Inductive versus Capacitive Reactance

First, let’s review some basic considerations regarding inductive and capacitive reactance. Recall that both inductive and capacitive reactance are not only dependent upon the value of the inductor or capacitor, but they are also dependent upon the applied frequency. This is evident in the equations shown, 14–1 for inductive reactance and 14–2 for capacitive reactance.

(14–1)

(14–2)

As was shown previously for an inductor, as the applied frequency increases, the reactance of the inductor (equation 14–1) increases if the value of the inductance remains constant.

This is shown again in Figure 14.1, a graph for a 10 millihenry inductor. For a capacitor, as the applied frequency increases, the reactance of the capacitor decreases if the value of the capacitor remains constant.

Figure 14.2 shows how capacitive reactance of a 4 microfarad capacitor decreases as the applied frequency increases.

Resonant Frequency

Suppose the 10 millihenry inductor and 4 microfarad capacitor are connected in series with one another as shown in Figure 14.3. Now, as shown in Figure 14.4, as frequency increases, the value of the inductive reactance increases, and the value of the capacitive reactance decreases. At one value of frequency the inductive reactance and capacitive reactance are equal. For this example, the frequency value is 796 hertz. At this frequency the circuit is said to be at resonance and the frequency is called the resonant frequency of this circuit. The value of the resonant frequency of this circuit can be determined mathematically by setting the capacitive reactance equal to the inductive reactance as shown in equation 14–3. This is the basic definition of resonance.

(14–3)

Substituting equation 14–1 for X_L and equation 14–2 for X_C into equation 14–3 gives,

By solving for f it is found that

where L is the value of the inductor in henrys, C is the value of the capacitor in farads, 2 π is the constant 6.28, and f is the resonant frequency in hertz. Since this is the resonant frequency, a subscript “r” is used to distinguish it from all other frequencies:

(14–4)

Equivalent Resonant Frequency Equations

For a specific inductance and capacitance, equation 14–4 is used to determine the frequency at which their reactances are equal in value — the resonant frequency.

Suppose, however, that you want to know what value of inductor will provide a certain resonant frequency when connected with a given capacitor or vice–versa. These values can be determined by rewriting equation 14–4 into equivalent forms;

(14–5)

or similarly,

(14–6)

Impedance

Now, consider the effect of resonance on a series RLC circuit such as the one illustrated in Figure 14.5. You know that the general expression for the impedance of any series RLC circuit may be expressed as:

(14–7)

At resonance, from equation 14–3,

and the net reactance of the circuit is zero. The impedance, then, at resonance, is

(14–8)

Thus, at resonance, the impedance of the series RLC circuit appears entirely resistive and is equal to the resistance in the circuit.

To further clarify this fact, note that the impedance phasor diagram for a series RLC circuit is as shown in Figure 14.6. If X_L is equal in value to X_C, as in the case at resonance, the net reactance of the circuit is zero, and the impedance phasor falls directly upon the resistance phasor. This indicates that the impedance is equal to the value of the resistance at resonance.

Minimum Impedance at Resonance

Since X_L equals X_C, the net reactance is zero and thus there is no reactive term in the impedance equation, equation 14–7. As shown in Figure 14.4 if the applied frequency to the circuit is varied either side of the resonant frequency, then a difference will exist between X_L and X_C. This difference would result in a net reactance and would add vectorially to the resistance in the circuit to create a larger impedance. Such circuits were the type discussed in previous chapters. Thus, it should be quite clear that at the resonant frequency of a series circuit the impedance expressed by equation 14–7 is at its lowest level, as expressed by equation 14–8.

Current Is Maximum at Resonance

Since the impedance of the circuit is at a minimum, it follows that the current flowing in the circuit must be a maximum. This can be seen by using Ohm’s law. Remember that for a series RLC circuit the total current is equal to the applied voltage divided by the circuit impedance:

(14–9)

At resonance, the impedance is equal to the resistance alone. Therefore, the total current is equal to the applied voltage divided by the resistance.

(14–10)

Since in this circuit the resistance alone represents a minimum impedance, the current must be maximum.

