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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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Problems