Answer:
Zeq = 1.80ohms
Vc = 1.11cos4t volts
Explanation:
Impedance of an AC circuit is defined as the opposition to the flow of current across the resistor, capacitor and inductor of an AC circuit. It is expressed mathematically as;
Zeq = √R²+(Xl-Xc)²
Where R is the resistance
Xl is the inductive reactance
Xc is the capacitive reactance
Xl = 2πfL
Source voltage Vs = Vcos2πft
Given Vs = 4 cos (4t)v
Comparing both source voltage to get the frequency f, we have;
2πft = 4t
2πf = 4...(1)
Given L = 0.5H
C = 0.5F
R = 1.0ohms
Xl = 4×0.5 (note that 2πf = 4 from equation 1)
Xl = 2.0ohms
Xc = 1/2πfC
Xc = 1/4(0.5)
Xc = 1/2
Xc = 0.5ohms
To get the impedance Z
Zeq = √1²+(2-0.5)²
Zeq = √1+1.5²
Zeq = √3.25
Zeq = 1.80ohms
The voltage across the capacitor is expressed as Vc = IXc
Since we know Xc = 0.5ohms
We need current I
To get current we use the relationship Vs = IZ
I = Vs/Z =
4 cos (4t)/1.8
I = 2.22cos4t A
Vc = 2.22cos4t × 0.5
Vc = 1.11cos4t volts
