A series RLC circuit is made with a C=100pF capacitor, an L=100μH inductor, and an R=30Ω resistor. The circuit is driven at 5×106rad/s with a 1V amplitude. What is the current oscillation amplitude in amps
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/rlcser.html
z = 5 * 10^5 ohms
i = 1/(5*`10^6) = .2 *10^-6 = 2 *10^-7
5x10^6 rad/s = 795774 Hz
Z = √(R^2 + (Xl-Xc)^2
Xc = 1/ωC = 1/(5*10^6 * 100*10^-12) = 1/(500*10^-6) = 2000
Xl = ωL = 5*10^6 * 100*10^-6 = 500
So, now you know R and Z. Go for it, and recall that I = E/Z
To find the current oscillation amplitude in a series RLC circuit, we need to calculate the impedance (Z) of the circuit at the given frequency and then use it to find the current amplitude.
1. Impedance Calculation:
The impedance of a series RLC circuit can be calculated using the following formula:
Z = √(R^2 + (Xl - Xc)^2)
Where:
R = resistance in ohms
Xl = inductive reactance in ohms
Xc = capacitive reactance in ohms
Resistance (R) = 30Ω (given)
Inductive Reactance (Xl) = 2πfL, where f is the frequency and L is the inductance
Capacitive Reactance (Xc) = 1 / (2πfC), where C is the capacitance
Given:
f = 5×10^6 rad/s
L = 100μH = 100×10^(-6) H
C = 100pF = 100×10^(-12) F
First, let's calculate the value of each reactance:
Xl = 2π × f × L
= 2π × (5×10^6) × (100×10^(-6))
= 10π × 10^(6-6)
= 10π ohms
Xc = 1 / (2π × f × C)
= 1 / (2π × (5×10^6) × (100×10^(-12)))
= 1 / (10π × 10^(-6))
= 0.1π × 10^6
= 0.1 × 3.14 × 10^6
= 314,000 ohms
Now, we can calculate the impedance (Z):
Z = √(R^2 + (Xl - Xc)^2)
= √(30^2 + (10π - 314,000)^2)
≈ √(900 + (10π - 314,000)^2)
≈ √(900 + 9π^2 - 628π + 314,000^2)
≈ √(900 + 9π^2 - 628π + 98,596,000)
≈ √(9π^2 - 628π + 98,596,900)
≈ √(9 × 9.87 - 628 × 3.14 + 98,596,900)
≈ √(88.83 - 1973.52 + 98,596,900)
≈ √(98,594.428)
≈ 314.02 ohms (approximately)
2. Current Amplitude Calculation:
The current amplitude (I0) in the circuit can be calculated using Ohm's Law: I0 = V0 / Z
Where:
V0 = voltage amplitude in volts
Z = impedance in ohms (calculated above)
Given:
V0 = 1V (given)
I0 = V0 / Z
= 1 / 314.02
≈ 0.0032 A (approximately)
Therefore, the current oscillation amplitude in amps is approximately 0.0032 A.