A series RCL circuit contains only a capacitor (C=6.60 �F),
an inductor (L=7.20 mH), and a generator (peak voltage=32.0 V,
frequency=1.50x10^3 Hz). When t � 0 s, the instantaneous value of
the voltage is zero, and it rises to a maximum one-quarter of a period
later. (a) Find the instantaneous value of the voltage across the capacitor/
inductor combination when t=1.20x10^-�4 s. (b) What is the
instantaneous value of the current when t=1.20x10^-�4 s? (Hint:
The instantaneous values of the voltage and current are, respectively, the
vertical components of the voltage and current phasors.)
To find the instantaneous value of the voltage across the capacitor/inductor combination and the current in the circuit at t=1.20x10^-4 s, we need to follow these steps:
Step 1: Find the angular frequency (ω) of the circuit using the given frequency (f).
ω = 2πf
ω = 2π(1.50x10^3 Hz)
ω ≈ 9.42x10^3 rad/s
Step 2: Calculate the impedance (Z) of the circuit using the values of the capacitor (C), inductor (L), and angular frequency (ω).
Z = sqrt((XL - XC)^2)
where XL and XC denote the inductive reactance and capacitive reactance, respectively.
XL = ωL
XL = (9.42x10^3 rad/s)(7.20x10^-3 H)
XL ≈ 67.78 Ω
XC = 1/(ωC)
XC = 1/(9.42x10^3 rad/s)(6.60x10^-6 F)
XC ≈ 1.79x10^3 Ω
Z = sqrt((67.78 Ω - 1.79x10^3 Ω)^2)
Z ≈ 1.79x10^3 Ω
Step 3: Calculate the magnitude (I0) and phase angle (ϕ) of the current in the circuit using Ohm's Law and the values of the peak voltage (V0) and impedance (Z).
I0 = V0 / Z
I0 = (32.0 V) / (1.79x10^3 Ω)
I0 ≈ 0.0178 A
ϕ = arctan((XL - XC) / R)
Since there is no resistance given in the problem statement, we assume the circuit has no resistance (R = 0).
ϕ = arctan((XL - XC) / 0)
ϕ = arctan((67.78 Ω - 1.79x10^3 Ω) / 0)
ϕ ≈ -π/2 (90 degrees)
Step 4: Express the voltage and current in the circuit as phasors.
The voltage phasor (Vp) is given by Vp = V0*exp(jϕ)
where j is the imaginary unit (√(-1)). In this case, the phase angle is already in radians.
The current phasor (Ip) is given by Ip = I0*exp(j(ωt + ϕ))
Step 5: Find the instantaneous values of voltage and current at t = 1.20x10^-4 s.
To find the instantaneous value, we need to substitute the given time value (t) into the phasor expressions.
V(t) = Re(Vp*exp(jωt))
where Re() denotes the real part of a complex number.
V(t) = Re(V0*exp(jϕ)*exp(jωt))
V(t) = Re(V0*exp(j(ωt + ϕ)))
V(t) = Re(32.0 V * exp(j(9.42x10^3 rad/s * 1.20x10^-4 s - π/2)))
Similarly, for the current:
I(t) = Re(Ip*exp(jωt))
I(t) = Re(I0*exp(j(ωt + ϕ)))
I(t) = Re(0.0178 A * exp(j(9.42x10^3 rad/s * 1.20x10^-4 s - π/2)))
To calculate the values of V(t) and I(t), you can use a scientific calculator or computer software with complex number support.