ECE 2300 – C
IRCUITA
NALYSISHOMEWORK #1
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i. Your name
ii. ECE 2300
iii. Section #
iv. Homework assignment number
v. Due date of homework assignment
Failure to follow the above instructions will result in a reduction of the grade for the
homework assignment in question.
Homework turned in after the due date and time will receive a grade of zero (0).
Please read the instructions on the next page concerning notation (Notation in Circuit
Analysis). These rules will be applied throughout the semester. In writing all your
homework assignments, quizzes, and exams, we expect you to follow them. There will be
significant credit deduction for not doing so.
Notation in Circuit Analysis
Getting notation right is an important part of what we want to teach you in ECE 2300. Below you will find rules concerning notation that we expect you to follow. Here we cover voltage, current, and power. In later homework assignments, we will review rules associated with other quantities. If these rules are not followed, you can expect to lose credit on homework, quizzes, and exams.
Circuit Variables
Voltage and current are the principal circuit variables we will encounter. The rules for labeling these are as follows. Examples are shown in the figure below.
• All voltages are labeled with a lower case v and a subscript; the subscript may be a letter or number. A “plus” and “minus” sign must be assigned to each voltage to show the reference polarity. The voltage indicator “v” must lie on a line with the “plus” and “minus” signs. • All currents are labeled with a lower case i and a subscript; the subscript may be a letter or
number. An arrow must be assigned to each current to show the reference direction. The current indicator “i” must be placed near the arrow.
• All voltages and currents used in equations must be labeled on a circuit diagram; otherwise, that variable is “undefined”.
Do not show a “v” without + and – signs; do not show + and – signs without a v. Do not show an “i” without an arrow; do not show an arrow without an “i”. These things have no meaning.
βiB vo RB1 RB2 RE RC vCE iC is iE iB vB2 + -vX +
-Power and Energy
Power is indicated by a lower case “p”. It is the product of voltage and current, but the interpretation of the results is also important, and must be indicated in the labeling.
When calculating power, your notation must show (i) whether the power you are calculating is absorbed or delivered and (ii) which circuit element you are calculating power for. This can be done as follows: if you are calculating the power delivered by current source iS1, you would indicate this as pdel by iS1; if you
are calculating the power absorbed by resistor R5, you could indicate this as pabs by R5.
Energy (indicated by lower case “w”) is the time integral of the power. In calculating energy you should follow the same notation rules as for power.
Notation
0.1a) Label voltages across devices A and B using labeling rules stated above. Note that there is more than one correct answer.
b) Label currents through devices A and B using labeling rules stated above. Note that there is more than one correct answer. c) Given the labels you provided, are the voltages across A and B the same, or of opposite value? How about the currents?
0.2
a) If vC =
-
5[V], is the reference polarity in the same direction asthe actual polarity?
b) If vC =
-
5[V], which of terminals 1 and 2 is at the higherpotential?
c) If vC = 5[V], which of terminals 1 and 2 is at the higher
potential?
0.3
a) Label a reference current through all elements using labeling rules provided.
b) Based on your labels, is the current going through device ___ necessarily the same as current through ___? (Y/N) A, B _____ B, D _____ A, C _____ B, C _____
A
B
C
iC 1 2v
C+
-B D A C
Units 0.4
Power has units of energy per time. Show that voltage [V] multiplied by current [A] gives power in Watts. Do this by breaking voltage, current, and power into more fundamental units.
Power and Charge 0.5
a) If for device D vD = 5[V] and iD = 10 [mA], what is the power
absorbed by D?
b) If for device D’ vD’ = 5[V] and iD’ = 10 [mA], what is the power
absorbed by D?
0.6
a) Is device A labeled in the passive or active sign convention?
b) Is device C labeled in the passive or active sign convention?
c) If vC = -3[V] and iC = 20[mA], what is
the sign convention for device C? d) If iA = - 100[mA], what is the sign
convention for device A?
D
iD 1 2v
D+
-D’
iD’ 1 2v
D’+
-C A B iA iC
v
C+
-v
A+
-0.7
a) In the diagram to the left, the current iD = 250[mA], and the
voltage vD =
-
25 [V]. How much charge passes through D in 3.6 [s]?If the current is being carried by electrons, which way are the electrons going?
b) For the current and voltage specified in part a), how much energy is delivered to D in 3.6 [s]?
0.8
Consider the device D shown in Problem 0.7.
a) Write an expression for the power delivered to D if iD = -100[mA] and
𝑣𝑣𝐷𝐷(𝑡𝑡) = (12[𝑉𝑉] − 1.6[𝑉𝑉𝑠𝑠−1]𝑡𝑡). The expression for vD(t) is considered valid for t > 0 [s].
b) Given the voltage and current specified in 0.8 a), find the energy delivered to D between t = 0 and t = 2 [s].
