1
Instructions for Using Circuit
System Design Method Cards
Neil E. Cotter, Member, IEEE, and Cynthia Furse, Fellow, IEEE
Abstract—The circuit design cards described here allow
students to design useful circuits by viewing circuits as
combinations of system building blocks portrayed on one side of
the cards. The other side of each card shows a circuit schematic
for the building block. Thevenin equivalents are the crucial
ingredient in translating from the circuit to the building block
and vice versa.
Index Terms—Linear circuit, system design, cards, Thevenin
equivalent
I INTRODUCTION
T
HE deck of Circuit System Design Method (CSDM) cards
allows users to design complete circuits by laying down
cards in sequence from left-to-right. The cards have two
sides: a system design side (with gray surrounding) and a
circuit schematic side (with gray strip on right side). On the
system design side, the processing of signals is represented as
mathematical operations such as addition or multiplication, or
logic operations such as And and Or. The user can design a
system using the system side of the cards and then flip the
cards over to see the circuit schematic and detailed formulas
for calculating additive factors and gains.
For circuits, the system design is more complicated than the
ideal case in which the behavior of each circuit system block
is the same regardless of the block that precedes it or follows
it. Instead, connecting a second block to the output of a first
block may cause loading that changes the output voltage of the
first block. A unique feature of the CSDM Cards is that they
account for these loading effects while retaining the system
design motif. For linear system design, the CSDM Cards give
exact formulas for calculating the voltages in a system in the
presence of loading, and the calculation proceeds left-to-right,
like the system design.
This paper discusses how the cards are used and the theory
on which they are based. Separate documents entitled Catalog
of Circuit System Design Method Cards [1] and Guide to
Individual Circuit System Design Method Cards [2] together
show each card with accompanying comments.
N. E. Cotter is with the Electrical and Computer Engineering Dept.,
University of Utah, Salt Lake City, UT 84112 USA (phone 801-581-8566, email: necotter@ece.utah.edu).
C. Furse is with the Department of Electrical and Computer Engineering,
University of Utah, Salt Lake City, UT 84112-9206 USA and also with
LiveWire Test Labs, Inc., Salt Lake City, UT 84117 USA (e-mail:
cfurse@ece.utah.edu).
II THEVENIN EQUIVALENTS
The key idea behind the CSDM cards is the Thevenin
equivalent. According to the theory of linear circuits, the
behavior of a linear circuit with respect to output terminals a
and b is indistinguishable from a circuit consisting of only a
voltage source, vTh, and a single resistor, RTh. Fig. 1 shows
the Voltage Reference card in the CSDM deck. The strip on
the right side on the card gives the Thevenin equivalent of the
circuit block. Thus, the circuit in black may be replaced by
the circuit in blue in the gray strip.
Fig. 1. Voltage Reference circuit card with Thevenin equivalent on right.
On the system side, as Fig. 2 shows, the Voltage Reference
card shows the output as
vTh = GVCC
(1)
where G is the multiplier of power supply VCC on the system
side of the card and is defined on the circuit side as
G=
Rb .
Ra + Rb
(2)
Fig. 2. Voltage Reference circuit card, system side.
This example incorporates two features that are essential to
the CSDM card system:
2
1) The system side output is the Thevenin voltage rather
than the circuit output voltage for that system block, and
2) The gain of the circuit block depends only on what is on
the card or on the preceding card but is unaffected by a
block connected to its output.
Feature (1) means that the voltages in the system design
may not be directly measurable in the circuit. This difficulty
can occur whenever one deals with a Thevenin equivalent, but
it may complicate debugging of the circuit. The alternative is
to require that all system blocks maintain their output voltage
regardless of what is connected to them. This requirement
would eliminate many simple circuit building blocks such as
the voltage reference.
Feature (2) means that a system may be designed by adding
blocks to the output of an existing system without altering
what the existing system does. Note that, although the output
voltage of an existing system may be loaded down by the
addition of another stage, the Thevenin equivalent voltage of
the existing stage is unaffected by what is connected to its
output. Furthermore, when we add a block to the output of an
existing system, we can find a new Thevenin equivalent for the
entire system in terms of the existing system's Thevenin
equivalent and the added block.
