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Unformatted text preview: SMALL for Big Things University at Buffalo SMALL for Big Things University at Buffalo nanobioSensors & MicroActuators Learning Lab The State University of New York nanobioSensors & MicroActuators Learning Lab The State University of New York EE 203 Circuit Analysis 2
Lecture 30
Chapter 14.3
High
HighPass Filters
(continue…)
Kwang W. Oh, Ph.D., Assistant Professor
SMALL (nanobioSensors and MicroActuators Learning Lab)
Department of Electrical Engineering
University at Buffalo, The State University of New York
215E Bonner Hall, SUNYBuffalo, Buffalo, NY 142601920
Tel: (716) 6453115 Ext. 1149, Fax: (716) 6453656
Email: kwangoh@buffalo.edu, http://www.SMALL.Buffalo.edu Recall Lecture 5
Equivalent Circuits for C and L as a Function of Frequency
1
I
jωC V= ∴Z = 1
jωC V = jωLI ∴ Z = jωL Z= 1
→ ∞ (open)
j (0)C Z = j (0) L → 0 (short ) Z= 1
→ 0 (short )
j ( ∞)C Z = j ( ∞) L → ∞ (open) EE 203 Circuit Analysis 2  Spring 2007  Prof. Kwang W. Oh  EE@SUNYBuffalo EE 203 Circuit Analysis 2  Spring 2008  Prof. Kwang W. Oh  EE@SUNYBuffalo Lecture 30  Chapter 14  3/5  1/13 EE 203 Circuit Analysis 2  Spring 2008  Prof. Kwang W. Oh  EE@SUNYBuffalo Lecture 05  Chapter 09  3/7  13/17 Lecture 30  Chapter 14  3/5  2/13 SMALL for Big Things University at Buffalo SMALL for Big Things University at Buffalo nanobioSensors & MicroActuators Learning Lab The State University of New York nanobioSensors & MicroActuators Learning Lab The State University of New York LowPass VS HighPass (a) A series
RL Lowpass
filter ∴Z = (b) at ω = 0 HighPass (b) at ω = 0 (c) at ω =∞ VS (a) A series
RC highpass
filter ∴ Z = jωL LowPass (c) at ω =∞ EE 203 Circuit Analysis 2  Spring 2008  Prof. Kwang W. Oh  EE@SUNYBuffalo 1
jωC Lecture 30  Chapter 14  3/5  3/13 EE 203 Circuit Analysis 2  Spring 2008  Prof. Kwang W. Oh  EE@SUNYBuffalo Lecture 30  Chapter 14  3/5  4/13 SMALL for Big Things University at Buffalo SMALL for Big Things University at Buffalo nanobioSensors & MicroActuators Learning Lab The State University of New York nanobioSensors & MicroActuators Learning Lab The State University of New York BandPass Filter (1)
EE 203 Circuit Analysis 2
Lecture 30
Chapter 14.4
BandPass Filters
Kwang W. Oh, Ph.D., Assistant Professor
SMALL (nanobioSensors and MicroActuators Learning Lab)
Department of Electrical Engineering
University at Buffalo, The State University of New York
215E Bonner Hall, SUNYBuffalo, Buffalo, NY 142601920
Tel: (716) 6453115 Ext. 1149, Fax: (716) 6453656
Email: kwangoh@buffalo.edu, http://www.SMALL.Buffalo.edu EE 203 Circuit Analysis 2  Spring 2008  Prof. Kwang W. Oh  EE@SUNYBuffalo Lecture 30  Chapter 14  3/5  5/13 The
The next filters we examine are those that pass voltages within a
band of frequencies to the output while filtering out voltages at
frequencies outside this band.
These filters are somewhat more complicated than the lowpass
and highpass filters of the previous sections.
Ideal bandpass filters have two cutoff frequencies, ωc1 and ωc2,
which identify the passband.
For realistic bandpass filters, these cutoff
frequencies are again defined as
the frequencies for which the magnitude of
the transfer function equals (1/√2)Hmax.
EE 203 Circuit Analysis 2  Spring 2008  Prof. Kwang W. Oh  EE@SUNYBuffalo Lecture 30  Chapter 14  3/5  6/13 SMALL for Big Things University at Buffalo SMALL for Big Things University at Buffalo nanobioSensors & MicroActuators Learning Lab The State University of New York nanobioSensors & MicroActuators Learning Lab The State University of New York BandPass Filter (2) Series RLC Circuits – Qualitative Analysis
At ω = 0 Center Frequency, ωo the frequency for which a circuit's transfer function is purely real.
