Current, Power, and Energy - Exercice

Assume that the voltage at the terminals of the element in Fig. 1.5, whose
current was defined in Assessment Problem 1.3, is
v=0t<0;v=10e−5000t kV, t≥0.
1. Calculate the power supplied to the element at 1 ms.
2. Calculate the total energy (in joules) delivered to the circuit element.

Solution

1. Since the current is entering the + terminal of the voltage drop defined
for the element in Fig. 1.5, we use a “+” sign in the power equation.
p = vi = (10,000e−5000t)(20e−5000t) = 200,000e
−10,000t W.p(0.001)= 200,000e − 10,000(0.0001) = 200,000e
−10= 200,000(45.4×10−6) = 9.08 W.
2. From the definition of power given in Eq. 1.3, the expression for energy
is
w(t)=∫0tp(x)dx.
To find the total energy delivered, integrate the expresssion for power
from zero to infinity. Therefore,
wtotal=∫0∞200,000e−10,000x dx=200,000e−10,000x−10,000|0∞
=−20e−∞−(−20e−0 )=0+20=20 J.
Thus, the total energy supplied to the circuit element is 20 J.

Assessment Problems

Objective 3—Know and use the definitions of power and energy;
Objective 4—Be able to use the passive sign convention

1. 1.5 Assume that a 20 V voltage drop occurs across an element from
terminal 2 to terminal 1 and that a current of 4 A enters terminal 2.
1. Specify the values of v and i for the polarity references shown in
Fig. 1.6(a)–(d).
2. Calculate the power associated with the circuit element.
3. Is the circuit element absorbing or delivering power?
Answer:
1. Circuit 1.6(a): v = −20 V,i=−4 A; circuit 1.6(b): v = −20 V,v=
−20 V,i=4 A; circuit 1.6(c): v = 20 V,i=−4 A; circuit 1.6(d): v =
20 V,i=4 A;
2. 80 W;
3. absorbing.
2. 1.6 The voltage and current at the terminals of the circuit element in Fig.
1.5 are zero for t<0. For t≥0, they are
v = 80,000t e −500t V, t≥0; i = 15t e −500t A, t≥0.
1. Find the time when the power delivered to the circuit element is
maximum.
2. Find the maximum value of power.
3. Find the total energy delivered to the circuit element.
Answer:
1. 2 ms;2. 649.6 mW;
3. 2.4 mJ.
3. 1.7 A high-voltage direct-current (dc) transmission line between Celilo,
Oregon, and Sylmar, California, is operating at 800 kV and carrying
1800 A, as shown. Calculate the power (in megawatts) at the Oregon
end of the line and state the direction of power flow.
Answer: 1440 MW, Celilo to Sylmar
SELF-CHECK: Also try Chapter Problems 1.15, 1.18, and 1.25.