Dr. Abdullah M. Elsayed

Department of Electrical Engineering

Damietta University – Egypt

010 60 79 1554

am.elsherif@du.edu.eg

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

Frequency Effects on AC circuits

1- RC Circuits

Frequency Effects on AC circuits

1- RC Circuits

𝑋𝐶 =1

𝜔𝐶=

1

2𝜋𝑓𝐶

𝐙T = 𝑅 +1

𝑗𝜔𝐶=1 + 𝑗𝜔𝑅𝐶

𝑗𝜔𝐶

𝜔 in logarithmic scale

Impedance magnetude

Frequency Effects on AC circuits

1- RC Circuits

1- For (ω very small)….

ω ≤ ωc/10 (or f ≤ fc/10) ZT can be expressed as

𝐙T =1 + 𝑗𝜔𝑅𝐶

𝑗𝜔𝐶=1 + 𝑗0

𝑗𝜔𝐶=

1

𝑗𝜔𝐶

𝜔 in logarithmic scale

Frequency Effects on AC circuits

1- RC Circuits

𝐙T =1 + 𝑗𝜔𝑅𝐶

𝑗𝜔𝐶=0 + 𝑗𝜔𝑅𝐶

𝑗𝜔𝐶= 𝑅

2- For (ω very large)….ω ≥ 10 ωc (or f ≥ 10 fc) ZT can be expressed as

𝜔 in logarithmic scale

Frequency Effects on AC circuits

1- RC Circuits

𝜔𝑐 =1

𝑅𝐶=1

𝜏(ra d s)

The cutoff or corner frequency for an RC circuit as (Corresponding to the RC circuit time constant

𝑓𝑐 =1

2𝜋𝑅𝐶(Hz)

ω ≥ 10 ωc (or f ≥ 10 fc)ω ≤ ωc/10 (or f ≤ fc/10)

𝜔 in logarithmic scale

ω =10 ωc

ω =ωc

𝐙T =1

𝜔𝐶𝐙T = 𝑅 𝐙T =

1 + 𝑗𝜔𝑅𝐶

𝑗𝜔𝐶

Frequency Effects on AC circuits

1- RC Circuits

𝐙T =𝐙𝑅𝐙𝐶

𝐙𝑅 + 𝐙𝐶=𝑅

1𝑗𝜔𝐶

𝑅 +1

𝑗𝜔𝐶

=

𝑅𝑗𝜔𝐶

1 + 𝑗𝜔𝑅𝐶𝑗𝜔𝐶

𝐙T =𝑅

1 + 𝑗𝜔𝑅𝐶

𝜔 in logarithmic scale

Impedance magnetude

Frequency Effects on AC circuits

1- RC Circuits

𝜔 in logarithmic scale

1- For (ω very small)….

ω ≤ ωc/10 (or f ≤ fc/10) ZT can be expressed as

𝐙T =𝑅

1 + 𝑗𝜔𝑅𝐶=

𝑅

1 + 0= 𝑅

Frequency Effects on AC circuits

1- RC Circuits

𝜔 in logarithmic scale

𝐙T =𝑅

1 + 𝑗𝜔𝑅𝐶=

𝑅

0 + 𝑗𝜔𝑅𝐶=

1

𝜔𝐶

2- For (ω very large)….ω ≥ 10 ωc (or f ≥ 10 fc) ZT can be expressed as

Frequency Effects on AC circuits

1- RC Circuits

𝜔𝑐 =1

𝑅𝐶=1

𝜏(ra d s)

The cutoff or corner frequency for an RC circuit as (Corresponding to the RC circuit time constant

𝑓𝑐 =1

2𝜋𝑅𝐶(Hz)

ω ≥ 10 ωc (or f ≥ 10 fc)ω ≤ ωc/10 (or f ≤ fc/10)

ω =10 ωcω =ωc

𝐙T = 𝑅𝐙T =

1

𝜔𝐶𝐙T =

𝑅

1 + 𝑗𝜔𝑅𝐶

Frequency Effects on AC circuits

2- RL Circuits

Frequency Effects on AC circuits

2- RL Circuits

𝜔 in logarithmic scale

𝐙T =𝐙𝑅𝐙𝐿

𝐙𝑅 + 𝐙𝐿=

𝑅 𝑗𝜔𝐿

𝑅 + 𝑗𝜔𝐿=

𝑗𝜔𝐿

1 + 𝑗𝜔𝐿𝑅

Frequency Effects on AC circuits

2- RL Circuits

𝜔 in logarithmic scale

1- For (ω very small)….

