Circuits II
EE221
Unit 7
Instructor: Kevin D. Donohue
Active Filters, Connections of Filters, and
Midterm Project
Useful Circuits:
Describe the transfer functions of these circuits:
Non-Inverting Op Amp: gain = (1 + Rf / R1)
R1
+
vo
vi
R1
-
Rf
vi
Rf
+
vo
-
Voltage Follower: gain = 1
vi
Inverting Op Amp: gain = (- Rf / R1)
+
vo
-
Cascading Filter Stages: no Loading Effects:
Given two active filter circuits with transfer functions Hˆ 1 ( s )
and Hˆ 2 ( s ) , their series combination has a transfer function equal
to the product of the individual transfer functions:
Hˆ T ( s )  Hˆ 1 ( s ) Hˆ 2 ( s )
provided that the connection between these circuits does not
significantly alter the output resistance of the first circuit or
the input resistance of the second circuit (i.e. no loading
effects).
V1o = V2i
VTi = V1i
Hˆ 1 ( s)
Hˆ 2 ( s )
V2o= VTo
ˆ
ˆ  Vˆ 

V
V
To
Hˆ T ( s ) 
  1o  2o   Hˆ 1 ( s ) Hˆ 2 ( s )
VˆTi  Vˆ1i  Vˆ2i 
Homework (1): In Matlab create bode plots for a first-order high-pass
filter (with G = 2 and fc = 1 kHz) in series with a second order bandreject filter (with f0 = 5000 Hz, B = 500 Hz, and GDC = G = 1 ).
Other Useful Circuits
The summing amp:
Rf
R1
vi1
The differential amp:
R2
vi2
+
+
vo
vi1
R1
-
RN
R2
R1
-
vi2
+
vo
R2
viN
Rf
Rf
 Rf

vo   vi1 
vi 2   
viN 
R2
RN 
 R1
vo 
R2
vi 2  vi1 
R1
-
Adding Outputs of Filter Stages
Given two active filter circuits with transfer functions Hˆ 1 ( s )
and Hˆ 2 ( s ) , the TF resulting from their parallel connection is the
sum of the TF (the complex TF values, not simply the
magnitudes), provided that the connection of these circuits
does not significantly alter the output resistance of the first
circuit or the input resistance of the second circuit (i.e. no
loading effects). If the outputs are connected through a
summing amp with gains 1 and 2 the combined TF become:
Hˆ 1 ( s)
VTi
Hˆ 2 ( s )
V1o

 Vˆ 
2
i
io
i 1
V2o
VˆTo  1Hˆ 1 (s)VˆTi   2 Hˆ 2 (s)VˆTi
ˆ
V
Hˆ T ( s)  To  1Hˆ 1 ( s)   2 Hˆ 2 ( s)
VˆTi
VTo
Adding Outputs of Filter Stages
Homework (2): In Matlab create Bode plots for 2 second order
bandpass filters (BPF1 with Go = 10dB and fo = 9 kHz, B = 3kHz, and
BPF2 with Go = 7dB and fo = 6 kHz, B = 800Hz) connected in parallel
through a summing amp with a gain of -0.5 for the output of BPF1 and
a gain of -3.5 for the output of BPF2. For the magnitude plots, plot
them in both dB (typical Bode plot) and linear (without the 20Log10
scaling).
Homework (3): Repeat Problem 2, with different gains for the
summing amp. Change the summer gain for the BPF1 node to -3 and
the gain for the BPF2 node to -0.5. Compare he Bode plots with those
in Homework Problem 2.
Homework (4): Compute the threshold of hearing curve for your ear
at the frequency points indicated in the hearing-aid project and scale
it as described in the project (and on the web page) and plot it against
the normal threshold of ear. Describe the environment where you did
the hearing test at (quiet office using speakers, … noisy lab with
headphones ….)
Midterm Project



In this project you will design a hearing aid to correct
your "client's" hearing to normal at the threshold of
hearing.
This project will combine your study of the AC circuit
frequency response, transfer functions, passive and active
filters, and filter combining to design a filters with a
transfer function magnitude of a specified shape.
The outcomes of this project involve the following
components:




Implementing a "Design Process"
Using a top-down approach to solve complex designs
Extending knowledge of filter design
Developing plans for efficient/effective teamwork
General Design Process
1.
2.
3.
4.
5.
6.
7.
Develop a set of specifications to formulate a clear and precise
problem statement.
Propose a set of potential solutions at the top-level.
Decide on the most feasible or promising design.
Develop solution(s) (build circuit protopypes or create computer
simulations for testing).
Optimize critical parameters of the design(s).
Test design(s) against specifications.
If design performance is not satisfactory, return to steps 2
through 6 (depending on the amount redesign necessary to
achieve satisfactory performance). If design performance is
satisfactory, complete documentation on the design process,
technical description of critical components, and performance
analysis.
General Design Process