Voltage Phase Relationships at Resonance

Recall that the voltage phasor diagram is proportional to the impedance phasor diagram by a factor of the total current. This is shown in Figure 14.7. At resonance, when X_L = X_C, then the voltage across the inductor, EL, must equal the voltage across the capacitor, E_C. If E_L = E_C, these voltages cancel because they are 180 degrees out of phase. As a result the applied voltage, E_A, is in phase with and equal to the voltage across the resistor, as shown in Figure 14.8.

Phase Angle Equals Zero

Assume the total current is taken as reference at zero degrees. Since the current is in phase with the voltage across the resistor, and at resonance the circuit becomes entirely resistive, then the applied voltage is in phase with the total current and the phase angle is zero. This also is shown in Figure 14.8.

Q of a Circuit

A circuit is sensitive to changes when used at the resonant frequency. This is determined by the Q or quality factor of the circuit. Recall that in the discussion of inductance in Chapter 8 that the Q of a circuit was defined as the ratio of the reactive power of a reactive component to the real power dissipated in the resistance in the circuit. This is expressed again in equation 14–11.

(14–11)

Let’s consider the reactive power of the inductor in a series RLC circuit, such as the one shown in Figure 14.5. The reactive power of the inductor is equal to the voltage across the inductor times the current through it:

(14–12)

Since E_L = I_LX_L, equation 14–12 can be rewritten as the inductive current squared times the value of inductive reactance. The result is equation 14–13.

(14–13)

The value of the real power dissipated by the resistance in the circuit is equal to the voltage across the resistance times the current through it (equation 14–14).

Using the same reasoning, this can be rewritten:

(14–14)

Substituting these two equations into equation 14.11 for Q,

(14–15)

In the series circuit, the current through the inductor equals the current through the resistor, thus,

(14–16)

As a result equation 14–15 simplifies to equation 14–17,

(14–17)

If the capacitive reactance had been used, the equation would be:

(14–18)

Thus, the Q of a resonant series RLC circuit is equal to the value of the inductive or capacitive reactance at the resonant frequency divided by the value of the series resistance in the circuit.

Voltage Magnification

Using the original equation 14–11 for Q,

and substituting the E–I equivalents for the power values, equation 14–19 results.

(14–19)

Since the current through the inductor, the current through the resistor, and the total current in the series circuit are the same, I_T = I_L = I_R, both I_L and I_R in equation 14–19 can be replaced by I_T.

(14–20)

Cancelling I_T, equation 14–20 simplifies to:

(14–21)

Q is equal to E_L divided by E_R.

Earlier you saw in the voltage phasor diagram of Figure 14.8 that the voltage across the resistor and the applied voltage in a series circuit were equal at resonance. Substituting E_R=E_Ainto equation 14–21 results in:

(14–22)

If both sides of equation 14–22 are multiplied by EA and rearranged, the result is equation 14–23:

(14–23)

Equation 14–23 relates the fact that at resonance, the voltage across the inductor will be Q times larger than the applied voltage. This is referred to as the voltage magnification in a series resonant circuit, and Q is sometimes referred to as the magnification factor.

Measuring Q

One of the easiest methods that can be used to measure the actual Q of a resonant circuit is based on the fact stated in equation 14–23. The first step in the method is to connect a frequency generator to the circuit as shown in Figure 14.9. Next, the circuit is brought to resonance by adjusting the frequency of the generator. With the circuit at resonance, a voltmeter is used to measure the applied voltage and the voltage across the inductor or capacitor as shown in Figure 14.9.(Either the inductor or capacitor can be used because the voltage across either is the same at resonance.) The value of Q is determined by equation 14–22 by simply dividing the voltage across the reactive component by the applied voltage.

The rise in voltage across the inductor or capacitor at resonance also provides a means of determining when a circuit is resonant. For example, if the voltage across the inductor is monitored as the frequency is varied, when the circuit becomes resonant the current reaches a peak value and the voltage across the inductor rises to a maximum of Q times the applied voltage.

Resonant Frequency

Since L is 10 millihenrys and C is 4 microfarads the resonant frequency is determined by using equation 14–4.

(14–4)

Substituting values,

The resonant frequency of the circuit is 796 hertz.

