Question

A sinusoidal voltage Δv = 35.0 sin 100t, where Δv is in volts and t is in seconds, is applied to a series RLC circuit with L = 165 mH, C = 99.0 µF, and R = 66.0 Ω. (a) What is the impedance of the circuit? Ω (b) What is the maximum current? A (c) Determine the numerical value for ω in the equation i = Imax sin (ωt − ϕ). rad/s (d) Determine the numerical value for ϕ in the equation i = Imax sin (ωt − ϕ).

Answer 1

(a) 107.2 Ω

The voltage in the circuit is written in the form

$V=V_0 sin(\omega t)$

where in this case

V0 = 35.0 V is the maximum voltage

$\omega=100 rad/s$

is the angular frequency

Given the inductance: L = 165 mH = 16.5 H, the reactance of the inductance is:

$X_L = \omega L=(100)(0.165)=16.5 \Omega$

Given the capacitance: $C=99\mu F=99\cdot 10^{-6} F$, the reactance of the capacitor is:

$X_C = \frac{1}{\omega C}=\frac{1}{(100)(99\cdot 10^{-6})}=101.0 \Omega$

And given the resistance in the circuit, R = 66.0 Ω, we can now find the impedance of the circuit:

$Z=\sqrt{R^2+(X_L-X_C)^2}=\sqrt{66^2+(16.5-101.0)^2}=107.2\Omega$

(b) 0.33 A

The maximum current can be calculated by using Ohm's Law for RLC circuit. In fact, we know the maximum voltage:

V0 = 35.0 V

The equivalent of Ohm's law for an RLC circuit is

$I=\frac{V}{Z}$

where

Z=107.2 Ω

is the impedance. Substituting these values into the formula, we find:

$I=\frac{35.0}{107.2}=0.33 A$

(c) 100 rad/s

The equation for the current is

$I=I_0 sin(\omega t-\Phi)$

where

I0 = 0.33 A is the maximum current, which we have calculated previously

$\omega$ is the angular frequency

$\Phi$ is the phase shift

In an RLC circuit, the voltage and the current have the same frequency. Therefore, we can say that

$\omega=100 rad/s$

also for the current.

(d) -0.907 rad

Here we want to calculate the phase shift $\Phi$, which represents the phase shift of the current with respect to the voltage. This can be calculated by using the equation:

$\Phi = tan^{-1}(\frac{X_L-X_C}{R})$

Substituting the values that we found in part a), we get

$\Phi = tan^{-1}(\frac{16.5-101.0}{66})=-52^{\circ}$

And the sign is negative, since the capacitive reactance is larger than the inductive reactance in this case. Converting in radians,

$\theta=-52 \cdot \frac{2\pi}{360}=-0.907 rad$

So the complete equation of the current would be

$I=0.33sin(100t+0.907)$

Answer 2

The impedance of the circuit is 107.2 Ω.

The maximum current is 0.33 A.

The numerical value for ω in the equation i = Imax sin (ωt − ϕ). rad/s is 100 rad/s.

The numerical value for ϕ in the equation i = Imax sin (ωt − ϕ) is -0.907 rad.

Calculations and Parameters:

1. To find the voltage, we already know that V0=35V

And the voltage in the circuit is V= V0sin(wt)

Hence, the angular frequency, w= 100 rad/s

Given the inductance is 16.5Ω and the capacitance is 101Ω.

And the resistance is 66Ω

Hence, the impedance of the circuit is $\sqrt{66^{2} + (16.5-101)^2} =107.2$Ω

2. To find the maximum current, since Z= 107.2Ω,

Recall, I = V/Z

Put the values into the formula

I = 0.33A.

3. To find the numerical value for ω in the equation i = Imax sin (ωt − ϕ)
We recall that I0 = 0.33A

In an RLC circuit, the voltage and the current have the same frequency. Therefore, we can say that

w= 100 rad/s.

4. To find the numerical value for ϕ in the equation i = Imax sin (ωt − ϕ)

Because the sign is in the negative and since the capacitive reactance is larger than the inductive reactance in this case.

Converting in radians,

θ = -52. 2π/360

= -0.907rad.

Hence, the complete equation would be:

I = 0.33sin(100t + 0.907)

Read more about impedance here:

https://brainly.com/question/13134405