2.4: Exact Equations
2. In this equation, M(x, y) = x2y+x4 cos x and N(x, y) = x3. Taking partial derivatives,
we obtain
M / y =(x2y+x4cosx) /y =x2= 3x2=N /x.
Therefore, according to Theorem 2, the eq
ECE 201
Exam #2
Review
Topics
Thevenins/Nortons Theorem
Maximum Power Transfer
Superposition
OP AMPs
Inverting Amplifier
Non Inverting Amplifier
Relaxation Oscillator
Comparator
Difference Amplifier
I
Problem # 4.93
Use the principle of superposition to find the voltage v 0 in the circuit shown.
The 240 V source acting alone.
'
0
v
(20 5)
(20 5) 12
(240V ) 60V
The 84 V source acting alone
"
0
v
(
Problem # 4.91
a) Use the principle of superposition to find the voltage v in the circuit shown.
b) Find the power dissipated in the 10 resistor.
Consider the voltage source acting alone.
Req (10 (2 1
PROBLEM # 6.28
Use realistic capacitor values from Appendix H to construct series and parallel
combinations of capacitors to yield the equivalent capacitances specified below.
Try to minimize the numb
PROBLEM # 6.3
The voltage at the terminals of the 200H inductor is shown below in (b). The inductor
current is known to be zero for t<=0.
a) Determine the expressions for i for t>=0.
vs (t ) =0, t 0
v
PROBLEM # 6.27
Find the equivalent capacitance with respect to the terminals ab for the circuit
shown.
1
1 1 5
= + =
Ceq1 4 6 12
Ceq1 =2.4 m F .12V
Ceq2 =1.6 m F +2.4 m F =4 m F .12V
1
1 1 16 1
= + =
PROBLEM # 6.32
Determine the equivalent circuit for a series connetion of ideal capacitors. Assume that
each capacitor has its own initial voltage.
The current in each capacitor is the same since they
PROBLEM # 6.42
The polarity markings on two coils are to be determined experimentally. The
experimental setup is shown below. Assume that the terminal connected to the negative
terminal of the battery
PROBLEM # 6.40
a) Show that the coupled coils shown below can be replaced by a single coil having an
inductance of Lab = L1 + L2 + 2M.
b) Show that if the connections to the terminals of the coil labe
PROBLEM # 7.34
a) Use component values fromAppendix H to create a firstorder RL circuit (see
Fig. 7.16) with a time constant of 8 s. Use a single inductor and a network of
resistors, if necessary. Dr
PROBLEM # 7.81
The voltage waveform is applied to the circuit shown. The initial voltage on the
capacitor is zero.
a) Calculate vo(t).
vs (t ) =0, 0 t
vs (t ) =50V , 0 t 1ms
vs (t ) =0,1ms t
For 0 < t
PROBLEM # 7.35
The switch in the circuit shown has been closed for a long time before opening at
t = 0.
a) find the numerical expressions for iL(t) and vO(t) for t>=0.
For t<0, the switch is closed, a
PROBLEM #7.55
Assume that the switch in the circuit shown has been in position a for a long time
and that at t = 0 it is moved to position b.
Find
a) vC(0+)
vC (0+ =vC (0 ) =50V
)
b) vC()
20
( 30)
Maximum Power Transfer
Maximize the power delivered to a
resistive load
ECE 201 Circuit Theory I
1
Consider the General Case
A resistive network contains independent
and dependent sources.
A load is
Thevenin and Norton Equivalent
Circuits
Voltage Source Model
Current Source Model
ECE 201 Circuit Theory I
1
Why do we need them?
Circuit simplification
Reduce the complicated circuit on the left to
Superposition
Principle of Superposition
When a linear circuit is excited by more than one
independent source of energy, the total response is
the sum of the responses to each source acting
individu
Problem # 4.6
Use the nodevoltage method to determine the voltages v 1 and v2 in the circuit
shown below.
The voltages have been identified for you show the ground connection
(voltage=0) and the curr
Alternator Problem
1.)
Identify the branch currents and node A.
2.)
Write equations for the branch currents and solve.
A
VA
+
I3
I2
I1
+

16 VA
I1
0.2
12.8 VA
I2
0.1
16 VA 12.8 VA
VA
0.2
0.1
16 VA
UNIVERSITY OF MASSACHUSETTS DARTMOUTH
DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING
ECE 201
CIRCUIT THEORY I
ALTERNATOR PROBLEM
An automobile alternator with an internal resistance of 0.2 develops
3.2: Compartmental Analysis
2. Let x(t) denote the mass of salt in the tank at time t with t = 0 denoting the moment
when the process started. Thus we have x(0) = 0.5 kg. We use the mathematical model
2.2: Separable Equations
2. This equation is separable because we can separate variables by multiplying both sides
by dx and dividing by 4y2 3y+1.
4. This equation is separable because
dy/dx= yex+y /
4.4: Nonhomogeneous Equations: The Method of Undetermined
Coefficients
t
(ln 3)t
rt
4. Rewriting the righthand side in the form 3 = e
= e , where r = ln 3, we
conclude that the method of undetermined
4.1: Introduction: The MassSpring Oscillator
2. (a) Substituting cy(t) into the equation yields
m(cy)+ b(cy)+ k(cy) = c(my+ by+ ky) = 0.
(b) Substituting y1(t) + y2(t) into the given equation, we obt
7.2: Definition of the Laplace Transform
2
2. L cfw_t (s) = 2/s3.
3t
4. L cfw_te (s) = 1/(s3)2.
s
10. L cfw_f (t) (s) = e /s2 + 1/s 1/s2.
12. L cfw_f (t) (s) = (1e
3(s2)
)/(s2) + e
3s
6
/s,
7.6: Transforms of Discontinuous and Periodic Functions
6. The function g(t) equals zero until t reaches 2, at which point g(t) jumps to t + 1. We write
g(t) = 00,2(t)+(t+1)u(t2) = (t+1)u(t2)
so then