Checking X_L= X_C at Resonance

Recall that it was stated that at the resonant frequency, the reactance of the inductor and the capacitor are equal. To prove this for the 10–millihenry inductor and the 4–microfarad capacitor, their individual reactances at the resonant frequency of 796 hertz are calculated.

The reactance of the inductor at 796 hertz is calculated using equation 14–1.

Next, the capacitive reactance at 796 hertz is calculated using equation 14–2.

Thus, at the resonant frequency, X_L = X_C = 50 ohms.

Impedance

The magnitude of the impedance of this circuit in its most general form is calculated using equation 14–7:

(14–7)

Substituting R = 10 ohms and X_L = 50 ohms and X_C = 50 ohms:

The impedance of this series circuit, when resonant, is 10 ohms, which is the value of the circuit resistance.

Current

The value of the series circuit current at resonance is determined by dividing the applied voltage by the impedance of the circuit:

Circuit Q

The Q of the circuit is found equal to 5 by using equation 14–17:

E_L and E_C

The voltage across the inductor equals the voltage across the capacitor at resonance and both are equal to Q times the applied voltage (equation 14–23):

To verify this, the value of this voltage in the traditional manner can be calculated. Recall in that calculation the voltage across the inductor equals the current through the inductor times the value of the inductive reactance:

Since the capacitor has the same reactance at resonance, the voltage drop across it will be identical to the voltage across the inductor.

Frequency Responses

In Figure 14.4, inductive reactance and capacitive reactance versus frequency were graphed. With resonant circuits, it is usually helpful to graph the response of the circuit current or impedance against frequency as shown in Figures 14.11 and 14.12. These graphs are called the frequency responses of the circuit.

As shown in Figure 14.11 the impedance, Z, of a series resonant circuit has a minimum value at the resonant frequency, f_r, when X_L = X_C, and the total reactance of the circuit is zero. The impedance increases on either side of the resonant frequency because X_L and X_C are not equal and do not result in a net reactance of zero. In the series resonant circuit just solved, the impedance had a value of 10 ohms at resonance and would be plotted on the impedance response graph as shown in Figure 14.11.

The current, on the other hand, has a maximum value at resonance and varies inversely with the impedance as shown in Figure 14.12. That is, as the impedance increases, the current decreases. For the series resonant circuit just solved the current frequency response appears as shown in Figure 14.12 with a maximum current of 5 amperes plotted at the resonant frequency, f_r.

Relationship of Bandwidth to Resonant Frequency and Circuit Q

An interesting fact is that a relationship also exists between the value of the Q, the resonant frequency, and the bandwidth of a circuit. This relationship is shown in equation 14–27:

(14–27)

Equation 14–27 means that if the resonant frequency and the Q of a circuit are known, the bandwidth can be calculated.

In the series circuit of Figure 14.10 solved previously, the resonant frequency was 796 hertz and the Q had a value of 5. The bandwidth of that circuit, then, is calculated using equation 14–27:

The upper and lower cutoff frequencies can be calculated using equations 14–25 and

These values are graphed in Figure 14.15. Recall that these frequencies define the points on the response curve at which the current is 70.7 percent of its maximum value. Therefore, in the series resonant circuit example of Figure 14.10 in which the current has a maximum value of 5 amperes, the value of the current at these cutoff frequencies is 3.535 amperes, calculated as follows:

(14–28)

Half–Power Points

The upper and lower frequency cutoff points on the response curve are sometimes referred to as the half-power points of the frequency response. This is because at these points on the response curve, the real power dissipation in the circuit is exactly one-half of what it is at the resonant frequency. To illustrate this, the series circuit of Figure 14.10, which is repeated in Figure 14.16, will be used. Recall that the real power is equal to the current squared times the value of the resistance:

(14–29)

In the example circuit the series resistance is 10 ohms, and at resonance the maximum current is 5 amperes. Therefore,

At resonance, the real power dissipated is 250 watts. At the cut–off frequency points the current from equation 14–28, is 3.535 amperes. Using this value the power is calculated as:

Therefore, at the cutoff frequencies, the real power dissipated in the circuit is 125 watts, one-half the real power dissipated at the resonant frequencies.