The proposal report involves completing steps 1 through 3 and
creating a time table for accomplishing steps 4 through 7.
The final documentation should be complete enough for someone
else to clearly compare the designed circuit’s performance with
the criterion in the problem statement, build your circuit, and
verify performance results.
Top-Down Design



The top-down method develops a solution by initially proposing a design in
terms of functional blocks. This first and most general block diagram is
called the top level. Then each functional block in the top level is broken
down into more detailed and specific functional blocks to create the
second level. Once the functional blocks are detailed enough so that a
circuit to perform the function is easily determined, the process is
finished.
Example: Build a hearing aid with the following specifications .....
The necessary functions (convert sound waves to electric signal,
preamplify signal, filter signal, convert amplified and filtered signal back
into sound waves). This suggests the following top-level
Microphone


Pre Amp
Custom
Filter
Power
Amp
Speaker
Specifications for the function and input-output characteristics must be
developed for each block.
Also note that “top” is a relative term. This is a top-level design if your
assignment was the whole hearing aid.
Top-Down Design

At the second design level (relative to the hearing aid as first level)
each of above functional blocks are broken down into finer detail.
For example, assume the transfer function of the filter block is
specified. The top-level design for the filter (this is top level for
your assignment, but second level the entire hearing aid) can be
specified as (parallel approach - something in the specifications may
suggest this combination):
Band-Pass
Filter 1
Band-Pass
Filter 3
Band-Reject
Filter 2
Band-Pass
Filter 4
Summing Amp
where
 The purpose of band-pass filter 1 is to help shape the transfer
function in the frequency range .... with a approximate gain of XX
dB, a resonant frequency near XX Hz, and approximate bandwidth
of XX Hz … (Do this type of description for each block at this
level).
Time Tables




Once the top-level diagrams for your circuit have been completed,
work can be assigned to determine the actual circuits, integrate
them at the top level, optimize performance, test against
specifications, redesign, document, and complete the final report.
The most efficient way to assign the work is to have as many
parallel tasks going on as possible. If someone is doing nothing until
some else finishes his or her task, the work should be redistributed.
Rather than have someone in the group remain inactive for an
extended time, you can at least assign several people to the same
task, or work independently on the same thing and take the best
result.
Regular group meetings should be scheduled to keep tabs on the
progress, to educate each other on findings, and make adjustments
if necessary (almost always necessary).
You should schedule to complete and test the design about a week
before the due date, so the last week can be devoted to minor
modifications (if needed) and completing the final report.
Engineering Notebooks
It is important to develop organized work habits. As tasks become
more complex, it is more difficult to remember results determined along
the way. Write down all details, even documenting a bad approach helps
organize your thinking and improve. This is also a critical record for
establishing patents and getting various types of approval. The notebook
should:
 Verify what you have accomplished and when
 Summarize the important results for future reference
 Record work useful for debugging and completing final
documentation
The notebook should be a regular bound book. Each time you work
on the project individually or as a group, you should enter the date into the
notebook, what you did and the results. Thus, all work that you do
regarding the project should appear in the notebook in chronological order.
If errors are made, don't tear out the page. Just put a line through it and
correct it in a later entry. Loose pages can be attached to pages in your
notebook, if work was done on different paper or printed out from a
computer.
Specifications and Problem Statement:



Unlike homework and test problems, design problems are typically
under-specified. This implies there can be many solutions. It
becomes difficult to judge a design in terms of right and wrong, but
instead the judgment is in more vague terms such as bad, good, and
better (best is often difficult to prove even for simple problems).
The given specifications should be organized at the beginning of the
project and stated in such a way that they can be measured when
the final design is tested. Sometimes the specifications are given in
terms of errors that should be minimized (or goodness maximized)
or bounds on the error that should be satisfied.
The problem statement should be as short as possible, while clearly
stating all the specified criteria, along with some additional criteria
if necessary to limit possibilities or create a useful way of
computing the error.
Order of Events


February 24 – Project assigned. Groups formed, notebook entries
started
February 24 - March 8






Meet as a team, identify tasks and time-table
Identify the design criteria
Come up with your preliminary top-level design for your client
Complete your preliminary report
Preliminary Report is due March 8
March 9 – April 12







Validate your top level design
Optimize your transfer function with Matlab.
Finalize your analyses and validate design approach with an error
analysis based on ideal transfer functions of proposed filters
Implement SPICE model for your optimized top-level design
Realize design, use off-the-shelf components.
Present final error and discuss how well design meets specifications.
Final Report due & Engineering notebooks due April 12