Name: ___SOLUTION_________________________
Student Number: ______________________
ELEC ENG 2CI5
DAY CLASS
DURATION OF EXAMINATION: 2 Hours
McMASTER UNIVERSITY FINAL EXAMINATION
Dr. P. M. Smith
DECEMBER 2016
THIS EXAMINATION PAPER INCLUDES 8 PAGES AND 5 QUESTIONS. YOU ARE
RESPONSIBLE FOR ENSURING THAT YOUR COPY OF THE PAPER IS COMPLETE. BRING
ANY DISCREPANCY TO THE ATTENTION OF YOUR INVIGILATOR.
Special Instructions:
Closed book examination.
Please answer all questions in the space provided.
Clearly explain and show your work for full marks.
Use the back of question pages or last two pages if you need
additional space.
Use of Casio FX-991 calculator only is allowed.
______________________________________________________________________________________________________
Q1
Q2
Q3
Q4
Q5
Total
Page 1 of 8
Name: ___SOLUTION_________________________
Student Number: ______________________
1. Find !! in the network below using nodal analysis. Clearly label your nodes in the
figure.
Answer:
[20 marks]
!
!
Define the voltage at the upper node as !! . Since !! = ! !!
, we have
!! − 12 !! − !!
!!
+
+
=0
1 kΩ
1 kΩ
1 kΩ
!! − 12 !! − !! 2!!
+
−
=0
1 kΩ
1 kΩ
1 kΩ
or
or
Solving,
1
1
1
+
+
1 kΩ 1 kΩ 1 kΩ
1
2
−
−
1 kΩ 1 kΩ
1
12
!
1 kΩ
∙ ! = 1 kΩ
1
1
!!
12
+
1 kΩ 1 kΩ
1 kΩ
−
3 −1 !!
12
∙
=
!!
−3 2
12
!!
12
=
!!
24
and
!! = 12 V
Page 2 of 8
Name: ___SOLUTION_________________________
Student Number: ______________________
2. Find !! in the circuit below using loop analysis. Clearly label your loop currents in the
figure.
Answer:
[20 marks]
Defining non-overlapping loop currents, we get
!! =
!!
4 kΩ !! − !!
=
= !! − !!
4000
4000
!! = 6 mA
4 kΩ !! − !! + 2 kΩ !! − !! + 4 kΩ !! = 0
Solving,
!! = 3 mA
!! = 6 mA
!! = 3 mA
Therefore,
!! = 4 kΩ !! = 12 V
Page 3 of 8
Name: ___SOLUTION_________________________
Student Number: ______________________
3. Find the value of !! that will maximize the power that it absorbs. What is that power?
Answer:
[20 marks]
We first need to obtain the Thevenin equivalent circuit, as seen by !! . If we opencircuit the load and perform a nodal analysis, we get
1
1
+
3 kΩ 2 kΩ
0
−
Simplifying,
1
2 kΩ
0
1
3V
2 kΩ
−1
mA
−
!!
2 kΩ
1
!
∙
1.5 mA
−
! =
1 kΩ
3V
!!
1
1
1
2 kΩ
+
+
1 kΩ 2 kΩ 1 kΩ
−
1
1 kΩ
1
−
1 kΩ
!!
5
0 −3
−15 A
0
6 −6 ∙ !! = 9 A
!!
−3 −6 15
9A
⇒
!!
−2.25 V
!! = 2.75 V
!!
1.25 V
Therefore
!!" = !! + 3 V − !! = −2 V
Since we only have independent sources, !!" can be found by removing the sources.
!!" = 1 kΩ + 2 kΩ // 4 kΩ = 2.33 kΩ
Maximum power will be delivered to !! when
!! = !!" = 2.33 kΩ
The maximum power will be
!! =
!
!!"
−2 !
=
= 0.43 mW
4!!" 4×2.33×10!
Page 4 of 8
Name: ___SOLUTION_________________________
Student Number: ______________________
4. Use the step-by-step method to find !! ! for ! > 0 in the circuit below. Sketch the
time response.
Answer:
Step 1:
[20 marks]
!
Assume !! ! = !! + !! ! !!
Step 2:
!! 0! = !! + !! =
Step 3:
If we replace the current source on the left with its Thevenin equivalent,
the voltage at the central node becomes
! ! → ∞ = 10 V
and
10
!! ! → ∞ = !! =
= 1.67 A
6
Step 4:
!=
Step 5:
24
=2A
12
!
4
4
=
= = 0.44 s
! 6 + 6//6 9
!
!! ! = 1.67 + 0.33 ! !!.!! (A)
Step 6: Since the same current is flowing through the resistor
!
!! ! = 6 Ω × !! ! = 10 + 2 ! !!.!! V
Page 5 of 8
Name: ___SOLUTION_________________________
Student Number: ______________________
5. Find !! ! in the network below. Sketch the phasor diagram showing the source and
!! ! .
Answer:
[20 marks]
This is a standard current divider-type circuit. Converting to phasors, the current
flowing in the right-most branch is
1
1
5+
!(10)(0.1)
!! =
× 5 ∠0!
1
1
1
1
5 + !(10)(0.4) + !(10)(0.02) +
5+
!(10)(0.1)
=
1
5 − !1
1
0.2 − !0.25 + !0.2 + 5 − !1
=
× 5 ∠0!
5 + !1
× 5 ∠0!
5.2 − !1.3 + 5 + !1
=
=
5 + !1
× 5 ∠0!
10.2 − !0.3
26 ∠11.3!
104
∠−1.7!
× 5 ∠0!
= 2.5 ∠13! (A)
where we have used ! = 10 rad/s, taken from the cosine function in the source.
Therefore,
and
!! = 5 Ω × !! = 12.5 ∠13! (V)
!! ! = 12.5 cos 10! + 13!
(V)
Page 6 of 8
Name: ___SOLUTION_________________________
Student Number: ______________________
(Please use this page for extra space. Clearly label the question you are answering)
5 (cont.) The phasor diagram is
Page 7 of 8
Name: ___SOLUTION_________________________
Student Number: ______________________
(Please use this page for extra space. Clearly label the question you are answering)
[ THE END ]
Page 8 of 8