Question

Given the below input signal, V????????(t), and the phasors of the transfer function at various frequencies, calculate the expression for the output signal, VO????T(t) in the form VO????T(t)=Acos(200πt+θ1)+Bcos(400πt+θ2). What is the value of A? V????????(t)=10cos(200πt+30∘)+20cos(400πt+45∘)

Answer 1

The question is incomplete! Complete question along with answer and step by step explanation is provided below.

Question:

1. Given the below input signal, VIN(t), and the phasors of the transfer function at various frequencies, calculate the expression for the output signal, VOUT(t) in the form VOUT(t)=Acos(200πt+θ1)+Bcos(400πt+θ2).

What is the value of A?

VIN(t)=10cos(200πt+30)+20cos(400πt+45)

H(f)=0.1∠10 f=100Hz

H(f)=1∠−20 f=200Hz

H(f)=2∠30 f=300Hz

H(f)=3∠−40 f=400Hz

H(f)=4∠50 f=500Hz

2. In the same question, what is the value of θ1 in degrees? Enter the value in the box below without the units.

3. In the same question, what is the value of B? Enter the value in the box below without the units.

4. In the same question, what is the value of θ2 in degrees? Enter the value in the box below without the units.

Answer:

Vout(t) = 10cos(200πt + 10°) + 60cos(400πt + 5°)

1. What is the value of A?

A = 10

2. In the same question, what is the value of θ1 in degrees?

θ₁ = 10°

3. In the same question, what is the value of B?

B = 60

4. In the same question, what is the value of θ2 in degrees?

θ₂ = 5°

Step-by-step explanation:

The output signal Vout(t) is in the form

Vout(t)=Acos(200πt + θ₁) + Bcos(400πt + θ₂)

The input signal Vin(t) is given as

Vin(t) = 10cos(200πt + 30°) + 20cos(400πt + 45°)

The output signal Vout(t) is found by

Vout(t) = H(f) × Vin(t)

Where H(f) is the transfer function at various frequencies and Vin(t) is the input signal.

H(200) = 1 ∠−20°

H(400) = 3 ∠−40°

converting the input signal into phasors

10cos(200πt + 30°) = 10 < 30°

20cos(400πt + 45°) = 20 < 45°

Vout(t) = H(200)×10cos(200πt + 30°) + H(400)×20cos(400πt + 45°)

Vout(t) = (1 ∠−20°)×10cos(200πt + 30°) + (3 ∠−40°)×20cos(400πt + 45°)

Vout(t) = (1 ∠−20°)×(10 < 30°) + (3 ∠−40°)×(20 < 45°)

Multipy the magnitude and add phase angles together

Vout(t) = (1×10 ∠−20° + 30°) + (3×20 ∠−40° + 45°)

Vout(t) = 10 ∠ 10° + 60 ∠5°

Vout(t) = 10cos(200πt + 10°) + 60cos(400πt + 5°)

Comparing it with the general form

Vout(t)=Acos(200πt + θ₁) + Bcos(400πt + θ₂)

1. What is the value of A?

A = 10

2. In the same question, what is the value of θ1 in degrees?

θ₁ = 10°

3. In the same question, what is the value of B?

B = 60

4. In the same question, what is the value of θ2 in degrees?

θ₂ = 5°