Another name for the center frequency is the resonant frequency.
The center frequency is the geometric center of the passband, that is, ωo = ωc1ωc 2
For bandpass filters, the magnitude of the transfer function is a maximum at the
center frequency (Hmax = H(jωo)) Bandwidth, β
the width of the passband. Quality Factor, Q
the ratio of the center frequency to the bandwidth.
The quality factor gives a measure of the width of the passband, independent of
independent
its location on the frequency axis.
It also describes the shape of the magnitude plot, independent of frequency.
EE 203 Circuit Analysis 2  Spring 2008  Prof. Kwang W. Oh  EE@SUNYBuffalo Lecture 30  Chapter 14  3/5  7/13 Capacitor behaves like an open circuit
Inductor behaves like a short circuit At ω = ∞ Capacitor behaves like a short circuit
Inductor behaves like an open circuit Between ω = 0 and ω = ∞
and Remember that Zcapacitor is negative,
whereas Zinductor is positive.
Th
Thus, at some frequency (ωo),
Zcapacitor and Zinductor have
equal magnitudes and opposite signs.
The two impedances cancel out,
causing v0 = vi.
On either side of ωo, the output voltage
is less than the source voltage.
EE 203 Circuit Analysis 2  Spring 2008  Prof. Kwang W. Oh  EE@SUNYBuffalo Lecture 30  Chapter 14  3/5  8/13 SMALL for Big Things University at Buffalo SMALL for Big Things University at Buffalo nanobioSensors & MicroActuators Learning Lab The State University of New York nanobioSensors & MicroActuators Learning Lab The State University of New York Series RLC Circuits – Quantitative Analysis (1) Series RLC Circuits – Quantitative Analysis (2)
Transfer Function R
( R / L) s
=2
sL + 1 / sC + R s + ( R / L) s + (1 / LC )
ω ( R / L)
H ( jω ) =
[(1 / LC ) − ω 2 ]2 + [ω ( R / L)]2 H (s) = R
( R / L) s
=
sL + 1 / sC + R s 2 + ( R / L) s + (1 / LC )
ω ( R / L)
H ( jω ) =
[(1 / LC ) − ω 2 ]2 + [ω ( R / L)]2 H ( s) = Center Frequency
The frequency for which the circuit’s transfer function is purely real
jω o L + 1
1
1
1
2
= 0 ⇒ jω o L − j
= 0 ⇒ ωo L =
⇒ ωo =
⇒∴ ωo =
jω o C
ωo C
ωo C
LC 1
LC The frequency for which the circuit’s transfer function has the maximum
H max = H ( jω0 ) = ω0 ( R / L ) [(1 / LC ) − ω0 ]2 + [ω0 ( R / L)]2
2 ∴[(1 / LC ) − ω0 ]2 = 0 ⇒ ω0 =
2 EE 203 Circuit Analysis 2  Spring 2008  Prof. Kwang W. Oh  EE@SUNYBuffalo Lecture 30  Chapter 14  3/5  9/13 =1 1