Effect of Changes in L and C

Equation 14–4 shows that the resonant frequency is dependent upon the

(14–4)

values of L and C. Changing either L or C changes the frequency response, but the value of either cannot be changed independently without changing the resonant frequency. However, since the resonant frequency is dependent upon the product of L and C, the resonant frequency can be held constant by changing both L and C without changing their product. For example, if a 6–henry inductor and a 2–farad capacitor are used initially, their LC product is 6 times 2 or 12 ((6H)(2F) = 12).

If the values are changed to a 4 henry inductor and a 3 farad capacitor, the LC product remains the same at 12 ((4H)(3F) = 12), and the resulting resonant frequency will be the same. Note that to keep the product the same and therefore, the resonant frequency, the value of the inductor was decreased while the value of the capacitor was increased. Obviously, many other possible combinations of L and C will yield the same product, and thus, the same resonant frequency.

L/C Ratio

Although changing the values of L and C in this manner keeps the resonant frequency constant, it does have a definite effect upon the frequency response of the circuit. The curvature (steepness) of the sides of the frequency reponse curve is changed. This change in curvature represents a change in bandwidth. In many cases, this change is referred to as “changing skirts” of the response curve.

For a series resonant circuit, the degree of steepness of the response curve (bandwidth) is determined by the L/C ratio. This is the ratio of the inductance divided by the capacitance:

(14–30)

When the value of L is increased, the ratio is increased; when the value of L is decreased, the ratio is decreased. Increasing L increases the L/C ratio and increases X_L, the inductive reactance (equation 14–1).

From equation 14–17, an increase in X_L will increase Q if the resistance of the circuit is held constant. Correspondingly, Q will decrease if X_L is decreased.

(14–17)

From equation 14–27, an increase or decrease in Q will affect the bandwidth. The resonant frequency, f_r, is being held constant, therefore, an increase in the value of Q will cause a decrease in bandwidth:

(14–27)

Therefore, when L is increased, X_L increases, the L/C ratio is increased, Q increases, the bandwidth is decreased and the sides of the response curve shown in Figure 14.17 become steeper. Conversely, when L is decreased, XL decreases, the L/C ratio is decreased, Q decreases, the bandwidth is increased, and the sides of the response curve become less steep as shown in Figure 14.18.

Effect of R

Another way to change the circuit’s frequency response is to change the circuit’s resistance. Recall from equation 14–4 that the resonant frequency of a series RLC circuit is not dependent on the value of the resistance, therefore, changing R will not change the resonant frequency. Changing R, affects the amount of maximum current at resonance. If L and C remain constant, a decrease in resistance causes an increase in the maximum current at resonance. As shown in Figure 14.19, this has the effect of increasing the slope of the sides of the response curve and decreasing the bandwidth. Conversely, an increase in resistance will result in a decrease of the maximum current value at resonance and a corresponding increase in bandwidth. This also is shown in Figure 14.19.

This phenomena can be explained mathematically by using equation 14–17 and recalling that as the resistance of the circuit is changed the value of Q is changed. If the resistance is increased, the Q of the circuit is decreased:

If the resistance is decreased, the Q of the circuit is increased:

As before, if the resonant frequency remains constant, a decrease in Q will cause an increase in bandwidth,

and an increase in Q will cause a decrease in bandwidth.

Consequently, by changing either the L/C ratio or the value of R, while holding the resonant frequency constant, the value of Q, and therefore the bandwidth and frequency response, can be changed.

Resonant Frequency

Thus far, the characteristics of a series RLC circuit at resonance have been discussed. Now let’s look at a parallel RLC circuit shown in Figure 14.20. It has a resistor, inductor, and capacitor all connected in parallel with a variable frequency power supply.

Recall that resonance was defined as the frequency at which X_L equals X_C. Because resonance is defined in this way, it makes no difference whether the inductance and capacitance of the circuit are connected in series or in parallel. Therefore, equation 14–14 for determining the resonant frequency is the same for a parallel circuit as it was for a series circuit:

(14–4)

At the resonant frequency for this parallel circuit, when X_L and X_C are equal, the inductive and capacitive branch currents will be equal, I_L = I_C, and 180 degrees out of phase.

Total Current

Recall that the total current, I_T, in a parallel RLC circuit is equal to the vector sum of the branch currents I_C, I_L, and I_R and it is calculated using equation 14–31.