Fig. 3. illustrates the concept (and notation) of collapsing a
Thevenin equivalent and a subsequent circuit system design
block into a new Thevenin equivalent. The equations needed
for finding the new Thevenin equivalent are encoded in the
strip on the right edge of the circuit side of the CSDM cards.
A left-pointing arrow (a narrow "<" symbol) in a subscript
means "from the card to the left". Thus, vTh< means
"Thevenin equivalent voltage from the card to the left," and
RTh< means "Thevenin equivalent resistance from the card to
the left." Using the equations in the gray strip on the right of
the V-Divider card, we obtain the Thevenin equivalent for
both cards shown in the lower rectangle.
The next section presents a more extensive example system
design and the calculation of successive Thevenin equivalents
in a system design.
Fig. 3. Collapsing Thevenin equivalents in CSDM notation. Gray strip on right side of V-Divider card specifies new Thevenin equivalent of entire circuit to
left, as shown in the lower rectangle.
III SYSTEM DESIGN EXAMPLE
Fig. 4 shows a system that lights an LED in accordance
with the temperature measured by a thermistor. On the system
side, we start with a reference, which represents zero volts.
On the circuit side, we must define a reference since voltages
have meaning only when they are compared to the circuit's
reference.
The next two cards on the system side constitute a sensor: a
voltage source that produces a quantity the system can
process, and a thermistor that has no immediate effect on the
voltage but changes the Thevenin resistance.
The final card of the system is an LED indicator that will
get brighter as temperature increases. To determine how
much current flows in the LED for a given temperature, we
work from left to right on the system side, writing an equation
for the Thevenin voltage as we go. For the successive cards,
we get the following formulas:
Card 1:
(3)
vTh = 0V
Card 2:
vTh = vs
(4)
3
Card 3:
Card 4:
vTh = vs
vTh = G1vs + G2 (1.5V)
(5)
increases with vTh.
(6)
The illumination of the LED depends on the current, which
(a)
(b)
Fig. 4. Temperature sensor circuit. (a) System desgin. (b) Circuit design.
To find the LED current precisely, we use the equations on
the circuit side of the cards to compute the Thevenin voltages
and resistances as we move from left to right, until the last
card where we use the formula for LED current. For the
successive cards, we get the following formulas:
Card 1:
(7)
vTh = 0V , RTh = 0 Ω
Card 2:
vTh = vs , RTh = 0 Ω
(8)
4
Card 3:
Card 4:
vTh = vs , RTh = RCdS
vTh = G1vs + G2 (1.5V) , RTh = RT || Rd
v − 1.5V
iLED = s
Rd + RT
(9)
(10)
(11)
The value of RCdS for a typical photocell, the CdS
photocell PDV-P9008 made by Advanced Photonix, Inc. [2],
might be 20 kΩ at 5 lux, dropping by a factor of about four
when the light level increases by a factor of 10. For 500 lux,
we would have about RCdS = 1.25 kΩ. For 100 klux, we
would have RCdS = 117 Ω. (500 lux is a typical light level for
an office, and 100 klux is a typical light level for sunlight [3]),
If we planned to connect something to the output of the
LED block, we would calculate the gains, G1 and G2. If we
chose vs = 5 V and Rd = 100 Ω, we would have
G1 =
100
100 + RCdS
(12)
G2 =
RCdS .
100 + RCdS
(13)
and
Note that the iLED of the LED block is valid here because
vTh< = vs = 5 V > 1.5 V and nothing is connected to the right
side of the LED block. Our systems ends with the LED, and
we are really only interested in the LED current. For vs = 5 V
and Rd = 100 Ω, we have a maximum current of
5V − 1.5V
iLED =
≈ 16.1mA .
217 Ω
3rd Card: vTh = Gvs =
R2
vs , RTh = R1 || R2 .