0.9
Repeat problem 0.8 for 𝑣𝑣𝐷𝐷(𝑡𝑡) = �12𝑒𝑒−0.2�𝑠𝑠−1�𝑡𝑡� [V], t > 0 [s].
D
iD1 2
v
D-Problems 1 and 2 below have been adapted from Nilsson and Riedel, 8 ed., Prentice Hall, 2008.
1. After a hard day at work, Dr. Dave and Dr. Trombetta go to their cars and find that one of them
has a dead battery! They decide to “jump” start the car that has the dead battery, which they do by connecting the batteries as shown in the diagram below. (Note the little half-circle where the battery cables cross near Dr. T’s battery. This indicates that there is no connection here – the cables simply cross over each other.) Dr. Dave happens to have a current meter on him, so he measures the current ijump in the figure, and finds that it is 30 [A].
a) Which car has the dead battery?
b) Assuming the current ijump and the battery voltages remain constant, how much energy is
transferred to the dead battery in 1 [min]?
Dr. Dave’s Battery BatteryDr. T’s
+
-12 [V]+
-12 [V]i
jump2. The manufacturer of a 9 [V] flashlight battery says that the battery will deliver 20 [mA] for 80
continuous hours. However, during that time the voltage will drop from 9 [V] to 6 [V]. Assume the voltage drop is linear in time, but that the current is constant.
a) How much energy does the battery deliver in 80 [h]?
b) Battery “capacity” can be stated in terms of the total charge the battery will deliver, with units of milliamp-hours [mA-h]. If the battery can be considered “dead” when the voltage reaches 6 [V], what is the capacity of the flashlight battery in [mA-h]?
3. The diagram in Figure P1.1 shows the electrical
connection of two devices. The current iX and the
voltage vX are shown. Assume that the current iX(t) is
a current made up of the flow of electrons. Assume that
(
-1)
475[s ]
( ) 42[mA] 25[mA] ; for 0, and ( ) 3.75[V]; for 0. t X X i t e t v t t -= - > = >
a) Is Device 1 in the active or passive sign convention with respect to iX and vX?
b) Which way are the electrons moving through Device 2 at t = 1[ms]? c) Which way are the electrons moving through Device 1 at t = 5[ms]? d) What is the power absorbed by Device 2 for t > 0?
e) Are the electrons gaining or losing energy as they move through Device 1, at t = 3[ms]? f) What is the energy delivered by Device 2 for the time period 0 < t < 2[ms]?
g) What is the energy delivered by Device 1 for the time period 2[ms] < t < 10[ms]?
4. The diagram shown in Figure P1.2 shows a model for a
car’s electrical system. The current iB and the voltage vB are
given as
(
-1)
2.3[s ]
5[A] 5[A] ; for 0, and rad 14[V]cos(3.7[ ] ); for 0. s t B B i e t v t t -= - > = >
a) Is the battery in the active or passive convention with respect to iB and vB?
b) How much energy is absorbed by the battery in the first second after
t
=
0
?Figure P1.1
5. The circuit shown in Fig. Pl.5 is a
combination of devices, with the voltages and currents as indicated.
a) Find the power associated with each device. b) Determine which of the devices is delivering
positive power, and add up all of those positive power values. Repeat the same process for the devices that absorb positive power.
c) The two sums that you determined in part a) should be equal. Check to see if this is true.
A B D E C F vX + - vW + -vY vT vZ + + + -iS iR iP iQ iN iM Figure P1.5 vT = 10.3[V], vW = -7.6[V], vX = -8.4[V], vY = -9.5[V], vZ = 1.9[V]
iM = 3.3[mA], iN = 24.1[mA], iP = 15.2[mA], iQ = -18.5[mA], iR = 5.6[mA], iS = -8.9[mA]
6. A student has made a series of voltage and current measurements on a circuit, and determined
the values shown in Figure P1.6. You have been asked to check the validity of these values. a) Determine whether these values can be valid by testing whether the energy is conserved in this
circuit.
b) There was one sign error that was made by the student in these measurements. Determine what the error was. Do not use Kirchhoff’s Laws in your solution.
