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blagie [28]
3 years ago
13

A series R-L circuit is given. Circuit is connected to an AC voltage generator. a) Derive equations for magnitude and phase of c

urrent and voltages on resistor and inductor in the phasor domain. Assume that the resistance of the resistor is R, inductance of the inductor is L, magnitude of the source voltage is Vm and phase of the source voltage is θ. Note that you don’t have numbers in this step, so to find the magnitude and phase for current I and voltages VR and VL you must first derive both numerator and denominator in polar form using variables R, omega, L, Vm, Vphase (do not use numbers). The solutions should look like equations in slide 24/27! b) In this step, assume that R
Engineering
1 answer:
igomit [66]3 years ago
6 0

Answer:

The equations for magnitude and phase of current and voltages on resistor and inductor are:

I=\frac{V_m}{\sqrt{R^2+(\omega L)^2}}\angle \theta - tan^{-1}(\omega L / R)

V_R=I\cdot Z_R=\frac{V_m \cdot R}{\sqrt{R^2+(\omega L)^2}}\angle \theta - tan^{-1}(\omega L / R)

V_L=I\cdot Z_L=\frac{V_m \cdot (\omega L)}{\sqrt{R^2+(\omega L)^2}}\angle \theta - tan^{-1}(\omega L / R)+90^{\circ}

Explanation:

The first step is to find the impedances of the resistance (R) and the inductor (L).

The impedance of the resistor is:

  • Rectangular form: Z_R=R
  • Polar form: Z_R=R\angle 0^{\circ}

The impedance of the inductor is:

  • Rectangular form: Z_L=j\omega L
  • Polar form: Z_L=\omega L \angle 90^{\circ}

Where \omega is the angular frequency of the source, and the angle is 90^{\circ} because a pure imaginary number is on the imaginary axis (y-axis).

The next step is to find the current expression. It is the same for the resistor and inductor because they are in series. The total impedance equals the sum of each one.

I=\frac{V}{Z_R+Z_L}

It is said that V=V_m\angle \theta, so, the current would be:

I=\frac{V_m\angle \theta }{R+j\omega L}

The numerator must be converted to polar form by calculating the magnitude and the angle:

  • The magnitude is \sqrt{R^2+(\omega L)^2}
  • The angle is tan^{-1}(\omega L / R)

The current expression would be as follows:

I=\frac{V_m\angle \theta }{\sqrt{R^2+(\omega L)^2}\, \angle tan^{-1}(\omega L / R)}

When dividing, the angles are subtracted from each other.

The final current expression is:

I=\frac{V_m}{\sqrt{R^2+(\omega L)^2}}\angle \theta - tan^{-1}(\omega L / R)

The last step is calculating the voltage on the resistor V_R and the voltage on the inductor V_L. In this step the polar form of the impedances could be used. Remember that V=I\cdot Z.

(Also remember that when multiplying, the angles are added from each other)

Voltage on the resistor V_R

V_R=I\cdot Z_R=\bigg( \frac{V_m}{\sqrt{R^2+(\omega L)^2}}\angle \theta - tan^{-1}(\omega L / R)\bigg) \cdot (R\angle 0^{\circ})

The final resistor voltage expression is:

V_R=I\cdot Z_R=\frac{V_m \cdot R}{\sqrt{R^2+(\omega L)^2}}\angle \theta - tan^{-1}(\omega L / R)

Voltage on the inductor V_L

V_L=I\cdot Z_L=\bigg( \frac{V_m}{\sqrt{R^2+(\omega L)^2}}\angle \theta - tan^{-1}(\omega L / R)\bigg) \cdot (\omega L \angle 90^{\circ})

The final inductor voltage expression is:

V_L=I\cdot Z_L=\frac{V_m \cdot (\omega L)}{\sqrt{R^2+(\omega L)^2}}\angle \theta - tan^{-1}(\omega L / R)+90^{\circ}

Summary: the final equations for magnitude and phase of current and voltages on resistor and inductor are:

I=\frac{V_m}{\sqrt{R^2+(\omega L)^2}}\angle \theta - tan^{-1}(\omega L / R)

V_R=I\cdot Z_R=\frac{V_m \cdot R}{\sqrt{R^2+(\omega L)^2}}\angle \theta - tan^{-1}(\omega L / R)

V_L=I\cdot Z_L=\frac{V_m \cdot (\omega L)}{\sqrt{R^2+(\omega L)^2}}\angle \theta - tan^{-1}(\omega L / R)+90^{\circ}

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Ad libitum [116K]
Don’t go on that file will give a virus! Sorry just looking out and I don’t know how to comment!
7 0
3 years ago
A rich industrialist was found murdered in his house. The police arrived at the scene at 11:00 PM. The temperature of the corpse
d1i1m1o1n [39]

Answer:

The dude was killed around 6:30PM

Explanation:

Newton's law of cooling states:

