An ac source of 50v,100÷πHz is connected in
series with an inductor of inductance L ad a
resistor of resistance R.the current in the circuit
is 2A ad p.d. across L and R are 30v ad 40v
respectively
1) calculate the average power dissipated ?
(2) calculate d power factor in the circuit?
average power: VI=40*2 watts
power factor
Vl and Vr are 90 degrees out of phase, so power factor=40/50=.8
Power factor is real power dissipated ratio to rms current times voltage of the circuit.
well, maybe V = 50 sin (2pift)
and i = I sin (2pift - phi)
then di/dt = 2pif I cos (2pift-phi)
with your numbers
2 pi f = 200
V = 50 sin 200 t
i = 20 sin (200 t-phi)
di/dt = 4000cos (200 t - phi)
now
VL = L di/dt
= L*4000 cos(200t-phi)
so 4000 L = 30
L = 3/400
Vr = iR =R*20 sin(200t-phi)
so 20 R = 40
R = 2 ohms
V = VL + VR
50 sin 200t = 30cos(200t-phi)+40sin(200t-phi)
30 cos(200t-phi) = 30cos200tcosphi+30sin200tsinphi
40sin(200t-phi) = 40sin200tcosphi-40cos200tsinphi
so
50 = 30 sin phi + 40 cos phi
and
00 = 30 cos phi - 40 sin phi
tan phi = .75
phi = 36.9 deg
multiply by pi/180 to get radians
sure enough 30*sin36.9+40*cos36.9 = 50 :)
so now I can do anything
V = 50 sin (200 t)
i = 20 sin (200 t-phi)
average power = average of V*i
power dissipated in R is
average of Vr*i
which is i^2 R
= 2^2 amp^2 * 2 ohms /2
we divide by 2 because average value of sin^2 or cos^2 is 1/2
= 4 watts
To calculate the average power dissipated in the circuit, you can use the formula:
Average Power (P) = (Current)^2 x Resistance
Given that the current in the circuit is 2A and the resistance is not directly mentioned, we need to find the resistance in order to proceed.
To find the resistance, we can use Ohm's law:
Resistance (R) = Voltage (V) / Current (I)
The voltage across the resistor (V_R) is given as 40V. So, the resistance can be calculated as:
R = V_R / I
R = 40V / 2A
R = 20Ω
Now that we have the resistance (R) as 20Ω, we can calculate the average power dissipated:
P = (I)^2 x R
P = (2A)^2 x 20Ω
P = 4A^2 x 20Ω
P = 80W
Therefore, the average power dissipated in the circuit is 80 Watts.
Moving on to the calculation of the power factor:
Power factor (PF) can be calculated using the formula:
Power factor (PF) = Cosine (θ)
where θ is the angle between the voltage and current waveforms.
Since only the frequency of the AC source is given (100÷πHz), we are missing the phase angle information. Without the phase angle, we cannot directly calculate the power factor.
In order to find the phase angle, we need more information, such as the inductive reactance (X_L) of the inductor. The given data does not provide this information, so we cannot find the power factor without it.
Hence, we cannot calculate the power factor in this circuit with the provided information.