LC EE 203 Circuit Analysis 2  Spring 2008  Prof. Kwang W. Oh  EE@SUNYBuffalo Lecture 30  Chapter 14  3/5  10/13 SMALL for Big Things University at Buffalo SMALL for Big Things University at Buffalo nanobioSensors & MicroActuators Learning Lab The State University of New York nanobioSensors & MicroActuators Learning Lab The State University of New York Series RLC Circuits – Quantitative Analysis (3) Series RLC Circuits – Quantitative Analysis (4) Cutoff Frequencies Center Frequency & Cutoff Frequencies at the ωc (ω1 or ω2) 1 =
[( 1 / LC ωc ( R / L) ωo = ωc1 ⋅ ωc 2 H(jωc)=(1/√2)Hmax ωc ( R / L)
1
1
H max =
=
=
2
2
2
[(1 / LC ) − ωc ]2 + [ωc ( R / L)]2 ωc2 + 1
ωc ( R / L )
2 )− = 1
1
2
{[(1 / LC ) − ωc ]2 + ωc ( R / L)]2 }
ωc 2 ( R / L ) 2 2 1 EE 203 Circuit Analysis 2  Spring 2008  Prof. Kwang W. Oh  EE@SUNYBuffalo 2 1
⎛R⎞ ⎛R⎞
= −⎜ ⎟ +⎜ ⎟ +
=
2 L ⎠ ⎝ 2 L ⎠ CL
⎝ 1
L
[
− ω c ]2 + 1
ωc RC
R 1
L
2
∴
− ωc = ±1 ⇒ ωc L m ωc R − 1 / C = 0
ωc RC
R
2
R
1
⎛R⎞
ωc > 0
ωc1 = −
+ ⎜ ⎟+
2L
⎝ 2 L ⎠ CL
2
R
1
⎛R⎞
2
+ ⎜ ⎟+
∴ ωc = m
R
1
⎛R⎞
2L
⎝ 2 L ⎠ CL ωc 2 =
+ ⎜ ⎟+
2L
⎝ 2 L ⎠ CL 2
2
⎡R
1 ⎤⎡ R
R⎞
1⎤
⎛R⎞
⎥
⎥⎢ + ⎛ ⎟ +
= ⎢−
+ ⎜ ⎟+
⎜
⎢ 2L
⎝ 2 L ⎠ CL ⎥ ⎢ 2 L
⎝ 2 L ⎠ CL ⎥
⎣
⎦
⎦⎣ Bandwidth 2 ωc1 = − R
1
⎛R⎞
+ ⎜ ⎟+
2L
⎝ 2 L ⎠ CL
2 ωc 2 = R
1
⎛R⎞
+ ⎜ ⎟+
2L
2 L ⎠ CL
⎝ 1
LC β = ωc 2 − ωc1
2
2
⎡R
1⎤ ⎡ R
1⎤ R
⎛R⎞
⎛R⎞
⎥ − ⎢−
⎥=
=⎢ + ⎜ ⎟ +
+ ⎜ ⎟+
⎢ 2L
⎝ 2 L ⎠ CL ⎥ ⎢ 2 L
⎝ 2 L ⎠ CL ⎥ L
⎣
⎦⎣
⎦ ax + bx + c = 0
2 2 ⇒ x= b
⎛b⎞ c
± ⎜ ⎟−
2a
⎝ 2a ⎠ a Lecture 30  Chapter 14  3/5  11/13 Quality Factor
Q= (1 / LC )
o
L
=
=
β
R/L
CR 2 EE 203 Circuit Analysis 2  Spring 2008  Prof. Kwang W. Oh  EE@SUNYBuffalo Lecture 30  Chapter 14  3/5  12/13 SMALL for Big Things University at Buffalo nanobioSensors & MicroActuators Learning Lab The State University of New York Series RLC Circuits – Quantitative Analysis (5)
ωc1 = − 2 β ⎛β ⎞
2
+ ⎜ ⎟ + ωo
2
⎝2⎠
2 β ⎛β ⎞
ωc 2 = + ⎜ ⎟ + ωo 2
2
⎝2⎠
⎡ 1
+
⎢ 2Q
⎣ ωc1 = ωo ⎢− ⎡
1
ωc 2 = ωo ⎢ +
⎢ 2Q
⎣ 2
⎤
⎛1⎞
⎟ + 1⎥
⎜
⎟
⎜
⎥
⎝ 2Q ⎠
⎦
2
⎤
⎛1⎞
⎜
⎟ + 1⎥
⎜ 2Q ⎟
⎥
⎝
⎠
⎦ EE 203 Circuit Analysis 2  Spring 2008  Prof. Kwang W. Oh  EE@SUNYBuffalo 2 ωc1 = − R
1
⎛R⎞
+ ⎜ ⎟+
2L
⎝ 2 L ⎠ CL
2 ωc 2 =
ωo =
β=
Q= R
1
⎛R⎞
+ ⎜ ⎟+
2L
⎝ 2 L ⎠ CL
1
LC R
L
L
CR 2 Lecture 30  Chapter 14  3/5  13/13 ...
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This note was uploaded on 10/19/2011 for the course EE 203 taught by Professor Staff during the Spring '08 term at SUNY Buffalo.
 Spring '08
 Staff

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