Three impedances zL=3+j4 ohms, zC=4-j4 ohms and zR=0+j3 ohms are connected in parallel. solve for the power factor of the combination.
zR = 0+j3. Did you mean zR = 3+j0?. The j operator is used with inductors and capacitors to
indicate a 90o phase difference.
To find the power factor of the combination, we first need to find the total impedance (Z) of the parallel combination.
In a parallel connection, the reciprocal of the total impedance (Z) is equal to the sum of the reciprocals of the individual impedances (ZL, ZC, and ZR).
1/Z = 1/ZL + 1/ZC + 1/ZR
Taking the reciprocals of the given impedances:
ZL = 3 + j4
ZC = 4 - j4
ZR = 0 + j3
1/ZL = 1/(3 + j4)
1/ZC = 1/(4 - j4)
1/ZR = 1/(0 + j3)
Now, let's calculate the reciprocal of each impedance:
1/ZL = (3 - j4) / (3 + j4) = (3 - j4) / (3 + j4) * (conjugate of the denominator) / (conjugate of the denominator)
= (3 - j4)(3 - j4) / [(3 + j4)(3 - j4)]
= (3 - j4)(3 - j4) / [(3 + j4)(3 - j4)]
= (9 - 6j + 6j - 16) / (9 - 0)
= -7 / 9 - (12j / 9)
Similarly, we can calculate the reciprocals of ZC and ZR:
1/ZC = (4 + j4) / (4 - j4) = (16 + 4j - 4j - j^2) / (16 - j^2^)
= (16 - 1) / 20
= 15 / 20
= 3 / 4
1/ZR = (0 - j3) / (0 + j3) = -1
Now, let's substitute the values into the equation for the parallel combination:
1/Z = 1/ZL + 1/ZC + 1/ZR
= -7/9 - (12j / 9) + 3/4 - 1
= -7/9 + 3/4 - 1 - (12j / 9)
To find the power factor, we need to find the angle (φ) of the impedance.
The power factor (PF) is given by:
PF = cos(φ)
Since we have the complex impedance (Z), we need to calculate its phase angle (φ).
To find φ, we can take the inverse tangent (tan^(-1)) of the imaginary part divided by the real part of the impedance.
tan^(-1)(Imaginary part / Real part)
φ = tan^(-1)(-12 / -7)
= tan^(-1)(12 / 7)
Using a calculator, we find that φ ≈ 59.04 degrees (approximately).
Now, we can find the power factor (PF) by taking the cosine of the angle:
PF = cos(φ)
= cos(59.04 degrees)
≈ 0.530
Therefore, the power factor of the combination is approximately 0.530.
To solve for the power factor of the combination, we need to find the impedance of the parallel combination of zL, zC, and zR.
Impedances connected in parallel can be calculated using the formula:
1/z_parallel = 1/zL + 1/zC + 1/zR
Let's calculate it step by step:
1/zL = 1/(3+j4) = (3-j4)/(3^2+4^2) = (3-j4)/25
1/zC = 1/(4-j4) = (4+j4)/(4^2+4^2) = (4+j4)/32
1/zR = 1/(0+j3) = (0-j3)/(0^2+3^2) = (0-j3)/9
Now, let's add these expressions together:
1/z_parallel = (3-j4)/25 + (4+j4)/32 + (0-j3)/9
To add the fractions, we need to have a common denominator. The common denominator in this case is 25*32*9 = 7200.
Multiplying each fraction by the appropriate factor to get the common denominator, we have:
1/z_parallel = (3-j4)*(32*9) + (4+j4)*(25*9) + (0-j3)*(25*32)
= (864-1152j) + (900+900j) + (0-2400j)
= 864-1152j + 900+900j - 2400j
Simplifying, we get:
1/z_parallel = 1764 - 1652j
Taking the inverse of this expression will give us the impedance of the parallel combination, z_parallel:
z_parallel = 1 / (1764 - 1652j)
= (1764 + 1652j) / (1764^2 + 1652^2)
= (1764/5498404) + (1652/5498404)j
= 0.0003202 + 0.0003003j
Now, we can calculate the power factor of the combination:
Power factor = cos(theta) = Re(z_parallel) / |z_parallel|
The real part of z_parallel is 0.0003202, and the magnitude of z_parallel is |z_parallel| = sqrt(Re(z_parallel)^2 + Im(z_parallel)^2) = sqrt((0.0003202)^2 + (0.0003003)^2).
Plugging in the values, we get:
Power factor = 0.0003202 / sqrt((0.0003202)^2 + (0.0003003)^2)
Evaluating this expression will give us the power factor of the combination.