Pregunta
If two electrical potential differences (measured in volts) are given by
50 cis -330 and 70 cis -40 , calculate the resulting potential difference
50 cis -330 and 70 cis -40 , calculate the resulting potential difference
Ask by Savage Clark. in South Africa
Feb 14,2025
Solución de inteligencia artificial de Upstudy
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The resulting potential difference is approximately 99 volts at an angle of –11.7 degrees.
Solución
We are given two phasors in the form “r cis θ”, where cis θ means cos θ + i sin θ. The two phasors are:
50 cis (–330°) and 70 cis (–40°)
To add these, it is easiest to convert them to rectangular form, add the real and imaginary parts separately, and then convert the result back to polar form.
Step 1. Convert each phasor to rectangular form.
For the first phasor, 50 cis (–330°):
Real part = 50 cos(–330°)
Imaginary part = 50 sin(–330°)
Imaginary part = 50 sin(–330°)
Note that cos(–330°) = cos(330°) and sin(–330°) = –sin(330°). Also, 330° = 360° – 30°, so cos330° = cos30° ≈ 0.8660 and sin330° = –sin30° = –0.5. Thus:
Real part ≈ 50 × 0.8660 = 43.30
Imaginary part ≈ 50 × (–(–0.5)) = 50 × 0.5 = 25.00
Imaginary part ≈ 50 × (–(–0.5)) = 50 × 0.5 = 25.00
So, 50 cis(–330°) ≈ 43.30 + j 25.00.
For the second phasor, 70 cis (–40°):
Real part = 70 cos(–40°) = 70 cos(40°)
Imaginary part = 70 sin(–40°) = –70 sin(40°)
Imaginary part = 70 sin(–40°) = –70 sin(40°)
Using cos 40° ≈ 0.7660 and sin 40° ≈ 0.6428:
Real part ≈ 70 × 0.7660 = 53.62
Imaginary part ≈ –70 × 0.6428 = –45.00
Imaginary part ≈ –70 × 0.6428 = –45.00
Thus, 70 cis(–40°) ≈ 53.62 – j 45.00.
Step 2. Add the rectangular forms.
Add the real parts:
Total real = 43.30 + 53.62 ≈ 96.92
Total real = 43.30 + 53.62 ≈ 96.92
Add the imaginary parts:
Total imaginary = 25.00 + (–45.00) = –20.00
Total imaginary = 25.00 + (–45.00) = –20.00
So the sum in rectangular form is:
96.92 – j 20.00
96.92 – j 20.00
Step 3. Convert the sum back to polar form.
Magnitude ®:
r = √(96.92² + (–20.00)²)
= √(9393.2 + 400)
= √(9793.2)
≈ 98.96
r = √(96.92² + (–20.00)²)
= √(9393.2 + 400)
= √(9793.2)
≈ 98.96
Angle (θ):
θ = arctan(imaginary/real) = arctan(–20.00/96.92)
≈ arctan(–0.2063)
≈ –11.7°
θ = arctan(imaginary/real) = arctan(–20.00/96.92)
≈ arctan(–0.2063)
≈ –11.7°
Thus, the resulting potential difference is approximately:
98.96 cis (–11.7°)
98.96 cis (–11.7°)
You can round this as needed (about 99 V ∠ –11.7°).
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Simplifique esta solución Beyond the Answer
To find the resulting potential difference, we can add the two given complex numbers that represent electrical potential differences. The first number is
Now, we add them together:
Real part:
Imaginary part:
Imaginary part:
Calculating these, we find the real part is approximately
Therefore, the resulting potential difference in rectangular form is approximately
To convert back to polar form, we find the magnitude:
And the angle (using tan inverse):
Thus, the resulting potential difference is approximately
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