Find the equivalent impedance Zeq seen by the source when Vs = 4 cos (4t) v, C = 0.5 F, R = 1 Ω and L = 0.5 H. (Give angles in degrees and round final answers to two decimal places.) Calculate the voltage across the capacitor. (Give angles in degrees and round final answers to two decimal places.)

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

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