R1 + R2
(15)
Equation (15) is the familiar Thevenin equivalent for a
voltage-divider, which is what the circuit up to the third card
has become. Adding the parallel resistor as the fourth card,
we apply the Thevenin equivalent equations from the right
edge of the circuit side to obtain the final Thevenin equivalent:
4th Card:
R
R2
⋅
vs ,
R + R1 || R2 R1 + R2
RTh = R || (R1 || R2 ) .
vTh =
(16)
After simplification, this becomes
R || R1 || R2
⋅ vs ,
R1
RTh = R || R1 || R2 .
vTh =
4th Crad:
(17)
The result in (17) should be the same as we would obtain
for the original voltage-divider with R2||R replacing R2 in
(15), since the parallel resistor is in parallel with R2 and could
be combined with it in the voltage-divider. Viewed in this
fashion, the following equivalent result is obtained, validating
the formulas on the card:
R || R2
R || R1 || R2 ,
vs =
vs
R1 + R || R2
R1
RTh = R1 || (R || R2 ) = R || R1 || R2 .
vTh =
(18)
V LOADING EFFECTS AND INPUT RESISTANCE
(14)
This value is under the current limit for readily available
LED's. The design is complete, (albeit without verification of
the resistance of the actual photocell to be used).
IV THEVENIN EQUIVALENT CALCULATION
The CSDM cards, in addition to facilitating circuit design,
may be used to find Thevenin equivalents of simple circuits.
Using the circuit side of the card, the user constructs the
circuit of interest and employs the equations for vTh and RTh
on the right edge of the cards to collapse the circuit into a
single Thevenin equivalent working, as always, from left to
right.
Fig. 5 shows an example of a voltage divider with a resistor
added to its output. (A more compact implementation of this
circuit would incorporate the Voltage Reference card from
Fig. 1, but the present example is chosen in order to show the
iterative nature of the calculations.)
The first two cards, according to the formulas on the right
edge of circuit side yield the Thevenin equivalents listed in (3)
and (4) earlier. Note that for Card 2, vTh< is vTh from the
first card, which is 0 V. When the V-Divider card is added as
the third card, vTh< is vTh from the second card, which is vs,
and RTh< is RTh from the second card, which is 0 Ω. Thus,
the new Thevenin equivalent becomes
A system design becomes simpler when the Theveninequivalent voltage of a system block is the same as the circuit
voltage at the output of the block. Otherwise, the Thevenin
equivalent voltage cannot be measured directly in the circuit,
leading to difficulties: debugging becomes problematic, the
design process becomes more complex, and the design
becomes harder to understand.
To explain the difference between the Thevenin equivalent
voltage and the actual circuit output voltage, we once again
invoke Thevenin equivalents, but this time they appear in a
different guise.
Fig. 6 illustrates how the input of a circuit block may be
viewed as a Thevenin equivalent. (Note that the Thevenin
equivalent of the input side of a circuit block is quite a
different entity than the Thevenin equivalent encoded in the
gray stripe on the circuit side of a card.) As Fig. 6 shows, we
may identify an input resistance and voltage source for a block
added to the output of an existing circuit system. In many
cases, the input voltage source is reference, and Fig. 6
becomes a simple voltage divider. For the sake of simplicity,
we discuss only the voltage divider here. The conclusions we
reach also apply to the more general case.
Three categories characterize the possible behavior of the
voltage divider:
1) Rout >> Rin (the output voltage is effectively shorted to
reference by the low input resistance)
5
2) Rout ≈ Rin (the output voltage is determined by the
voltage divider)
3) Rout << Rin (the output voltage is effectively the same as
the Thevenin equivalent voltage on the left)
The latter case means the output of a block acts like a voltage
source, which simplifies system designs; the voltage measured
in the circuit at the junction between blocks becomes the same
as the Thevenin equivalent output voltage that appears in the
system design.
