Chapter 1 – Electric Circuit Variables
Exercises
Exercise 1.2-1 Find the charge that has entered an element by time t when i = 8t2 – 4t A, t ≥ 0.
Assume q(t) = 0 for t < 0.
8
Answer: q(t ) = t 3 − 2t 2 C
3
Solution:
i (t ) = 8 t 2 − 4 t A
t t 8 3 2 t 8
q(t ) = ∫0 i dτ + q(0) = ∫0 τ − τ τ + = τ − τ = t3 −2 t 2 C
2
(8 4 ) d 0 2
3 0 3
Exercise 1.2-2 The total charge that has entered a circuit element is q(t) = 4 sin 3t C when t ≥ 0
and q(t) = 0 when t < 0. Determine the current in this circuit element for t > 0.
d
Answer: i (t ) = 4sin 3t = 12 cos 3t A
dt
dq d
Solution: i (t ) = = 4sin 3t = 12 cos 3t A
dt dt
Exercise 1.3-1 Which of the three currents, i1 = 45 μA, i2 = 0.03 mA, and i3 = 25 × 10–4 A, is
largest?
Answer: i3 is largest.
Solution:
i1 = 45 μA = 45 × 10-6 A < i2 = 0.03 mA = .03 × 10-3 A = 3 × 10-5 A < i3 = 25 × 10-4 A
1-1
,Exercise 1.5-1 Figure E 1.5-1 shows four circuit elements identified by the letters A, B, C, and
D.
(a) Which of the devices supply 12 W?
(b) Which of the devices absorb 12 W?
(c) What is the value of the power received by device B?
(d) What is the value of the power delivered by device B?
(e) What is the value of the power delivered by device D?
Answers: (a) B and C, (b) A and D, (c)–12 W, (c) 12 W, (e)–12 W
+ 4V – – 2V + + 6V – – 3V +
3A 6A 2A 4A
(A) (B) (C) (D)
Figure E 1.5-1
Solution:
(a) B and C. The element voltage and current do not adhere to the passive convention in B and C
so the product of the element voltage and current is the power supplied by these elements.
(b) A and D. The element voltage and current adhere to the passive convention in A and D so the
product of the element voltage and current is the power delivered to, or absorbed by these
elements.
(c) −12 W. The element voltage and current do not adhere to the passive convention in B, so the
product of the element voltage and current is the power received by this element: (2 V)(6 A) =
−12 W. The power supplied by the element is the negative of the power delivered to the element,
12 W.
(d) 12 W
(e) –12 W. The element voltage and current adhere to the passive convention in D, so the product
of the element voltage and current is the power received by this element: (3 V)(4 A) = 12 W. The
power supplied by the element is the negative of the power received to the element, −12 W.
1-2
,Problems
Section 1-2 Electric Circuits and Current Flow
P1.2.1 The total charge that has entered a circuit element is q(t) = 1.25(1–e–5t) when t ≥ 0 and
q(t) = 0 when t < 0. Determine the current in this circuit element for t ≥ 0.
Answer: i(t) = 6.25e–5t A
d
Solution: i (t ) = 1.25 (1 − e−5t ) = 6.25 e−5t A
dt
P 1.2-2 The current in a circuit element is i(t) = 4(1–e–5t) A when t ≥ 0 and i(t) = 0 when t < 0.
Determine the total charge that has entered a circuit element for t ≥ 0.
t 0
Hint: q (0) = ∫ i (τ ) = ∫ 0dτ = 0
−∞ −∞
Answer: q(t) = 4t + 0.8e–5t– 0.8 C for t ≥ 0
Solution:
4 4
q ( t ) = ∫ i (τ ) dτ + q ( 0 ) = ∫ 4 (1 − e−5τ ) dτ + 0 = ∫ 4 dτ − ∫ 4 e−5τ dτ = 4 t + e−5t − C
t t t t
0 0 0 0 5 5
P 1.2-3 The current in a circuit element is i(t) = 4 sin 3t A when t ≥ 0 and i(t) = 0 when t < 0.