(14–31)

The phasor diagram is shown in Figure 14.21. I_R is the current in the resistive branch, I_L is the current in the inductive branch, and I_C is the current in the capacitive branch. At resonance, when I_L and I_C are equal, the net reactive current is equal to zero and equation 14–31 becomes,

(14–32)

and can be rewritten as

(14–33)

Solving for I_T results in equation 14–34.

(14–34)

At resonance the total current in a parallel resonant RLC circuit is equal to the resistive branch current because as shown in Figure 14.22a, I_L = I_C and is 180 degrees out of phase with I_C. Since at resonance the total current seen by the power source is resistive current, the current must be at its lowest level, at its minimum. If the applied frequency is varied either side of the resonant frequency, as shown by the phasor diagrams of Figure 14.22, then a difference will exist between the value of I_L and I_C. This difference will result in a net reactive current which would add vectorially to the resistive current of the circuit and create a larger value of total current.

The phasor diagram of the parallel circuit at resonance is shown in Figure 14.23. Since the applied voltage is taken as reference at zero degrees, and since at resonance the resistive current is in phase with the applied voltage, the total current is in phase with the applied voltage and the phase angle is zero.

Impedance

The total impedance of the parallel circuit is equal to the applied voltage divided by the total current.

(14–35)

Since the total current in the circuit is a minimum at resonance, the impedance must be at its maximum value. From equation 14–34, the total current is equal to the resistive branch current, and therefore, equation 14–35 can be rewritten:

By Ohm’s law, the applied voltage divided by the resistive current is equal to resistance:

(14–36)

In other words, the impedance at resonance is simply equal to the value of the resistance in the resistive branch and is a maximum at resonance.

Circuit Q of a Parallel–Resonant Circuit

Recall from equation 14–11 that the Q of a resonant circuit is defined as the ratio of the reactive power to the resistive power.

(14–11)

The reactive power is for either the inductor or capacitor since they are equal at resonance.

To illustrate this, let’s again use the reactive power of the inductor as an example. The reactive power of the inductor is equal to the voltage across the inductor times the current through it. Because , the equation can be rewritten as the voltage across the inductor squared divided by the value of the inductive reactance (equation 14–37).

(14–37)

The real power dissipated by the resistor is equal to the voltage across the resistor times the current through it. Because , the equation can be rewritten as the voltage across the resistor squared divided by the value of the resistor (equation 14–38).

(14–38)

Substituting equations 14–37 and 14–38 into equation 14–11,

(14–39)

In the parallel circuit, E_L, the voltage across the inductor equals E_R, the voltage across the resistor. As a result equation 14–39 simplifies to equation 14–40.

(14–40)

The Q of a parallel resonant circuit of the type shown in Figure 14.20 is equal to the value of the parallel circuit resistance divided by the value of the parallel circuit inductive reactance calculated at the resonant frequency. Since the capacitive reactance equals the inductive reactance at resonance, the same value of Q may be obtained by dividing the resistance by the capacitive reactance:

(14–41)

Current Magnification

Now, let’s take the original equation for Q, equation 14–11, and use it in a slightly different way. In equation 14–42, the E and Iequivalents are substituted for the power values, P_X and P_R.

(14–42)

Since the voltage across the inductor, E_L, and voltage across the resistor, E_R, are the same in the parallel circuit under consideration, and they both equal E_A, they cancel each other and the equation simplifies to equation 14–43.

(14–43)

Q equals the inductive branch current divided by the resistive branch current.

Earlier you saw in the current phasor diagrams of Figure 14.22 and proven by equation 14–34 that the current through the resistor and the total current were equal at resonance. As a result of the substitution I_R = I_T, equation 14–43 for Q can be rewritten:

(14–44)

If both sides of equation 14–44 are multiplied by I_T and rearranged, the equation can be rewritten as:

I_T cancels on the right side, therefore,

(14–45)

The equation relates the fact that at resonance, the current through the inductor will be Q times larger than the total current.

Since the capacitive current is equal to the inductive current, the capacitive current is also Q times the total circuit current at resonance (equation 14–46).

(14–46)

This is referred to as the current magnification in a parallel resonant circuit. Q, again, is known as the magnification factor.

