equal to the source voltage, at resonance, the voltage across the inductor leads the current by 90° and the voltage across the capacitor lags the current by 90° [see Fig. 2-18(b)]. Therefore, the inductive and reactive voltages are equal but 180° out of phase. As a result, when added, they cancel each other, leaving a total reactive voltage of 0. This means that the entire applied voltage appears across the circuit resistance.

The resonant step-up voltage across the coil or capacitor can be easily computed by multiplying the input or source voltage by Q: VL = VC = QVs In the example in Fig. 2-18, VL = 50(2) = 100 V.

This interesting and useful phenomenon means that small applied voltages can essentially be stepped up to a higher voltage—a form of simple amplification without active circuits that is widely applied in communication circuits.

**Example 2-19** A series resonant circuit has a Q of 150 at 3.5 MHz. The applied voltage is 3 μV.

What is the voltage across the capacitor?

A parallel resonant circuit is formed when the inductor and capacitor are connected in parallel with the applied voltage, as shown in Fig. 2-19(a). In general, resonance in parallel tuned circuits can also be defined as the point at which the inductive and capacitive reactances are equal. The resonant frequency is therefore calculated by the resonant frequency formula given earlier. If we assume lossless components in the circuit (no resistance), then the current in the inductor equals the current in the capacitor:

IL = IC

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, parallel tuned circuits appear 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 the capacitor. Energy is being stored and transferred between the inductor and capacitor. Because such a circuit acts as a kind of storage vessel for electric energy, it is often referred to as a tank circuit and the circulating current is referred 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). At resonance, where XL = XC, 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

**Z = Vs/ IT **

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 translated to a parallel resistance that gives the same overall results, as shown in Fig. 2-21. The equivalent inductance Leq and resistance Req are calculated with the formulas

where RW is the coil winding resistance. If Q is high, usually more than 10, Leq 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=Req**

**Example 2-20** What is the impedance of a parallel LC circuit with a resonant frequency of 52 MHz

and a Q of 12? L 5 0.15 μH.

If the Q of the parallel resonant circuit is greater than 10, the following simplified formula can be used to calculate the resistive impedance at resonance:

**Z = L/CRW**

The value of RW is the winding resistance of the coil.

**Example 2-21** Calculate the impedance of the circuit given in Example 2-20 by using the formula

Z = L/CR.

This is close to the previously computed value of 592 V. The formula Z = L/CRW is an approximation.

A frequency and phase response curve of a parallel resonant circuit is shown 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 to the applied voltage. The phase angle of the impedance will be leading below resonance and lagging above resonance. 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 very high resistance, and the small line current is in phase with the applied voltage.

**Note** that the Q of a parallel circuit, which was previously expressed as Q 5 XL /RW, can also be computed with the expression

Q = RP/XL

where RP is the equivalent parallel resistance, Req in parallel with any other parallel resistance, and XL is the inductive reactance of the equivalent inductance Leq. You can set the bandwidth of parallel tuned circuits by controlling Q. The Q can be determined by connecting an external resistor across the circuit. This has the effect of lowering RP and increasing the bandwidth.d

**Example 2-22** What value of the parallel resistor is needed to set the bandwidth of a parallel tuned

circuit to 1 MHz? Assume XL = 300 V, RW = 10 V, and fr = 10 MHz.

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