Determine the total charge that has entered a circuit element for t ≥ 0.
t 0
Hint: q (0) = ∫ i (τ )dτ = ∫ 0 dτ = 0
−∞ −∞
Solution:
t t 4 4 4
q ( t ) = ∫ i (τ ) dτ + q ( 0 ) = ∫ 4sin 5τ dτ + 0 = −
t
cos 3τ 0 = − cos 3 t + C
0 0 5 5 5
1-3
, ⎧0 t < 2
⎪
⎪2 2 < t < 4
P 1.2-4 The current in a circuit element is i (t ) = ⎨ where the units of current are A
⎪ −1 4 < t < 8
⎪⎩0 8 < t
and the units of time are s. Determine the total charge that has entered a circuit element for t ≥ 0.
Answer:
⎧0 t<2
⎪2t − 4 2<t <4
⎪
q (t ) = ⎨ where the units of charge are C.
⎪8 − t 4<t <8
⎩⎪0 8<t
Solution:
t t
q ( t ) = ∫ i (τ ) dτ = ∫ 0 dτ = 0 C for t ≤ 2 so q(2) = 0.
−∞ −∞
t t
q ( t ) = ∫ i (τ ) dτ + q ( 2 ) = ∫ 2 dτ = 2τ
t
2
= 2t−4 C for 2 ≤ t ≤ 4. In particular, q(4) = 4 C.
2 2
t t
q ( t ) = ∫ i (τ ) dτ + q ( 4 ) = ∫ −1 dτ + 4 = − τ 4 + 4 = 8−t C for 4 ≤ t ≤ 8. In particular, q(8) = 0 C.
t
4 4
t t
q ( t ) = ∫ i (τ ) dτ + q ( 8 ) = ∫ 0 dτ + 0 = 0 C for 8 ≤ t .
8 8
P 1.2-5 The total charge q(t), in coulombs, that enters the terminal of an element is
⎧0 t<0
⎪
q (t ) = ⎨2t 0≤t ≤2
⎪3 + e −2t (t −2) t>2
⎩
Find the current i(t) and sketch its waveform for t ≥ 0.
Solution:
⎧0 t <0
dq ( t ) ⎪
i (t ) = i (t ) = ⎨2 0< t < 2
dt ⎪ −2( t − 2 )
⎩−2e t >2
1-4
Exercises
Exercise 1.2-1 Find the charge that has entered an element by time t when i = 8t2 – 4t A, t ≥ 0.
Assume q(t) = 0 for t < 0.
8
Answer: q(t ) = t 3 − 2t 2 C
3
Solution:
i (t ) = 8 t 2 − 4 t A
t t 8 3 2 t 8
q(t ) = ∫0 i dτ + q(0) = ∫0 τ − τ τ + = τ − τ = t3 −2 t 2 C
2
(8 4 ) d 0 2
3 0 3
Exercise 1.2-2 The total charge that has entered a circuit element is q(t) = 4 sin 3t C when t ≥ 0
and q(t) = 0 when t < 0. Determine the current in this circuit element for t > 0.
d
Answer: i (t ) = 4sin 3t = 12 cos 3t A
dt
dq d
Solution: i (t ) = = 4sin 3t = 12 cos 3t A
dt dt
Exercise 1.3-1 Which of the three currents, i1 = 45 μA, i2 = 0.03 mA, and i3 = 25 × 10–4 A, is
largest?
Answer: i3 is largest.
Solution:
i1 = 45 μA = 45 × 10-6 A < i2 = 0.03 mA = .03 × 10-3 A = 3 × 10-5 A < i3 = 25 × 10-4 A
1-1
,Exercise 1.5-1 Figure E 1.5-1 shows four circuit elements identified by the letters A, B, C, and
D.
(a) Which of the devices supply 12 W?
(b) Which of the devices absorb 12 W?
(c) What is the value of the power received by device B?
(d) What is the value of the power delivered by device B?
(e) What is the value of the power delivered by device D?
Answers: (a) B and C, (b) A and D, (c)–12 W, (c) 12 W, (e)–12 W
+ 4V – – 2V + + 6V – – 3V +
3A 6A 2A 4A
(A) (B) (C) (D)
Figure E 1.5-1
Solution:
(a) B and C. The element voltage and current do not adhere to the passive convention in B and C
so the product of the element voltage and current is the power supplied by these elements.