You may find this unusual but, the current as expressed by equations 14–45 and 14–46 is much larger in the inductive and capacitive branches of a parallel circuit at resonance than the total current. This is shown in Figure 14.24. The larger current in the inductive-capacitive branches is a result of the continual charging and discharging of the capacitor and the continual building-up and collapsing of the magnetic field of the inductor at the resonant frequency. It is a characteristic of resonance and is known as the circulating current. The parallel combination of the inductor and capacitor is often called a tank circuit, often shortened simply to tank. Although the circulating current is large, the total current drawn by the circuit has a minimum value at resonance.

Frequency Response

If the total current in a parallel resonant circuit is plotted versus frequency it appears as is shown in the solid-line curve in Figure 14.25. It has a minimum value at resonance.

Circuit impedance is shown by the dotted-line curve in Figure 14.25. Since total current is minimum at resonance, circuit impedance is at maximum at resonance.

The specifications that describe the parallel circuit frequency response, such as bandwidth, the 70.7 percent half-power points, and upper and lower cutoff frequencies are determined exactly as they were for a series resonant circuit.

Resonant Frequency

In this circuit where L is 10 millihenrys and C is 4 microfarads, the resonant frequency is:

At the resonant frequency of 796 hertz, the value of the inductive and capacitive reactance are equal and, as before, both are equal to 50 ohms.

Branch Currents

The branch current can be calculated using Ohm’s law for each branch. The voltage across each branch is E_A which is equal to 100 volts. I_L is calculated first.

In a similar manner, the capacitive branch current I_C, is determined:

The resistive branch current is the last to be calculated.

Total Current

The total current for the example circuit in its most general form is:

As was determined previously for parallel RLC circuits the total current at resonance is simply equal to the resistive branch current. In this case it has a minimum value of 100 milliamperes.

Impedance

The value of the impedance is equal to the applied voltage divided by the total current or,

The circuit impedance at resonance is a maximum value and equal to 1 kilohm, the value of the parallel branch resistance.

Circuit Q

The Q of the parallel circuit can now be determined by substituting R = 1 kilohm and X_L = 50 ohms.

Circulating Current

It was established previously that the capacitive or inductive circulating current was Q times the total current at resonance. It is calculated as follows:

This 2 amperes is identical to the value calculated for the inductive and capacitive branch currents determined earlier and verifies that calculation.

Frequency Response and Bandwidth

The frequency response of the circuit is shown in Figure 14.27. It shows that impedance is at a maximum of 1 kilohm at resonance: 796 hertz. The bandwidth from equation 14–27 is equal to the resonant frequency divided by the Q of the circuit. It is calculated as follows:

The upper and lower cut-off frequencies are calculated using equations 14–25 and 14–26. From equation 14–25,

From equation 14–26,

Remember that these upper and lower cut-off frequencies are the frequencies at which the frequency response is down to 70.7 percent of its maximum value. In this example, the value of the impedance at these cutoff frequencies is equal to 70.7 percent of 1 kilohm.

All of these values are presented as they apply to the frequency response of the impedance in Figure 14.28.

Effect of Changing L/C Ratio

Impedance plotted against frequency is shown in Figure 14.29. It also demonstrates the effect of changing the L over C ratio on the frequency response of a parallel resonant circuit. The effect of increasing the L/C ratio for a parallel resonant circuit has the opposite effect as it did for a series resonant circuit. An increase in the L/C ratio causes a decrease in Q, a decrease in the steepness of the response curve, and an increase in bandwidth.

This effect can be substantiated mathematically by reasoning that in increasing the L/C ratio, L is increased causing an increase in the inductive reactance at the resonant frequency. Assuming the applied voltage is held constant, an increase in X_L in this parallel circuit causes a decrease in the inductive branch current, I_L, indicating a decrease in the circuit Q.

From equation 14–40 an increase in X_Lcauses a decrease in the Q of the circuit, and from equation 14–27 a decrease in Q causes an increase in the bandwidth. Conversely, a decrease in the L/C ratio causes an increase in Q, an increase in the steepness of the response curve, and a decrease in bandwidth as shown in Figure 14.30.

Effect of Changing R Value

The effect of changing the resistance of the circuit on the frequency response of a parallel resonant circuit is shown in Figures 14.31 and 14.32.