ω ≤ ωc/10 (or f ≤ fc/10)

𝐙T =𝑗𝜔𝐿

1 + 𝑗𝜔𝐿𝑅

=𝑗𝜔𝐿

1 + 0= 𝑗𝜔𝐿

Frequency Effects on AC circuits

2- RL Circuits

𝜔 in logarithmic scale

2- For (ω very large)….

ω ≥ 10 ωc (or f ≥ 10 fc)

𝐙T =𝑗𝜔𝐿

1 + 𝑗𝜔𝐿𝑅

= 𝑅

Frequency Effects on AC circuits

2- RL Circuits

𝜔𝑐 =𝑅

𝐿=1

𝜏(rad/s)

The cutoff or corner frequency for an RL circuit as (Corresponding to the RL circuit time constant

𝑓𝑐 =𝑅

2𝜋𝐿(Hz)

ω ≥ 10 ωc (or f ≥ 10 fc)ω ≤ ωc/10 (or f ≤ fc/10)

ω =10 ωcω =ωc

𝐙T = 𝑗𝜔𝐿 𝐙T = 𝑅𝐙T =𝑗𝜔𝐿

1 + 𝑗𝜔𝐿𝑅

Frequency Effects on AC circuits

2- RL Circuits

Homework

Frequency Effects on AC circuits

3- RLC Circuits

Frequency Effects on AC circuits

3- RLC Circuits

ZT = R + jXL – jXC

= R + j(XL – XC)

At very low frequencies, the inductor will appear as a

very low impedance while the capacitor will appear as a

very high impedance (effectively an open circuit).

As the frequency increases, the inductive reactance

increases, while the capacitive reactance decreases. At

some frequency, f0, the inductor and the capacitor will

have the same magnitude of reactance. At this frequency,

the reactances cancel, resulting in a circuit impedance

which is equal to the resistance value.

As the frequency increases still further, the inductive

reactance becomes larger than the capacitive reactance.

The circuit becomes inductive and the magnitude of the

total impedance of the circuit again rises.

Series Resonance

Series Resonance

Tacoma Narrows Bridge during collapse, Tacoma, Washington 1940

Series Resonance

ZT = R + jXL – jXC

= R + j(XL – XC)

Resonance occurs when the reactance of the circuit is effectivelyeliminated, resulting in a total impedance that is purely resistive.

XL= ωL =2πfL

𝑋𝐶 =1

𝜔𝐶=

1

2𝜋𝑓𝐶

𝜔𝐿 =1

𝜔𝐶𝜔2 =

1

𝐿𝐶

𝜔𝑠 =1

𝐿𝐶rad/s

𝑓𝑠 =1

2𝜋 𝐿𝐶(Hz)

Series Resonance

VL = I XL May be very large

which may damage the coils

VC = I XC May be very large

which often damage the capacitors

Series Resonance

ZT = R + j(XL – XC)

𝐈 =𝐄

𝐙𝐓=

𝐸∠0o

𝑅∠0o=

𝐸

𝑅∠0o (A)

VR = I R∠0°

VL = I XL∠90°

VC = I XC∠−90°

Series Resonance

ZT = R + j(XL – XC)

𝐈 =𝐄

𝐙𝐓=

𝐸∠0o

𝑅∠0o=

𝐸

𝑅∠0o (A)

PR = I 2R (W)

QL = I 2XL (VAR)

QC = I 2XC (VAR)

Quality Factor, Q

Quality Factor, Q

For any resonant circuit, we define the quality factor, Q, as the ratio of

reactive power to average power, namely,

𝑄 =reactive power

average power

𝑄𝑠 =𝐼2 𝑋𝐿𝐼2 𝑅

The reactive power of the inductor is equal to the reactive power of thecapacitor at resonance

𝑄𝑠 =𝑋𝐿𝑅

=𝜔𝐿

𝑅𝑄𝐶𝑜𝑖𝑙 =

𝑋𝐿𝑅𝐶𝑜𝑖𝑙

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

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

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

Quality Factor, Q

Example 6–1

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

b. Total impedance at resonance.

Quality Factor, Q

Example 6–1

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

c. Current at resonance.

Quality Factor, Q

Example 6–1

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

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

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

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

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

f. Quality factor of the circuit, Qs.

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

Week Required1st 2nd 3rd

Chapter (1)

Methods of AC Analysis

4th Chapter (2)

Graphical Solution of DC Circuits Contains Nonlinear

Elements5th Chapter (3)

Exam-1

Circle Diagrams6th 7th

Chapter (4)

Transient Analysis of Basic Circuits

8th 9th Chapter (5)

Mid Term

Harmonics10th 11th

Chapter (6)

Resonance12th 13th

Chapter (7)

Passive Filters