(b) A and D. The element voltage and current adhere to the passive convention in A and D so the
product of the element voltage and current is the power delivered to, or absorbed by these
elements.
(c) −12 W. The element voltage and current do not adhere to the passive convention in B, so the
product of the element voltage and current is the power received by this element: (2 V)(6 A) =
−12 W. The power supplied by the element is the negative of the power delivered to the element,
12 W.
(d) 12 W
(e) –12 W. The element voltage and current adhere to the passive convention in D, so the product
of the element voltage and current is the power received by this element: (3 V)(4 A) = 12 W. The
power supplied by the element is the negative of the power received to the element, −12 W.
1-2
,Problems
Section 1-2 Electric Circuits and Current Flow
P1.2.1 The total charge that has entered a circuit element is q(t) = 1.25(1–e–5t) when t ≥ 0 and
q(t) = 0 when t < 0. Determine the current in this circuit element for t ≥ 0.
Answer: i(t) = 6.25e–5t A
d
Solution: i (t ) = 1.25 (1 − e−5t ) = 6.25 e−5t A
dt
P 1.2-2 The current in a circuit element is i(t) = 4(1–e–5t) A when t ≥ 0 and i(t) = 0 when t < 0.
Determine the total charge that has entered a circuit element for t ≥ 0.
t 0
Hint: q (0) = ∫ i (τ ) = ∫ 0dτ = 0
−∞ −∞
Answer: q(t) = 4t + 0.8e–5t– 0.8 C for t ≥ 0
Solution:
4 4
q ( t ) = ∫ i (τ ) dτ + q ( 0 ) = ∫ 4 (1 − e−5τ ) dτ + 0 = ∫ 4 dτ − ∫ 4 e−5τ dτ = 4 t + e−5t − C
t t t t
0 0 0 0 5 5
P 1.2-3 The current in a circuit element is i(t) = 4 sin 3t A when t ≥ 0 and i(t) = 0 when t < 0.
Determine the total charge that has entered a circuit element for t ≥ 0.
t 0
Hint: q (0) = ∫ i (τ )dτ = ∫ 0 dτ = 0
−∞ −∞
Solution:
t t 4 4 4
q ( t ) = ∫ i (τ ) dτ + q ( 0 ) = ∫ 4sin 5τ dτ + 0 = −
t
cos 3τ 0 = − cos 3 t + C
0 0 5 5 5
1-3
, ⎧0 t < 2
⎪
⎪2 2 < t < 4
P 1.2-4 The current in a circuit element is i (t ) = ⎨ where the units of current are A
⎪ −1 4 < t < 8
⎪⎩0 8 < t
and the units of time are s. Determine the total charge that has entered a circuit element for t ≥ 0.
Answer:
⎧0 t<2
⎪2t − 4 2<t <4
⎪
q (t ) = ⎨ where the units of charge are C.
⎪8 − t 4<t <8
⎩⎪0 8<t
Solution:
t t
q ( t ) = ∫ i (τ ) dτ = ∫ 0 dτ = 0 C for t ≤ 2 so q(2) = 0.
−∞ −∞
t t
q ( t ) = ∫ i (τ ) dτ + q ( 2 ) = ∫ 2 dτ = 2τ
t
2
= 2t−4 C for 2 ≤ t ≤ 4. In particular, q(4) = 4 C.
2 2
t t
q ( t ) = ∫ i (τ ) dτ + q ( 4 ) = ∫ −1 dτ + 4 = − τ 4 + 4 = 8−t C for 4 ≤ t ≤ 8. In particular, q(8) = 0 C.
t
4 4
t t
q ( t ) = ∫ i (τ ) dτ + q ( 8 ) = ∫ 0 dτ + 0 = 0 C for 8 ≤ t .
8 8
P 1.2-5 The total charge q(t), in coulombs, that enters the terminal of an element is
⎧0 t<0
⎪
q (t ) = ⎨2t 0≤t ≤2
⎪3 + e −2t (t −2) t>2
⎩
Find the current i(t) and sketch its waveform for t ≥ 0.
Solution:
⎧0 t <0
dq ( t ) ⎪
i (t ) = i (t ) = ⎨2 0< t < 2
dt ⎪ −2( t − 2 )
⎩−2e t >2
1-4
