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1.45
Fig. 1.10-2
Load current
15 A
1 2 3 4 5 6 7 8 9
Time
4. The current through a two-terminal element is given by
i t
k
t
t
s
( )
cos
. sin
.
=
+
+
≤ ≤
2
100
0 5
200
0 035
0
p
p
A for 0 t
for all other
tt
{
.
Find the constant k if the total charge
that went through the element in the interval [0,0.05 s] is zero.
5. The voltage across an ideal two-terminal passive element is v(t)
=
10e
-
100t
V for t
≥
0 and
zero for t
<
0. The current through the element is i(t)
=
0.1e
-
100t
A for t
≥
0 and zero for t
<
0.
(a) Identify the element and its parameter value. (b) What is the amount of charge that went
through the element in the time interval [0.01 sec, 0.05 sec]? (c) What is the amount of charge
that went through the element in [0,
∞
] time-interval? (c) What is the ratio of instantaneous
power delivered to it at 0.01 sec to the corresponding value at 0 sec? (d) What is the total
energy delivered to the element? (e) What is the time at which the energy delivered to it reached
99% of total energy delivered to it? Assume that v(t) and i(t) given are as per passive sign
convention.
6. The voltage across an ideal two-terminal passive element is v(t)
=
10(1
-
e
-
1000t
) V for t
≥
0
and zero for t
<
0. The current through the element is i(t)
=
0.001e
-
1000t
A for t
≥
0 and zero
for t
<
0. (a) Identify the element and its parameter value. (b) What is the amount of charge
that went through the element in [0,
∞
] time-interval? (c) What is the total energy delivered
to the element? (d) What is the time instant at which the power delivered to the element
is a maximum? What is the value of this maximum power? What is the value of energy
delivered to the element till that instant? Assume that v(t) and i(t) given are as per passive
sign convention.
7. The voltage across an ideal two-terminal passive element is v(t)
=
600 cos (100 t) V for t
≥
0
and zero for t
<
0. The current through that element as per passive sign convention is i(t)
=
10 sin (100 t) A for t
≥
0 and zero for t
<
0. (a) Identify the element and its parameter value.
(b) Find an expression for the charge that goes through the element as a function of time. (c) Find
an expression for instantaneous power delivered to the element as a function of time. (d) Show
that the energy delivered to the element till t is non-negative for all t.
8. The current that flows through an ideal independent voltage source with v(t)
=
12 V is i(t)
=
10
+
10 cos 100
p
t A for t
≥
0 and 0 A for t
<
0. Assume passive sign convention. (a) What is
the power delivered by the source at t
=
1 sec? (b) What is the change in energy storage in the
source between t
=
0 and t
=
1 sec? Does the energy storage in source increase with time or
decrease with time?
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1.46
CircuitVariablesandCircuitElements
9. The voltage across a two-terminal element is v(t)
=
10 sin 1000
p
t V. The current that flows into
the element as per passive sign convention is
i t
k
t
t
s
t
( )
. cos(
)
.
=
+
+
≤ ≤
1 5
100
3
0 05
0
p
p
A for 0
A for all other
Find k if the energy delivered to the element in the time interval [0,0.1 s] is zero.
10. The voltage across a two-terminal element is v(t)
=
10 sin 1000
p
t V
and current in that element is
i t
k
t
t
s
t
( )
. cos(
)
.
=
+
+
≤ ≤
1 5
100
3
0 05
0
p
p
A for 0
A for all other
Assume passive sign convention. Find the average power delivered to the element over any time
interval of width equal to the period of the voltage and current waveforms.
11. The voltage across a two-terminal resistor is v(t)
=
10 sin 1000
p
t V. Find the value of a DC
voltage that will deliver a power that is equal to the average power delivered by this voltage source
over any time interval equal to the period of the voltage waveform.
12. The value of resistance of a resistor is measured to be 10
W
at room temperature of 35
0
C.
Temperature coefficient of this resistance is 0.004. A constant current source of 0.25A is connected
across the resistance. The resistance attains a steady temperature after some time. The temperature
rise in the resistor after the temperature has reached a steady-state is given by 100p where p is
the power dissipated in the resistor in Watts. (i) Find the steady-state temperature, corresponding
resistance value and the power dissipated in the resistor under steady-state condition. (ii) Find the
critical value of current at which the temperature of the resistor increases without any limit and
it burns out.
13. A DC voltage source of 2.5 V is connected across the resistor in Problem 12. (i) Find the steady-
state temperature, corresponding resistance value and the power dissipated in the resistor under
steady-state condition. (ii) Find the critical value of applied DC voltage (if such a value exists) at
which the temperature of the resistor increases without any limit and it burns out.
14. There are only three elements in an isolated circuit. Assume passive sign convention. The terminal
voltage and current of first element are given by
v t
e
t
t
t
1
100
5 5 1
0
0
0
( )
(
)
=
+
−
≥
<
−
V for
V for
and
i t
e
t
t
1
100
0
0
0
( )
=
≥
<
−
A for
A for
Corresponding variables for the second element are v
2
(t)
=
v
1
(t) and i
2
(t)
=
-
2 A. The voltage
across the third element is v
3
(t)
=
v
1
(t). Identify the third element assuming that it is a passive
element, find its parameter value and the current through the third element as a function of time.
[Hint: Sum of power delivered by all elements in a circuit is zero.]
15. The v – i characteristic of a passive two-terminal element as per passive sign convention is
v(t)
=
100i(t)
+
20i(t)|i(t)| V. (a) Show that this element is nonlinear. (b) Show that this element
is a passive element. (c) Show that it is a bilateral element. (d) Find the current flow through the
element when the voltage across it is a constant at 100 V. There are two possible values for the
current. How do you choose the correct one?
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