dr. abdullah m. elsayeddr. abdullah m. elsayed department of electrical engineering damietta...

Dr. Abdullah M. Elsayed

Department of Electrical Engineering

Damietta University – Egypt

010 60 79 1554

[email protected]

mailto:[email protected]

mailto:[email protected]

Lecture - 13

Course Content

Chapter (6)

Resonance

6.1 Introduction

6.2 Frequency Effects on AC circuits

6.3 Series Resonance

6.4 Quality Factor, Q

6.5 Impedance of a Series Resonant Circuit

6.6 Power, Bandwidth, and Selectivity of a Series Resonant

Circuit

6.7 Series-to-Parallel RL and RC Conversion

6.8 Parallel Resonance

Quality Factor, Q

Example 6–1

Find the indicated quantities for the circuit of the following figure.

a. Resonant frequency expressed as ω(rad/s) and f(Hz).

b. Total impedance at resonance.

c. Current at resonance.

d. VL and VC.

e. Reactive powers, QC and QL.

f. Quality factor of the circuit, Qs.

Quality Factor, Q

Example 6–1


a. Resonant frequency expressed as ω(rad/s) and f(Hz).

Quality Factor, Q

Example 6–1


b. Total impedance at resonance.

Quality Factor, Q

Example 6–1


c. Current at resonance.

Quality Factor, Q

Example 6–1


d. VL and VC.

Notice that the voltage across the reactive elements is ten

times greater than the applied signal voltage.

Quality Factor, Q

Example 6–1


e. Reactive powers, QC and QL.

e. Although we use the symbol Q to designate both reactive power and

the quality factor, the context of the question generally provides us with a

clue as to which meaning to use.

Quality Factor, Q

Example 6–1


f. Quality factor of the circuit, Qs.

P=I2*R = VI QL=I2*XL

Impedance of a Series Resonant Circuit


The total impedance of a simple series resonant circuit is written as


When ω = ωs (Resonace frequency)

ZT = R (XL=XC) and θ = tan-10 = 0°

This result is consistent with the results obtained in the previous

section.


When ω < ωs:

As we decrease ω from resonance, ZT will get larger until ω = 0. At this

point, the magnitude of the impedance will be undefined, corresponding

to an open circuit.

The angle θ will occur between of 0° and −90° since the numerator of the

argument of the arctangent function will always be negative, (XL<XC)

corresponding to an angle in the fourth quadrant. Because the angle of

the impedance has a negative sign, we conclude that the impedance must

appear capacitive in this region.


When ω > ωs:

As ω is made larger than resonance, the impedance ZT will increase due

to the increasing reactance of the inductor.

For these values of ω, the angle θ will always be within 0° and +90°

because both the numerator and the denominator of the arctangent

function are positive (XL > XC). Because the angle of ZT occurs in the

first quadrant, the impedance must be inductive.

Power, Bandwidth, and Selectivity of a

Series Resonant Circuit

Power, Bandwidth, and Selectivity of a Series

Resonant Circuit

The total power dissipated by the circuit at any frequency is given as

At resonance, it follows that the power must similarly

be maximum at resonance. The maximum power

dissipated by the series resonant circuit is therefore

given as


Resonant Circuit

The power response of a series resonant

circuit has a bell-shaped curve called the

selectivity curve, which is similar to the

current response.


Resonant Circuit

We define the bandwidth, BW, of the resonant circuit to be the difference

between the frequencies at which the circuit delivers half of the maximum

power. The frequencies ω1 and ω2 are called the half-power frequencies, the

cutoff frequencies, or the band frequencies.


Resonant Circuit

If the bandwidth of a circuit is kept very narrow, the circuit is said to have a

high selectivity, since it is highly selective to signals occurring within a very

narrow range of frequencies. On the other hand, if the bandwidth of a circuit is

large, the circuit is said to have a low selectivity.


Resonant Circuit

The elements of a series resonant circuit determine not only the

frequency at which the circuit is resonant, but also the shape (and hence

the bandwidth) of the power response curve.

1- Consider a circuit in which the resistance, R, and the resonant

frequency, ωs, are held constant. We find that by increasing the ratio of

L/C, the sides of the power response curve become steeper.

L/C ≥ 100Rcoil


Resonant Circuit

The elements of a series resonant circuit determine not only the frequency at

which the circuit is resonant, but also the shape (and hence the bandwidth) of

the power response curve.

2- If, on the other hand, L and C are kept constant, we find that the bandwidth

will decrease as R is decreased and will increase as R is increased.


Resonant Circuit


Resonant Circuit

For the series resonant circuit the power at any frequency is determined

as


Resonant Circuit

The magnitude of current at the half-power frequencies is


Resonant Circuit

ω1, ω2=?????

The cutoff frequencies are found by evaluating the frequencies at which

the power dissipated by the circuit is half of the maximum power.


Resonant Circuit

When ω < ωs, the term ω2LC must be less than

1. In this case the solution is determined as

follows:


Resonant Circuit

In a similar manner, for ω > ωs, the upper half-

power frequency is


Resonant Circuit

Taking the difference between ω1 and ω2, we find the bandwidth of the

circuit as

and since Qs = ωsL/R we further simplify the

bandwidth as


Resonant Circuit

Taking the difference between ω1 and ω2, we find the bandwidth of the

circuit as

If Q ≥ 10, the actual half-power frequencies are

very nearly equal to the resonance frequency;

Example 6–3


Resonant Circuit

Refer to the circuit of the following figure.

a. Determine the maximum power dissipated by the circuit.

b. Use the results obtained from Example 6–1 to determine the bandwidth of the

resonant circuit and to arrive at the approximate half-power frequencies, ω1 and

ω2.

c. Calculate the actual half-power frequencies, ω1 and ω2, from the given component

values. Show two decimal places of precision.

d. Solve for the circuit current, I, and power dissipated at the lower halfpower

frequency, ω1, found in Part (c).

Example 6–3


Resonant Circuit


a. Determine the maximum power dissipated by the circuit.

Example 6–3


Resonant Circuit


b. Use the results obtained from Example 6–1 to determine the bandwidth of the

resonant circuit and to arrive at the approximate

half-power frequencies, ω1 and ω2.

Example 6–3


Resonant Circuit


c. Calculate the actual half-power frequencies, ω1 and ω2, from the given component

values. Show two decimal places of precision.

Example 6–3


Resonant Circuit


d. Solve for the circuit current, I, and power dissipated at the lower halfpower

frequency, ω1, found in Part (c).

Example 6–4


Resonant Circuit


a. Calculate the values of RL and C for the circuit to have a resonant frequency of

200 kHz and a bandwidth of 16 kHz.

b. Use the designed component values to determine the power dissipated by the

circuit at resonance.

c. Solve for vout(t) at resonance.

Example 6–4


Resonant Circuit


a. Calculate the values of RL and C for the circuit to have a resonant frequency of

200 kHz and a bandwidth of 16 kHz.

Example 6–4


Resonant Circuit


b. Use the designed component values to determine the power dissipated by the

circuit at resonance.

dr. abdullah m. elsayeddr. abdullah m. elsayed department of electrical engineering damietta...

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