---
title: "Three-phase connections and series RC phasor"
subject: "Fundamental of Electrical and Electronics Engineering"
chapter: 3
key: "semester_1/fundamental_of_electrical_and_electronics_engineering:three-phase-connections-series-rc"
source: https://www.bctengineer.com/q/semester_1/fundamental_of_electrical_and_electronics_engineering/three-phase-connections-series-rc
site: BCT Engineering
---

# Three-phase connections and series RC phasor

Fundamental of Electrical and Electronics Engineering · Chapter 3: AC Circuit Analysis

- Asked in: 2081 Baishakh (new course), 2+4 marks
- Video: [29.RC Series Circuit | 1-Phase AC Circuits  | BEE | BEEE | JNTU | Autonomous](https://www.youtube.com/watch?v=M-B05JM8BO8) — VMR ACADEMY
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/fundamental_of_electrical_and_electronics_engineering/three-phase-connections-series-rc
- Whole chapter: https://www.bctengineer.com/md/semester_1/fundamental_of_electrical_and_electronics_engineering/chapter-3

## Question

What are the two ways of connecting a 3-phase system and mention their relation between phase and line quantities. Derive the expression for series connected RC with phasor diagram.

## Short answer

Star (Y): V_L = √3 V_ph, I_L = I_ph. Delta (Δ): V_L = V_ph, I_L = √3 I_ph. In a series RC circuit, common current I is reference. V_R = IR in phase, V_C = I X_C lags 90°; total V = √(V_R²+V_C²) = I√(R²+X_C²). Impedance Z = R − jX_C, current leads voltage by φ = tan⁻¹(X_C/R).

## Answer

#### Two ways of connecting a 3-phase system

1. **Star (Y) connection:** one end of each of the three phase windings is joined to a common point called the neutral (N); the other three ends go to the lines.
   - V_L = √3 V_ph (line voltage leads the phase voltage by 30°)
   - I_L = I_ph
2. **Delta (Δ) connection:** the three windings are joined end to end in a closed loop; the lines are taken from the three junctions.
   - V_L = V_ph
   - I_L = √3 I_ph (line current lags the phase current by 30°)

In both cases the total power is P = √3 V_L I_L cos φ.

#### Series RC circuit

A resistance R and capacitance C are in series across v = Vm sin ωt. The same current I flows through both, so I is taken as the reference phasor.

```text
 o---[ R ]---| C |---o
     V_R      V_C
 <-------- V -------->
```

- Voltage across R: V_R = I R, **in phase** with I.
- Voltage across C: V_C = I X_C, **lagging** I by 90°, where X_C = 1/(ωC) = 1/(2πfC).

Phasor sum:

```text
V   = V_R − j V_C = I (R − j X_C)
|V| = √(V_R² + V_C²) = I √(R² + X_C²)
Z   = V/I = R − j X_C
|Z| = √(R² + X_C²)
φ   = tan⁻¹(X_C/R)   (current leads voltage)
```

**Phasor diagram:**

```text
        I, V_R
  o------------->       (reference)
  |\
  | \  φ
  |  \
V_C   \ V
  |    \
  v     v
```

The voltage V lags the current I by φ (i.e. current leads voltage).

If v = Vm sin ωt, then i = Im sin(ωt + φ) with Im = Vm/|Z|.

**Power:** power factor cos φ = R/|Z| (leading). Average power P = V I cos φ = I²R; the capacitor takes no average power, only reactive power Q = I² X_C.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. In a star connection, the relations between line and phase quantities are:
   A. V_L = V_ph, I_L = sqrt3 I_ph
   B. V_L = sqrt3 V_ph, I_L = sqrt3 I_ph
   C. V_L = V_ph, I_L = I_ph
   D. V_L = sqrt3 V_ph, I_L = I_ph
   Answer: D. Each line is in series with one phase (equal currents), and the line voltage is the phasor difference of two phase voltages. (Key point: Star relations)

2. In a delta connection, which is true?
   A. V_L = V_ph and I_L = sqrt3 I_ph
   B. V_L = sqrt3 V_ph and I_L = I_ph
   C. V_L = V_ph/sqrt3
   D. I_L = I_ph/sqrt3
   Answer: A. Each phase is directly across two lines; line current is the phasor difference of two phase currents. (Key point: Delta relations)

3. In a series RC circuit with current as reference, V_C is:
   A. Leading current by 90 degrees
   B. Lagging current by 90 degrees
   C. In phase with current
   D. Lagging current by 180 degrees
   Answer: B. Voltage across a capacitor lags its current by 90 degrees; V_R is in phase. (Key point: Capacitor voltage phase)

4. For series RC, what are |Z| and the current-voltage relation?
   A. |Z| = R + X_C; current lags voltage
   B. |Z| = sqrt(R^2 - X_C^2); current in phase
   C. |Z| = sqrt(R^2 + X_C^2); current leads voltage by tan^-1(X_C/R)
   D. |Z| = sqrt(R^2 + X_C^2); current lags voltage
   Answer: C. V = I(R - jX_C), so the voltage lags current by phi, meaning current leads; pf = R/|Z| leading. (Key point: Impedance and phase angle)

## Related in Fundamental of Electrical and Electronics Engineering

- [Methods of three-phase connection](https://www.bctengineer.com/question.md?key=semester_1%2Ffundamental_of_electrical_and_electronics_engineering%3Athree-phase-connection-methods) — Fundamental of Electrical and Electronics Engineering, chapter 3
- [Emitter-bias BJT Q point and load line](https://www.bctengineer.com/question.md?key=semester_1%2Ffundamental_of_electrical_and_electronics_engineering%3Aemitter-bias-q-point) — Fundamental of Electrical and Electronics Engineering, chapter 5
- [Voltage-divider bias load line and Q point](https://www.bctengineer.com/question.md?key=semester_1%2Ffundamental_of_electrical_and_electronics_engineering%3Avoltage-divider-bias-q-point) — Fundamental of Electrical and Electronics Engineering, chapter 5
- [Power factor and series-parallel circuit calculation](https://www.bctengineer.com/question.md?key=semester_1%2Ffundamental_of_electrical_and_electronics_engineering%3Apower-factor-series-parallel-circuit) — Fundamental of Electrical and Electronics Engineering, chapter 3

## Related in other subjects

- [Three-phase transformer connections](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines_i%3Athree-phase-transformer-connections) — Electrical Machines I, chapter 2
- [Three phase transformer connections and applications](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines%3Athree-phase-transformer-connections) — Electrical Machines, chapter 2
- [Scott connection for three-phase to two-phase](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines_i%3Ascott-connection) — Electrical Machines I, chapter 2
- [Derive structural quantities using ILD](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Aild-derive-structural-quantities) — Theory of Structures I, chapter 5, asked 4×