    T = T_m + (T_0-T_m)e^{kt}

where,

T_0 = initial temp

T_m = temp of room

T = temp after t hours

k = how fast the temp is changing

t = time (hours)

T_0 = 31     because the body was initlally 31ºC when the police found it

T_m = 22   because that was the room temp

T = 30  because the body temp drop to 30ºC after 1 hour

t = 1 because that's the time it took for the body temp to drop to 30ºC

k=???   we don't know k so we must solve for this

rearrange the equation to solve for k

T = T_m + (T_0-T_m)e^{kt}

T - T_m= (T_0-T_m)e^{kt}

\frac{T - T_m}{(T_0-T_m)}= e^{kt}

ln(\frac{T - T_m}{T_0-T_m})=kt

\frac{ln(\frac{T - T_m}{T_0-T_m})}{t}=k

plug in the numbers to solve for k

k = \frac{ln(\frac{T - T_m}{T_0-T_m})}{t}

k = \frac{ln(\frac{30 - 22}{31-22})}{1}

k=ln(\frac{8}{9})

Now that we know the value for k, we can find the moment the murder occur. A crucial information that the question left out is the temperature of a human body when they're still alive. A living human body is about 37ºC. We can use that as out initial temperature to solve this problem because we can assume that the freshly killed body will be around 37ºC.

T_0 = 37     because the body was 37ºC right after being killed

T_m = 22   because that was the room temp

T = 31  because the body temp when the police found it

k=ln(\frac{8}{9})   we solved this earlier

t = ???   we don't know how long it took from the time of the murder to when the police found the body

Rearrange the equation to solve for t

T = T_m + (T_0-T_m)e^{kt}

T - T_m= (T_0-T_m)e^{kt}

\frac{T - T_m}{(T_0-T_m)}= e^{kt}

ln(\frac{T - T_m}{T_0-T_m})=kt

\frac{ln(\frac{T - T_m}{T_0-T_m})}{k}=t

plug in the values

t=\frac{ln(\frac{T - T_m}{T_0-T_m})}{k}

t=\frac{ln(\frac{31 - 22}{37-22})}{ln(8/9)}

t=\frac{ln(3/5)}{ln(8/9)}

t=\frac{ln(3/5)}{ln(8/9)}

t ≈ 4.337 hours from the time the body was killed to when the police found it.

The police found the body at 11:00PM so subtract 4.337 from that.

11 - 4.33 = 6.66 ≈ 6:30PM

7 0
3 years ago
Define the word vision statement​
Elena L [17]

Answer:

A vision statement describes what a company desires to achieve in the long-run, generally in a time frame of five to ten years, or sometimes even longer. It depicts a vision of what the company will look like in the future and sets a defined direction for the planning and execution of corporate-level strategies.

Explanation:

While companies should not be too ambitious in defining their long-term goals, it is critical to set a bigger and further target in a vision statement that communicates a company’s aspirations and motivates the audience. Below are the main elements of an effective vision statement:

-Forward-looking

-Motivating and inspirational

-Reflective of a company’s culture and core values

-Aimed at bringing benefits and improvements to the organization in the future

-Defines a company’s reason for existence and where it is heading

7 0
3 years ago
A tubular reactor has been sized to obtain 98% conversion and to process 0.03 m^3/s. The reaction is a first-order irreversible
Zolol [24]

Answer:

The conversion in the real reactor is = 88%

Explanation:

conversion = 98% = 0.98

process rate = 0.03 m^3/s

length of reactor = 3 m

cross sectional area of reactor = 25 dm^2

pulse tracer test results on the reactor :

mean residence time ( tm) = 10 s and variance (∝2) = 65 s^2

note:  space time (t) =

t = \frac{A*L}{Vo}   Vo = flow metric flow rate , L = length of reactor , A = cross sectional area of the reactor

therefore (t) = \frac{25*3*10^{-2} }{0.03} = 25 s

since the reaction is in first order

X = 1 - e^{-kt}

e^{-kt} = 1 - X

kt = In \frac{1}{1-X}

k = In \frac{1}{1-X} / t  

X = 98% = 0.98 (conversion in PFR ) insert the value into the above equation then  

K = 0.156 s^{-1}

Calculating Da for a closed vessel

; Da = tk

      = 25 * 0.156 = 3.9

calculate Peclet number Per using this equation

0.65 = \frac{2}{Per} - \frac{2}{Per^2} ( 1 - e^{-per})

therefore

\frac{2}{Per} - \frac{2}{Per^2} (1 - e^{-per}) - 0.65 = 0

solving the Non-linear equation above( Per = 1.5 )

Attached is the Remaining part of the solution

3 0
3 years ago
Which of the following circumstances call for a greater than normal following distance?
mariarad [96]

Explanation:

The three-second rule is recommended for passenger vehicles during ideal road and weather conditions. Slow down and increase your following distance even more during adverse weather conditions or when visibility is reduced. Also increase your following distance if you are driving a larger vehicle or towing a trailer.

7 0
2 years ago
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