This document summarizes a lecture about filters and quality factors. It introduces low pass, high pass, band pass and band stop filters which are built by combining impedances from inductors, resistors and capacitors. It also discusses the quality factor Q of a filter, which determines its selectivity. Q is defined as the center frequency divided by the bandwidth, and depends on the ratio of an inductor's impedance to a resistor's impedance in a series RLC filter. A higher Q indicates a more selective filter.

1. 6.002 Fall 2000 Lecture 18
Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.002 CIRCUITS AND
ELECTRONICS
Filters

2. 6.002 Fall 2000 Lecture 18
Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
Review
+
–
CvIv +
–
R
C
Reading: Section 14.5, 14.6, 15.3 from A & L.
+
–
cViV +
–
RZ
CZ
i
RC
C
c V
ZZ
Z
V ⋅
+
=
RCj1
1
R
Cj
1
Cj
1
V
V
i
c
ω
ω
ω
+
=
+
=

3. Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.002 Fall 2000 Lecture 18
A Filter
RCj1
1
V
ZZ
Z
V i
RC
C
c
ω+
=⋅
+
=
“Low Pass Filter”
ω
1
( )
i
c
V
V
H =ω
Demo
with audio
+
–
cViV +
–
RZ
CZ

4. Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.002 Fall 2000 Lecture 18
Quick Review of Impedances-
Just as
21
ab
ab
AB RR
I
V
R +==
LjR
I
V
Z 1
ab
ab
AB ω+==
1R
abI +
–
abV
2R
A
B
1R
abI +
–
abV
Ljω
A
B

5. Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.002 Fall 2000 Lecture 18
Quick Review of Impedances
Similarly
L2C1AB ZR||ZRZ ++=
L
2C
2C
1 Z
RZ
RZ
R +
+
+=
Lj
CRj1
R
R
2
2
1 ω
ω
+
+
+=
1R
L
A
B
2RC

6. Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.002 Fall 2000 Lecture 18
We can build other filters by
combining impedances
( )ωZ
L
R
C
ω
Z

7. Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.002 Fall 2000 Lecture 18
We can build other filters by
combining impedances
HPF
High Pass Filter
ω
( )ωH
ω
( )ωH
LPF
Low Pass Filter
ω
( )ωH
HPF
( )ωZ
L
R
C ω
Z
+
–
+
–
+
–

8. Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.002 Fall 2000 Lecture 18
Check out:
R
Cj
1
Lj
R
V
V
i
r
++
=
ω
ω
RCjLC1
RCj
2
ωω
ω
+−
=
( ) ( )222
i
r
RCLC1
RC
V
V
ωω
ω
+−
=
ω
+
–
L C
R
+
–
rViV
LC
1
o =ω
At resonance,
ω = ωo
and
ZL + ZC = 0,
so Vi sees
only R!
More later…
Intuitively:
i
r
V
V
1
L blocks high freq
C blocks low freq

9. Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.002 Fall 2000 Lecture 18
What about:
+
–
L C
R
+ –lcV
iV
Band Stop Filteri
lc
V
V
ω
1 C open L open
Check out Vl and Vc in the lab.

10. Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.002 Fall 2000 Lecture 18
Another example:
+
–
+
– L
R
iV C oV
i
o
V
V
oω ω
BPF
C shortL short
Application: see AM radio coming up shortly

11. Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.002 Fall 2000 Lecture 18
AM Radio Receiver
crystal radio demo
Thévenin
antenna
model
+
– L
R
iV C
demodulator
amplifier
antenna

12. Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.002 Fall 2000 Lecture 18
AM Receiver
“Selectivity” important —
relates to a parameter “Q” for the filter. Next…
+
– L
R
iV C
demodulator
amplifier
f
signal
strength
540 …1000 1010 1020 1030 … 1600 KHz
10 KHz
filter WBZ
News
Radio

13. Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.002 Fall 2000 Lecture 18
Recall,
Selectivity:
Look at series RLC in more detail
+
–
L C
R
+
–
rViV
Cj
1
LjR
R
V
V
i
r
ω
ω ++
=
i
r
V
V
ω
oω
2
1
higher Q
1
Define quality factor
Δ
=Q o
ω
ω
ωΔ
bandwidth
⇒Qhigh more selective

14. Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.002 Fall 2000 Lecture 18
ω
ω
Δ
= o
Q
LC
1
o =ω
Quality Factor Q
⎟
⎠
⎞
⎜
⎝
⎛ −+
=
++
=
CR
1
R
L
j1
1
Cj
1
LjR
R
iV
rV
ω
ω
ω
ω
?ωΔ
at =0ωο
ωο:

15. Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.002 Fall 2000 Lecture 18
Note that abs magnitude is
2
1
when
1j1
1
CR
1
R
L
j1
1
V
V
i
r
±
=
⎟
⎠
⎞
⎜
⎝
⎛
−+
=
ω
ω
i.e. when 1
CR
1
R
L
±=−
ω
ω
0
LC
1
L
R2
=−
ω
ω ∓
:ωΔ
ω
ω
Δ
= o
Q
Quality Factor Q
Looking at the roots of both equations,
LC
4
L
R
2
1
L2
R
2
2
1 ++=ω
LC
4
L
R
2
1
L2
R
2
2
2 ++−=ω
L
R
=−=Δ 21 ωωω

16. Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.002 Fall 2000 Lecture 18
R
L
L
R
Q oo ωω
==
The lower the R (for series R),
the sharper the peak
ω
ω
Δ
= o
Q
Quality Factor Q
LC
1
o =ω

17. Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.002 Fall 2000 Lecture 18
Another way of looking at Q :
cycleperlostenergy
storedenergy
2π=Q
0
2
r
2
r
2
RI
2
1
IL
2
1
2
ω
π
π=
R
L
Q oω
=
Quality Factor Q