As you know by equation 14–4 the resonant frequency is not dependent upon the value of the resistance. Therefore, the resistance can be changed without changing the resonant frequency.

(14–4)

Changing the circuit resistance, however, does change the frequency response. You have seen by equation 14–36 that the circuit impedance at resonance is equal to R, and that the impedance is a maximum at resonance.

(14–36)

Changing the value of R changes the value of the maximum impedance at resonance. With L and C held constant, as R is decreased, the circuit impedance at resonance decreases (equation 14–36), the Q of the circuit decreases (equation 14–40), causing a corresponding increase in bandwidth (equation 14–27) and the steepness of the response curve decreases as shown in Figure 14.31. This may be confirmed mathematically:

(14–36)

(14–40)

(14–27)

Conversely, as shown in Figure 14.32, an increase in R causes an increase of circuit impedance at resonance along with an increase in the Q of the circuit, a corresponding decrease in bandwidth, a resulting increase in the steepness of the response curve.

1. Calculate the resonant frequency of the following L-C combinations:

Solutions:

2. Determine the value of the inductor needed to provide the resonant frequency specified if used in conjunction with the capacitor given:

Solutions:

3. Determine the value of capacitor needed to provide the resonant frequency specified if used in conjunction with the inductor given:

Solutions:

4. For the following circuit, determine the resonant frequency of the circuit; then determine the values specified at the resonant frequency.

Solutions:

5. Sketch the frequency response of I versus frequency for the circuit of Problem 4. Show the location of the resonant frequency, upper and lower edge (cutoff) frequencies, bandwidth, and values of current at resonance and at the edge (cutoff) frequencies.

6. For the following circuit, determine the resonant frequency of the circuit; then determine the values specified at the resonant frequency.

Solutions:

7. Sketch the frequency response of Z versus frequency for the circuit of Problem 6. Show the location of the resonant frequency, upper and lower edge (cutoff) frequencies, bandwidth, and values of impedance at resonance and at the edge (cutoff) frequencies.

1. Calculate the resonant frequency of the following L-C combinations:

2. Determine the value of the inductor needed to provide the resonant frequency specified if used in conjunction with the capacitor given:

3. Determine the value of the capacitor needed to provide the resonant frequency specified if used in conjunction with the inductor given:

4. For the following circuit, determine the resonant frequency; then determine the values specified at the resonant frequency:

5. For the following circuit, determine the resonant frequency; then determine the values specified at the resonant frequency.

6. Sketch the frequency responses of I versus frequency for the circuits of Problems 4 and 5. Show the location of the resonant frequency, upper and lower edge (cutoff) frequencies, bandwidth, and values of current at resonance and at the edge (cutoff) frequencies.

7. For the following circuit, determine the resonant frequency; then determine the values specified at the resonant frequency.

8. For the following circuit, determine the resonant frequency; then determine the values specified at the resonant frequency.

9. Sketch the frequency responses of Z versus frequency for the circuits of Problems 7 and 8. Show the location of the resonant frequency, upper and lower edge (cutoff) frequencies, bandwidth, and values of impedance at resonance and at the edge (cutoff) frequencies.

10. In the circuit shown below, it is desired to be able to change its resonant frequency from 540 kHz to 1,600 kHz by adjusting the value of the capacitor. Determine the range of values over which the capacitor must be variable to achieve this.

1. Determine the value of the inductor needed to provide the resonant frequency specified if used in conjunction with the capacitor given:

2. Determine the value of the capacitor needed to provide the resonant frequency specified if used in conjunction with the inductor given:

3. For the following circuit, determine the resonant frequency; then determine the values specified at the resonant frequency.

4. Sketch the frequency response of current versus frequency for the circuit of Question 3. Show the location of the resonant frequency, upper and lower edge (cutoff) frequencies, bandwidth, and values of current at resonance and at the edge (cutoff) frequencies.

5. For the following circuit, determine the resonant frequency; then determine the values specified at the resonant frequency.

6. Sketch the frequency response of impedance versus frequency for the circuit of Question 5. Show the location of the resonant frequency, upper and lower edge (cutoff) frequencies, bandwidth, and values of impedance at resonance and at the edge (cutoff) frequencies.