Dr. Abdullah M. Elsayed - du.edu.eg
Transcript of Dr. Abdullah M. Elsayed - du.edu.eg
Dr. Abdullah M. Elsayed
Department of Electrical Engineering
Damietta University – 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
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