When desired, we can tack on the "Op-Amp Buffer (VFollower)" card to a circuit to achieve a low output resistance
without altering the system's output voltage. The buffer block
has a gain of unity and a high input resistance, making it ideal
for extracting the Thevenin output voltage.
Fig. 5. Example of extending Thevenin equivalent.
A high input resistance for a stage breaks the chain of
Thevenin-equivalent resistance calculations, and a low output
resistance starts a new chain of Thevenin resistance values at
(approximately) zero. Thus, the relative size of input
6
resistance and output resistance is of interest.
This
information is encoded on the system side of each card by
stylized wires in the gray "buffer zone" around the outside
edge. Fig. 7 shows the location of the output and input
resistance indicators. Fig. 8 shows the definitions of the wire
symbols, along with the result of the various possible
combinations of output and input resistances. The wire
symbols may be interpreted as follows:
Thick Wire = low resistance (big wire)
Normal wire = intermediate resistance
hollow wire = high resistance (open circuit)
Fig. 8 shows that, when output resistance is low or input
resistance is high, the Thevenin-equivalent voltage is the same
as the output voltage for a block. The system design is also
however, only occur in practice for open circuit outputs, which
just means the output signal is cut off from the succeeding
stage. Conversely, circuit blocks with desirable input or
output resistances are numerous in the CSDM deck.
Fig. 7. Meaning of wiring symbols in gray buffer area on system side of
card.
Fig. 6. Thevenin equivalent of output of left block drives Thevenin
equivalent of input of right block.
simplified by the elimination of RTh< terms from RTh
calculations.
Cards with desirable low output resistance include op-amp
circuits, comparators, and logic gates. Op-amp circuits can
also be linear, which helps to account for their ubiquity. Cards
with desirable high input resistance include comparators, logic
gates, and op-amp circuits where inputs are connected to the +
input of the op-amp. These examples all involve integrated
circuits intentionally designed to have high input resistance
and low output resistance.
A normal-sized wire at the input or output on the system
side of a card indicates an intermediate resistance. In this
case, the equations for the Thevenin resistance from the circuit
side of the card capture the loading effect of the second card
on the first card. The reader is cautioned, however, that the
equations for the Thevenin resistance explicitly encode only
the output resistance. Explicit information about input
resistance is available only in the qualitative wire symbols on
the system side of the card.
In the CSDM card deck, low input-resistance blocks are of
minimal concern. Worst-case scenarios occur when an output
resistance is high or an input resistance is low. In such
circumstances, the high output resistance effectively
disconnects the input card's Thevenin voltage-source from its
output. Likewise, the low input resistance effectively shorts
the input to the Thevenin equivalent voltage source on the
output side, which is often reference. In other words, the
signal is shorted to reference and lost. Such undesirable cases,
Fig. 8. Output and input resistance combinations' effect on vout.
VI OTHER CARD FEATURES
This section lists miscellaneous features of the cards
unmentioned previously.
1) Cards come in five "suits" listed in Table I. The suits are
convenient groupings of functions, and the suits are
convenient for identifying cards.
2) Cards for wires and meters are printed on transparent
stock. This tactic allows the wires to be flipped and
rotated, if necessary, and it allows the meters to be placed
on top of cards without obscuring them.
3) Although the figures in this paper show the cards laid out
vertically, they may be placed side-by-side, eliminating
the needed for additional connecting lines.
4) Redundancy in the CSDM deck allows many circuits to
be realized in multiple ways. This redundancy is helpful
when more than one circuit of a given type is needed.
5) The generic "Op-Amp" card presents linear, negativefeedback op-amp circuits in a universal way. To wit, one
may reduce the circuitry driving the + and – inputs of the
op-amp to Thevenin equivalents and use the formula for
vTh on the circuit side to find the formula for the output
of the op-amp. For students in an introductory circuits
course, this approach allows them to easily derive all of
the standard op-amp gain formulas.
