---
title: "Series connection of two full converters"
subject: "Power Electronics"
chapter: 2
key: "semester_7/power_electronics:series-full-converters-waveforms"
source: https://www.bctengineer.com/q/semester_7/power_electronics/series-full-converters-waveforms
site: BCT Engineering
---

# Series connection of two full converters

Power Electronics · Chapter 2: Single phase AC to DC conversion

- Asked in: 2079 Bhadra, 8 marks; 2070 Asar, 8 marks (asked 2 times)
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/power_electronics/series-full-converters-waveforms
- Whole chapter: https://www.bctengineer.com/md/semester_7/power_electronics/chapter-2

## Question

Explain the series connection of two single phase full converter with necessary circuit diagram and waveforms.

## Short answer

Two single-phase full converters with dc outputs in series give high dc voltage and better power factor. Total output is (2Vm/pi)(cos alpha1 + cos alpha2); with one converter at zero and the other controlled, Vo = (2Vm/pi)(1+cos alpha2), maximum 4Vm/pi. Input primary current is a stepped wave with displacement factor cos(alpha/2).

## Answer

Two single-phase full converters are fed from two separate transformer secondaries and their d.c. outputs are connected **in series**. This is used for high-voltage d.c. output (for example traction) and, with sequence control, to improve the input power factor and reduce harmonics.

#### Circuit

```text
              +------------------+
  ~ ---+  S1  |  Full converter 1|--+ +
       |  ))  |  T1 T3 / T4 T2   |  |  Vo1
  pri  |      +------------------+  |
 (one  |                            +----+
 trans-|      +------------------+  |    | load
 former|  S2  |  Full converter 2|--+    | (RLE)
  two  |  ))  |  T1'T3'/T4'T2'   |  |    |
  sec.)|      +------------------+  |    |
  ~ ---+                            +----+
                                       -
```

Each secondary has rms voltage $V_s$ and peak $V_m$. The total output is
$$
V_o = V_{o1} + V_{o2} = \frac{2V_m}{\pi}(\cos\alpha_1 + \cos\alpha_2)
$$
The maximum output is $\frac{4V_m}{\pi}$ (both at $\alpha = 0$).

#### Rectification mode (power flow a.c. to d.c.)

Converter 1 is kept at $\alpha_1 = 0^\circ$ (output $+\frac{2V_m}{\pi}$) and converter 2 is phase-controlled, $\alpha_2$ from $180^\circ$ down to $0^\circ$:
$$
V_o = \frac{2V_m}{\pi}(1 + \cos\alpha_2)
$$
At $\alpha_2 = 180^\circ$ the two outputs cancel ($V_o = 0$); at $\alpha_2 = 0^\circ$, $V_o = \frac{4V_m}{\pi}$. So the output is varied smoothly from 0 to $\frac{4V_m}{\pi}$ with power flowing from the a.c. supply to the load.

#### Inversion mode (power flow d.c. to a.c.)

The load must contain a d.c. source $E$ of reversed polarity (for example a motor in regenerative braking). One converter is held at $\alpha_1 = 180^\circ$ (full negative voltage, $-\frac{2V_m}{\pi}$) and the other is controlled between $0^\circ$ and $180^\circ$:
$$
V_o = \frac{2V_m}{\pi}(\cos\alpha_2 - 1)
$$
so $V_o$ ranges from 0 to $-\frac{4V_m}{\pi}$; current direction stays the same, so power flows back to the a.c. supply. (In practice the angle is kept below about $165^\circ$ to allow a margin for commutation.)

#### Waveforms (rectification, $\alpha_1 = 0$, $\alpha_2 = \alpha$, constant load current $I_o$)

```text
 vo1  /\  /\  /\     full-wave, alpha1 = 0
 ----/--\/--\/--\--
 vo2   _    _    _   full converter, alpha2,
 -----|-\-.|-\-.|-\  negative from pi to pi+a
 is1  ____      ____
 ----|    |____|      square, in phase
 is2    ____      __
 ------|    |____|    square, lags by alpha
 is   (sum, on primary):
          ____      level 2Io from a to pi
 ----.___|    |___.   level 0 from 0 to a
```

- $v_o = v_{o1} + v_{o2}$: during $0$ to $\alpha$ of each half cycle $v_{o2}$ is negative and reduces $v_o$; from $\alpha$ to $\pi$ both add.
- The primary current $i_s = i_{s1} + i_{s2}$ (equal turns) is a stepped wave: zero from $0$ to $\alpha$ and $2I_o$ from $\alpha$ to $\pi$. Its fundamental lags by only $\alpha/2$, so the displacement factor $\cos(\alpha/2)$ is better than $\cos\alpha$ of a single converter.

#### Advantages

- High output voltage with devices of half the voltage rating.
- Better input power factor and less harmonic content than a single converter of the same rating, because only one converter is phase-controlled at a time (the other runs at $0^\circ$ or $180^\circ$).

## Check yourself

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

1. With alpha1 = 0 and alpha2 = alpha, what does vo2 do from 0 to alpha of each half cycle?
   A. It is positive and adds to vo1
   B. It is negative and reduces vo
   C. It is zero
   D. It equals vo1
   Answer: B. During 0 to alpha, the second converter's output is negative, so vo = vo1 + vo2 is reduced; from alpha to pi both add. (Key point: Output voltage waveform)

2. What is the shape of the primary current is = is1 + is2 with equal turns?
   A. A sine wave in phase with the supply
   B. A square wave of amplitude Io
   C. Stepped: 0 from 0 to alpha and 2Io from alpha to pi
   D. A triangular wave
   Answer: C. is1 is in phase and is2 lags by alpha, so the sum is 2Io only where both flow. (Key point: Primary current waveform)

3. What is the maximum output voltage and what advantage does the series connection give?
   A. 2Vm/pi, with devices of twice the voltage rating
   B. 4Vm/pi, with devices of twice the voltage rating
   C. 2Vm/pi, with higher harmonic content
   D. 4Vm/pi, with devices of half the voltage rating
   Answer: D. Both at alpha = 0 give 4Vm/pi, and each converter handles only half the voltage. (Key point: Maximum output and rating)

## Related in Power Electronics

- [Series full converters: rectification and inversion](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_electronics%3Aseries-full-converters-rect-inversion) — Power Electronics, chapter 2, asked 4×
- [Single-phase full converter, 10 A, α = 30°](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_electronics%3Afull-converter-10a-30deg) — Power Electronics, chapter 2, asked 4×
- [Series operation of two full converters](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_electronics%3Aseries-full-converters-high-voltage) — Power Electronics, chapter 2
- [Single-phase full converter: rectification and inversion](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_electronics%3Afull-converter-rect-inversion) — Power Electronics, chapter 2
- [Series full converters in rectification mode](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_electronics%3Aseries-full-converters-rectification) — Power Electronics, chapter 2
- [Series full converters: average and rms output](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_electronics%3Aseries-full-converters-avg-rms) — Power Electronics, chapter 2

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