In the circuit we have E=48V, R=6 ohm and L=4H. We close the switch at the instant t=0, the current I in the circuit grows from zero to its maximum value Io, and the permanent phase is established.
A) determine the value of Io.
B) calculate the values of the potential difference across R and across the coil at the instant t where I=5A.
C) calculate the potential difference across the coil for i=8A. Explain this result
I just need to know what formula i need to use
I assume in series
E = i R + L di/dt
let i = a (1- e^-kt)
then di/dt = a*k e^-kt
when t =oo , E=i R
i = E/R = a
so a = E/R
so
i = (E/R)(1-e^-kt)
di/dt = (E/R) k e^-kt
when t-->0, i = 0 and di/dt = a k
so
E = L di/dt = L a k = L(E/R)k
so
k = R/L
so
In the end all that we need is:
i = (E/R)(1 - e^-Rt/L)
di/dt = (E/R)(R/L)e^-Rt/L
A))Io is current when t-->oo
= E/R (1) = 48/6 = 8 amps
B) i = 8 (1-e^-1.5 t)
Vresistor = 6 i
you can put t = 5 in
Vcoil = 48 - Vresistor but check with
V = L di/dt :)
C) LOL, no di/dt at t=oo so no V across coil
The entire 48 volts is across resistor b then
To determine the values requested in this circuit, we can use the concepts of Kirchhoff's laws and the time-dependent behavior of inductors.
A) To calculate the maximum current Io, we need to find the steady-state solution for the circuit. This means that the transient behavior has settled, and the circuit is in a permanent phase.
The time constant of the circuit is given by the formula τ = L/R, where L is the inductance of the coil and R is the resistance. In this case, τ = 4H / 6Ω = 2/3 seconds.
The maximum current Io can be found using the formula Io = E/R, where E is the voltage source. In this case, Io = 48V / 6Ω = 8A.
B) To calculate the potential difference across R and across the coil at the instant t where I = 5A, we need to consider the behavior of the transient response.
The transient response of the current can be described by the equation I(t) = (Io - Ie^(-t/τ)), where I is the current at time t, Io is the maximum current, and τ is the time constant. Plugging in the values, we have I(t) = (8A - 5Ae^(-t/(2/3))).
At the instant when I = 5A, we can plug in this value on the equation to find the time t. Once we find t, we can substitute it back into the equation to calculate the potential difference across R using Ohm's law (V = IR) and across the coil using Faraday's law (V = L(dI/dt)).
C) To calculate the potential difference across the coil for I = 8A, we can use the same approach as in part B. We will plug in the value I = 8A into the equation I(t) = (8A - 5Ae^(-t/(2/3))), and solve for t. Once we have t, we can substitute it into Faraday's law (V = L(dI/dt)) to calculate the potential difference across the coil.
By following these steps, you will be able to solve parts A, B, and C of the problem.