Dr. Abdullah M. Elsayed
Department of Electrical Engineering
Damietta University โ Egypt
010 60 79 1554
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
๐๐ถ =1
๐๐ถ=
1
2๐๐๐ถ
๐T = ๐ +1
๐๐๐ถ=1 + ๐๐๐ ๐ถ
๐๐๐ถ
๐ in logarithmic scale
Impedance magnetude
Frequency Effects on AC circuits
1- RC Circuits
1- For (ฯ very small)โฆ.
ฯ โค ฯc/10 (or f โค fc/10) ZT can be expressed as
๐T =1 + ๐๐๐ ๐ถ
๐๐๐ถ=1 + ๐0
๐๐๐ถ=
1
๐๐๐ถ
๐ in logarithmic scale
Frequency Effects on AC circuits
1- RC Circuits
๐T =1 + ๐๐๐ ๐ถ
๐๐๐ถ=0 + ๐๐๐ ๐ถ
๐๐๐ถ= ๐
2- For (ฯ very large)โฆ.ฯ โฅ 10 ฯc (or f โฅ 10 fc) ZT can be expressed as
๐ in logarithmic scale
Frequency Effects on AC circuits
1- RC Circuits
๐๐ =1
๐ ๐ถ=1
๐(ra d s)
The cutoff or corner frequency for an RC circuit as (Corresponding to the RC circuit time constant
๐๐ =1
2๐๐ ๐ถ(Hz)
ฯ โฅ 10 ฯc (or f โฅ 10 fc)ฯ โค ฯc/10 (or f โค fc/10)
๐ in logarithmic scale
ฯ =10 ฯc
ฯ =ฯc
๐T =1
๐๐ถ๐T = ๐ ๐T =
1 + ๐๐๐ ๐ถ
๐๐๐ถ
Frequency Effects on AC circuits
1- RC Circuits
๐T =๐๐ ๐๐ถ
๐๐ + ๐๐ถ=๐
1๐๐๐ถ
๐ +1
๐๐๐ถ
=
๐ ๐๐๐ถ
1 + ๐๐๐ ๐ถ๐๐๐ถ
๐T =๐
1 + ๐๐๐ ๐ถ
๐ in logarithmic scale
Impedance magnetude
Frequency Effects on AC circuits
1- RC Circuits
๐ in logarithmic scale
1- For (ฯ very small)โฆ.
ฯ โค ฯc/10 (or f โค fc/10) ZT can be expressed as
๐T =๐
1 + ๐๐๐ ๐ถ=
๐
1 + 0= ๐
Frequency Effects on AC circuits
1- RC Circuits
๐ in logarithmic scale
๐T =๐
1 + ๐๐๐ ๐ถ=
๐
0 + ๐๐๐ ๐ถ=
1
๐๐ถ
2- For (ฯ very large)โฆ.ฯ โฅ 10 ฯc (or f โฅ 10 fc) ZT can be expressed as
Frequency Effects on AC circuits
1- RC Circuits
๐๐ =1
๐ ๐ถ=1
๐(ra d s)
The cutoff or corner frequency for an RC circuit as (Corresponding to the RC circuit time constant
๐๐ =1
2๐๐ ๐ถ(Hz)
ฯ โฅ 10 ฯc (or f โฅ 10 fc)ฯ โค ฯc/10 (or f โค fc/10)
ฯ =10 ฯcฯ =ฯc
๐T = ๐ ๐T =
1
๐๐ถ๐T =
๐
1 + ๐๐๐ ๐ถ
Frequency Effects on AC circuits
2- RL Circuits
Frequency Effects on AC circuits
2- RL Circuits
๐ in logarithmic scale
๐T =๐๐ ๐๐ฟ
๐๐ + ๐๐ฟ=
๐ ๐๐๐ฟ
๐ + ๐๐๐ฟ=
๐๐๐ฟ
1 + ๐๐๐ฟ๐
Frequency Effects on AC circuits
2- RL Circuits
๐ in logarithmic scale
1- For (ฯ very small)โฆ.
ฯ โค ฯc/10 (or f โค fc/10)
๐T =๐๐๐ฟ
1 + ๐๐๐ฟ๐
=๐๐๐ฟ
1 + 0= ๐๐๐ฟ
Frequency Effects on AC circuits
2- RL Circuits
๐ in logarithmic scale
2- For (ฯ very large)โฆ.
ฯ โฅ 10 ฯc (or f โฅ 10 fc)
๐T =๐๐๐ฟ
1 + ๐๐๐ฟ๐
= ๐
Frequency Effects on AC circuits
2- RL Circuits
๐๐ =๐
๐ฟ=1
๐(rad/s)
The cutoff or corner frequency for an RL circuit as (Corresponding to the RL circuit time constant
๐๐ =๐
2๐๐ฟ(Hz)
ฯ โฅ 10 ฯc (or f โฅ 10 fc)ฯ โค ฯc/10 (or f โค fc/10)
ฯ =10 ฯcฯ =ฯc
๐T = ๐๐๐ฟ ๐T = ๐ ๐T =๐๐๐ฟ
1 + ๐๐๐ฟ๐
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 โ jXC
= R + j(XL โ 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 โ jXC
= R + j(XL โ XC)
Resonance occurs when the reactance of the circuit is effectivelyeliminated, resulting in a total impedance that is purely resistive.
XL= ฯL =2ฯfL
๐๐ถ =1
๐๐ถ=
1
2๐๐๐ถ
๐๐ฟ =1
๐๐ถ๐2 =
1
๐ฟ๐ถ
๐๐ =1
๐ฟ๐ถrad/s
๐๐ =1
2๐ ๐ฟ๐ถ(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 โ XC)
๐ =๐
๐๐=
๐ธโ 0o
๐ โ 0o=
๐ธ
๐ โ 0o (A)
VR = I Rโ 0ยฐ
VL = I XLโ 90ยฐ
VC = I XCโ โ90ยฐ
Series Resonance
ZT = R + j(XL โ XC)
๐ =๐
๐๐=
๐ธโ 0o
๐ โ 0o=
๐ธ
๐ โ 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,
๐ =reactive power
average power
๐๐ =๐ผ2 ๐๐ฟ๐ผ2 ๐
The reactive power of the inductor is equal to the reactive power of thecapacitor at resonance
๐๐ =๐๐ฟ๐
=๐๐ฟ
๐ ๐๐ถ๐๐๐ =
๐๐ฟ๐ ๐ถ๐๐๐
Quality Factor, Q
Example 6โ1
Find the indicated quantities for the circuit of the following figure.
a. Resonant frequency expressed as ฯ(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 6โ1
Find the indicated quantities for the circuit of the following figure.
a. Resonant frequency expressed as ฯ(rad/s) and f(Hz).
Quality Factor, Q
Example 6โ1
Find the indicated quantities for the circuit of the following figure.
b. Total impedance at resonance.
Quality Factor, Q
Example 6โ1
Find the indicated quantities for the circuit of the following figure.
c. Current at resonance.
Quality Factor, Q
Example 6โ1
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 6โ1
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 6โ1
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