7
TABLE I
CSDM CARD SUITS
Symbol
Suit
# of Cards
Measurement
Wire
Component
Amplifier
Gate
4
8
8
15
12
Description
Meters for v and i values
Wires, including reference
Power sources and resistors
Blocks with voltage gains
Nonlinear blocks
VII CONCLUSION
The CSDM cards allow the user to design simple circuit
systems by laying down cards from left to right representing
common circuit building blocks such as voltage sources,
resistors, op-amps, logic gates, and LED's. On the system
side, the cards show Thevenin equivalent voltages being
altered by additions and multiplications to create a final output
value that may be displayed on an LED or used to run an
output device. On the circuit side, the cards show the
formulas, in terms of component values, for the additions and
multiplications from the system side. These formulas allow
the user to complete a circuit design.
Examples presented in this paper illustrate the design
process for an LED temperature indicator and show how the
cards may be used to find Thevenin equivalents of basic
circuits. In each case, the key idea is to collapse a circuit into
a new Thevenin equivalent after adding a system (or circuit)
block to its output. Formulas on the right side of the circuit
side of the cards indicate how to do this.
By processing Thevenin equivalent voltages rather than the
output voltages from each stage, the CSDM cards achieve a
generality that extends beyond conventional approaches that
require blocks with ideal input or output resistances.
ACKNOWLEDGMENT
The authors thank the Electrical and Computer Engineering
Department at the University of Utah for its support of
innovative teaching methods such as team teaching, the
flipped classroom, USB lab instrument modules, and system
design in early courses for electrical and computer engineering
students.
REFERENCES
[1]
[2]
[3]
[4]
N. E. Cotter and C. Furse. (2014, July 30) Catalog of Circuit System
Design Method Cards [Online]. Available: http://www.atm.com
N. E. Cotter and C. Furse. (2014, July 30) Guide to Individual Circuit
System Design Method Cards [Online]. Available: http://www.atm.com
Advanced Photonix, Inc. CdS Photoconductive Photocells, PDV-P9008.
Available:
http://www.advancedphotonix.com/wp-content/uploads/PDV-P9008.pdf
The Engineering ToolBox, Illuminance—Recommended Light Levels,
Available:
http://www.engineeringtoolbox.com/light-level-rooms-d_708.html
Neil E. Cotter (M’84) earned an M.S. degree in mathematics and a Ph.D. in
electrical engineering from Stanford University, Palo Alto, CA, in 1986.
He is currently Associate Professor (Lecturer) in the Electrical and
Computer Engineering Department at the University of Utah in Salt Lake
City, Utah, where he has worked since 1999. He previously worked as a
Design Engineer on speech recognition algorithms at Fonix, Inc. and on
control systems at Geneva Steel Corp. Prior to that, he published papers on
artificial neural networks as an Assistant Professor at the University of Utah.
Dr. Cotter was awarded a millennium medal by IEEE in 2000.
Cynthia Furse (M’87–SM’99–F’08) received the Ph.D. degree from the
University of Utah, Salt Lake City, in 1994.
She is the Associate Vice President for Research at the University of Utah
and Professor in the Electrical and Computer Engineering Department. Her research focuses on imbedded antennas and sensors in complex environments,
such as telemetry systems in the human body, and sensors for location of
faults on aging aircraft wiring. She has directed the Utah “Smart Wiring”
program, sponsored by NAVAIR and USAF, since 1998. She is Chief
Scientist for LiveWire Test Labs, Inc., a spin off company commercializing
devices to locate intermittent faults on live wires. She teaches
electromagnetics, wireless communication, computational electro- magnetics,
microwave engineering, and antenna design.
Dr. Furse was the Professor of the Year in the College of Engineering,
Utah State University for the year 2000, Faculty Employee of the Year 2002,
a National Science Foundation Computational and Information Sciences and
Engineering Graduate Fellow, IEEE Microwave Theory and Techniques
Graduate Fellow, and President’s Scholar at the University of Utah. She also
received the Distinguished Young Alumni Award from the Department of
Electrical and Computer Engineering, University of Utah and the College of
Engineering Distinguished Engineering Educator Award.