Question

The voltage across and current through a certain device are given by v(t)=100(1−exp−0.2t)V; i(t)=30exp−0.2tmA

a) Find the instantaneous power p(t) at t=0 and t=3s
b) Find the cumulative energy delivered to the device from t=0 to t=00

Answer 1

a) The instantaneous power at t = 0 is 3W and at t = 3s is 0.562W. b) The cumulative energy delivered to the device from t = 0 to t = ∞ cannot be determined, but the energy delivered from t = 0 to t = 3s can be found by evaluating an integral.

Step-by-step explanation:

a) To find the instantaneous power at t = 0, we substitute t = 0 into the given equations for voltage and current:

v(0) = 100(1 − exp⁡(−0.2(0))) = 100V

i(0) = 30exp⁡(−0.2(0)) = 30mA

Using the formula for power, P = IV, the instantaneous power at t = 0 is:

p(0) = v(0) * i(0) = 100V * 30mA = 3W

To find the instantaneous power at t = 3s, we substitute t = 3 into the given equations for voltage and current:

v(3) = 100(1 − exp⁡(−0.2(3))) = 75.26V

i(3) = 30exp⁡(−0.2(3)) = 7.47mA

Using the formula for power, P = IV, the instantaneous power at t = 3s is:

p(3) = v(3) * i(3) = 75.26V * 7.47mA = 0.562W

b) The cumulative energy delivered to the device from t = 0 to t = ∞ can be found using the formula for energy, E = ∫p(t)dt. Integrating the expression for power, p(t) = v(t) * i(t), over the interval [0, ∞]:

E = ∫[0,∞] v(t) * i(t) dt

However, since the upper limit for integration is infinity, we cannot compute the exact cumulative energy delivered to the device. We can only determine the energy delivered over a finite interval. To find the energy delivered from t = 0 to t = 3s, we evaluate the integral:

E = ∫[0,3] v(t) * i(t) dt