A B D E C F + -+ -27.4[V] + + + -2.6[A] G 13.0[V] 18.9[V] 78.8[V] 57.3[V] + 38.4[V] --2.6[A] 5.7[A] 4.1[A] -1.5[A] 3.1[A] Figure P1.6
7. The voltage in a wall socket has been measured, and
found to be equal to
A blender is connected to this wall socket, and the current in the blender is measured and found to be
The diagram shows the voltage and current polarities in this connection.
a) Find the power absorbed by the blender at t = 0. b) Find the power absorbed by the blender at t = 4.3[ms]. c) Find the power absorbed by the blender at t = 8.6[ms].
d) Find the energy absorbed by the blender over one cycle of the sinusoid.
(PWA Problem 1 in Module 1)
8. The diagram below shows a model for a car’s
electrical system. The current iB and the voltage vB are given as
(
1)
2[ ]
10
5 [A]; for
0, and
14[V]; for
0.
s t B Bi
e
t
v
t
-= -
+
>
=
>
The variable t is given in [s].
a) With respect to iB and vB, is the battery in the active or passive convention?
b) How much energy is absorbed by the battery in the first second after
t
=
0
? (PEQWS Problem 2 in Module 1)Blender Wall Socket vS(t) + -iB(t) Rest of the Car iB(t) vB(t) + -Battery
9. The device shown in Figure P1.3a has the current iQ(t) as
shown in Figure P1.3b, and the voltage vQ(t) as shown in Figure
P1.3c.
a) The energy delivered by Device Q during the time period 1[s] < t < 4[s].
b) Find the power absorbed by Device Q at t = 3.5[s].
10. The terminal voltage vB(t), and the terminal current iB(t), for the device
shown in Figure P1.4a are changing with time. Plots of these two variables are shown in Figures P1.4b and P1.4c.
a) Find the numerical expressions for the power delivered by this device for the time interval 1[s] < t < 6[s].
b) Calculate the energy delivered by this device during the time interval 1[s] < t < 6[s].
11. A device is shown, with reference polarities for voltage and
current, in Figure 1. A plot of the voltage across the device, vX, is
shown in Figure 2. A plot of the current through the device, iX, is
shown in Figure 3.
a) Find the power absorbed by the device at t = 8[s].
b) Find the energy absorbed by the device in the time period 0 < t < 12[s].
12. Six components, labeled A, B, C, D, E, and F, are connected as shown in the circuit diagram
given in Figure 1. Some voltages and currents are defined in the equations below. The currents in this circuit are made up of the flow of electrons.
a. Find the direction that the electrons are moving through component D at t = 1[ms]. Your answer should be either left to right, or right to left. Explain how you got your answer.
b. Find the power delivered by component C at t = 2[ms].
c. Find the energy absorbed by component B during the time period from
t = 1[ms] to t = 4[ms].
d. Find the energy delivered by component B during the third [millisecond], counting [milliseconds] starting at t = 2[ms].
( )
[ ]
( )
[ ]
( )
[ ]
( )
[ ]
kV 3.5 5.3 V s kV 4.6 6.4 V s kA 5.7 7.5 A s kA 6.8 8.6 A s X Q X Q v t t v t t i t t i t t = + = - = - - = - + B D C A + -iA E -+ vX vQ iQ iX F Figure 1Selected Numerical Solutions for ECE 2300 Homework Set #1.
1) b) 21.6 [kJ] 2) a) 43.2 [kJ]
b) 1600 [mA-h]
3) a) passive sign convention
b) the electrons are moving down through Device 2, from the top to the bottom c) the electrons are moving down through Device 1, from the top to the bottom
d)
(
-1)
475[s ] . . 2 157.5 t 93.75 [mW], for 0. ABS BY DEV p = - e- + t> e) gaining f) 15.84[ J]µ g) 625[ J]µ4) a) active sign convention b) -18.5[J]
5) a)
∑
pDEL =315.87[mW],∑
pABS =315.87[mW]. b) Yes, the two power summations are equal. 6) Solution omitted here.7) a) 746.9 [W] b) -1.39 [W] c) 746 [W] d) 6.224 [J]
8) a) active sign convention b) -9.47 [J]
9) Solution omitted here, but required for you. 10) a) Solution omitted here, but required for you. b) -41.67[J]
11) Solution omitted here. 12) d) wDEL.BY.B = -317.2[mJ].
Problem 7) is worked out in detail in the Dr. Dave Project files, as PWA Problem 1 in
Module 1. This is a PowerPoint file, and is intended to work as a tutor to help you on these problems. These files are available on the course web page.
Problem 8) is worked out in detail in the Dr. Dave Project files, as PEQWS Problem 2 in
Module 1. This is a Word file, and is intended to serve as a practice examination question, to help you prepare to solve these kinds of problems. These files are available on the course web page.
