'a) Bus 1 is connected to bus 2 by a short (less than 50 miles) and lossless transmission line modeled by a complex series impedance Z = jX, where X is the line reactance. (The real transmission line has 3 phases a,b,c, but in the balanced situation of this question, we decouple the phases and analyze the single phase a). The phasor voltage at bus 1 is Vị = |Vi|e3®i_and the phasor voltage at bus 2 is V2 = |V2|e3®2. We already know the formula for the real power sent by bus 1 |Vi||V2 , P12 sin 0 X (1) Take |V1| = |V2| = 1.0 per unit. Take X = 0.5 per unit. Sketch a graph of P12 versus 0. What is the real power flow P12 on the line in per unit when 0 = 10°? (b) [so called “DC power flow" approximation] Take the same transmission line as in part (a) and make the approximations |V1| = |V2| = 1 and 0 is small so that sin 0 = 0. Let B = 1/X. Use the approximations to obtain the formula for the real power P12 in terms of B %3D and 0. (c) Use the formula in (b) to compute the power flow P12 when 0 = 10°. Compare your answer to the answer in part (a).
'a) Bus 1 is connected to bus 2 by a short (less than 50 miles) and lossless transmission line modeled by a complex series impedance Z = jX, where X is the line reactance. (The real transmission line has 3 phases a,b,c, but in the balanced situation of this question, we decouple the phases and analyze the single phase a). The phasor voltage at bus 1 is Vị = |Vi|e3®i_and the phasor voltage at bus 2 is V2 = |V2|e3®2. We already know the formula for the real power sent by bus 1 |Vi||V2 , P12 sin 0 X (1) Take |V1| = |V2| = 1.0 per unit. Take X = 0.5 per unit. Sketch a graph of P12 versus 0. What is the real power flow P12 on the line in per unit when 0 = 10°? (b) [so called “DC power flow" approximation] Take the same transmission line as in part (a) and make the approximations |V1| = |V2| = 1 and 0 is small so that sin 0 = 0. Let B = 1/X. Use the approximations to obtain the formula for the real power P12 in terms of B %3D and 0. (c) Use the formula in (b) to compute the power flow P12 when 0 = 10°. Compare your answer to the answer in part (a).
Introductory Circuit Analysis (13th Edition)
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Having some trouble understanding a-c
![**Problem Description:**
Bus 1 is connected to bus 2 by a short (less than 50 miles) and lossless transmission line, modeled by a complex series impedance \( Z = jX \), where \( X \) is the line reactance. For the purposes of this question, the phases are decoupled and only a single phase is analyzed.
- The phasor voltage at bus 1 is \( V_1 = |V_1|e^{j\theta_1} \).
- The phasor voltage at bus 2 is \( V_2 = |V_2|e^{j\theta_2} \).
The formula for the real power \( P_{12} \) sent by bus 1 is given by:
\[
P_{12} = \frac{|V_1||V_2|}{X} \sin \theta
\]
**Task:**
1. **Given Values:**
- Assume \(|V_1| = |V_2| = 1.0\) per unit.
- Assume \( X = 0.5 \) per unit.
2. **Graphical Representation:**
- Graph \( P_{12} \) versus \( \theta \).
3. **Real Power Flow Calculation:**
- Compute the real power flow \( P_{12} \) when \( \theta = 10^\circ \).
4. **DC Power Flow Approximation:**
- Use \(|V_1| = |V_2| = 1\) and small \( \theta \) such that \( \sin \theta = \theta \).
- Let \( B = 1/X \).
5. **Formula Derivation and Application:**
- Derive the formula for real power \( P_{12} \) using the approximation and express it in terms of \( B \) and \( \theta \).
- Compare the results with part (a).
**Steps:**
- Use approximations to derive a formula for the real power \( P_{12} \) and evaluate the real power flow at \( \theta = 10^\circ \).
- Compare computed values from the approximation with those from the actual setup.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Feb010926-0d81-478d-9736-679f94816a5d%2F1fd71e4c-8a2c-4ad2-8409-db2cae15297a%2Fujq7o5_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Description:**
Bus 1 is connected to bus 2 by a short (less than 50 miles) and lossless transmission line, modeled by a complex series impedance \( Z = jX \), where \( X \) is the line reactance. For the purposes of this question, the phases are decoupled and only a single phase is analyzed.
- The phasor voltage at bus 1 is \( V_1 = |V_1|e^{j\theta_1} \).
- The phasor voltage at bus 2 is \( V_2 = |V_2|e^{j\theta_2} \).
The formula for the real power \( P_{12} \) sent by bus 1 is given by:
\[
P_{12} = \frac{|V_1||V_2|}{X} \sin \theta
\]
**Task:**
1. **Given Values:**
- Assume \(|V_1| = |V_2| = 1.0\) per unit.
- Assume \( X = 0.5 \) per unit.
2. **Graphical Representation:**
- Graph \( P_{12} \) versus \( \theta \).
3. **Real Power Flow Calculation:**
- Compute the real power flow \( P_{12} \) when \( \theta = 10^\circ \).
4. **DC Power Flow Approximation:**
- Use \(|V_1| = |V_2| = 1\) and small \( \theta \) such that \( \sin \theta = \theta \).
- Let \( B = 1/X \).
5. **Formula Derivation and Application:**
- Derive the formula for real power \( P_{12} \) using the approximation and express it in terms of \( B \) and \( \theta \).
- Compare the results with part (a).
**Steps:**
- Use approximations to derive a formula for the real power \( P_{12} \) and evaluate the real power flow at \( \theta = 10^\circ \).
- Compare computed values from the approximation with those from the actual setup.
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