QuangAnhcons power-quality monograph (13 October 2026). Full five chapters, 6 computational diagrams, derivations, appendices and sources. Synthetic figures are not field measurements.

Download complete PDF study (26 pages, optimized format)

Scope, notation and
research questions

Harmonics are not an independent substance called “dirty
electricity”. They are frequency components of voltage and current
arising from the interaction of sources, load characteristics and
network impedance. A variable-speed drive can draw a substantially
distorted current while its supply-bus voltage remains approximately
sinusoidal if the source is sufficiently stiff. Conversely, a modest
harmonic-current injection can produce a large harmonic voltage near a
network-impedance resonance. A useful investigation therefore asks not
only “What is the THD?” but also “Which quantity, at which location,
with which denominator, under which operating condition and through
which return path?” [R1, R7, R8]

This monograph connects three analytical layers: Fourier frequency
decomposition, the spatial structure of three-phase quantities through
symmetrical components, and the frequency response of the network from
500 kV to low voltage. The derivations and numerical cases are developed
from explicitly stated assumptions. They are not measurements from a
particular factory or substation. The monetary values in the economic
example are calculation assumptions, not equipment quotations or current
electricity tariffs.

Two corrections are essential. First, in a three-phase system
balanced by cyclic time displacement, every harmonic order
divisible by three is zero sequence, including the 6th and
12th. The usual absence of even harmonics does not turn them into
positive- or negative-sequence components. Second, a delta
winding is not a universal harmonic filter. It can support
internal zero-sequence circulation, but it does not necessarily prevent
zero-sequence transfer between two grounded-wye windings. Chapters 2 and
3 establish these conclusions and their limits. [R2, R3]

The assumed fundamental is
$f_1=50\,\mathrm{Hz}$,
with
$T=1/f_1$
and
$\omega_1=2\pi f_1$.
Upper-case
$V_h$
and
$I_h$
denote RMS harmonic magnitudes; an underline denotes a phasor.
Subscripts
$+,-,0$
indicate positive, negative and zero sequence. Line-to-line and phase
voltages are distinguished wherever used. Short-circuit power and
capacitor rating must refer to the same bus and compatible units.

Chapter 1.
Physical and mathematical foundations

1.1. From
nonlinear circuits to nonsinusoidal waveforms

An ideal constant resistance supplied by a sinusoidal voltage draws
sinusoidal current at the same frequency. In a diode rectifier with a
DC-link capacitor, conduction takes place only when the instantaneous
source voltage exceeds the relevant DC-side voltage and circuit drops.
The AC current is consequently concentrated into conduction intervals
rather than being continuously proportional to voltage. Drives with
diode front ends, controlled rectifiers, DC chokes and active front ends
need not have the same current spectrum. A harmonic spectrum cannot be
inferred from the equipment’s kW rating alone. [R7]

For a periodic waveform that is integrable and square-integrable over
a period, the Fourier representation is

$$f(t)=\frac{a_0}{2}+\sum_{h=1}^{\infty} \left[a_h\cos(h\omega_1t)+b_h\sin(h\omega_1t)\right].$$

\[

f(t)=\frac{a_0}{2}+\sum_{h=1}^{\infty}
\left[a_h\cos(h\omega_1t)+b_h\sin(h\omega_1t)\right].

\]

Its coefficients are

$$a_0=\frac{2}{T}\int_0^T f(t)\,dt,\qquad a_h=\frac{2}{T}\int_0^T f(t)\cos(h\omega_1t)\,dt,$$

\[

a_0=\frac{2}{T}\int_0^T f(t)\,dt,\qquad
a_h=\frac{2}{T}\int_0^T f(t)\cos(h\omega_1t)\,dt,

\]

$$b_h=\frac{2}{T}\int_0^T f(t)\sin(h\omega_1t)\,dt.$$

\[

b_h=\frac{2}{T}\int_0^T f(t)\sin(h\omega_1t)\,dt.

\]

An electrical-engineering form uses RMS amplitudes:

$$f(t)=F_{\mathrm{dc}}+\sqrt2\sum_{h=1}^{\infty}F_h\cos(h\omega_1t+\varphi_h), \qquad F_h=\frac{\sqrt{a_h^2+b_h^2}}{\sqrt2}.$$

\[

f(t)=F_{\mathrm{dc}}+\sqrt2\sum_{h=1}^{\infty}F_h\cos(h\omega_1t+\varphi_h),
\qquad F_h=\frac{\sqrt{a_h^2+b_h^2}}{\sqrt2}.

\]

Under the usual convergence conditions, the series takes the average
of the one-sided limits at a discontinuity. Gibbs oscillations near an
idealized switching edge are not proof that the physical converter
generates exactly the overshoot of a truncated series. The physical
waveform, the sampling process and the finite-harmonic reconstruction
must be distinguished.

A harmonic has frequency
$f_h=hf_1$,
where
$h$
is a positive integer. A component not equal to an integer multiple of
the fundamental is an interharmonic; it should not be
assigned arbitrarily to the nearest integer order. DC is considered
separately. Fluctuating arcs, power modulation and asynchronous
switching can create a spectrum broader than the characteristic orders
of an ideal rectifier. A single THD value cannot describe all of this
frequency and time information. Contemporary power-quality measurement
methods address both harmonics and interharmonics. [R4]

1.2. RMS
values, THD and the information lost in aggregation

Orthogonality gives Parseval’s relation:

$$F_{\mathrm{rms}}^2=F_{\mathrm{dc}}^2+\sum_{h=1}^{\infty}F_h^2.$$

\[

F_{\mathrm{rms}}^2=F_{\mathrm{dc}}^2+\sum_{h=1}^{\infty}F_h^2.

\]

The conventional harmonic-distortion indices, excluding DC, are

$$\mathrm{THD}_V=\frac{\sqrt{\sum_{h=2}^{H}V_h^2}}{V_1}\,100\%, \qquad \mathrm{THD}_I=\frac{\sqrt{\sum_{h=2}^{H}I_h^2}}{I_1}\,100\%.$$

\[

\mathrm{THD}_V=\frac{\sqrt{\sum_{h=2}^{H}V_h^2}}{V_1}\,100\%,
\qquad
\mathrm{THD}_I=\frac{\sqrt{\sum_{h=2}^{H}I_h^2}}{I_1}\,100\%.

\]

The highest included order
$H$,
measurement window, spectral grouping and transducer performance must be
documented. Equal THD values do not imply equal consequences: a spectrum
dominated by the 5th and one dominated by the 25th can produce different
heating, resonance and control effects. THD also discards phase angles.
Equal-magnitude harmonic currents may reinforce or partly cancel when
they meet at a common bus. [R1, R4]

The numerical example used throughout this monograph is

$$I_1=1000\,\mathrm A,\quad I_5=200\,\mathrm A,\quad I_7=140\,\mathrm A,\quad I_{11}=90\,\mathrm A,\quad I_{13}=70\,\mathrm A.$$

\[

I_1=1000\,\mathrm A,\quad I_5=200\,\mathrm A,\quad
I_7=140\,\mathrm A,\quad I_{11}=90\,\mathrm A,\quad I_{13}=70\,\mathrm A.

\]

Hence

$$I_{\mathrm{harm}}=\sqrt{200^2+140^2+90^2+70^2}=269.444\,\mathrm A,$$

\[

I_{\mathrm{harm}}=\sqrt{200^2+140^2+90^2+70^2}=269.444\,\mathrm A,

\]

$$\mathrm{THD}_I=26.944\%,\qquad I_{\mathrm{rms}}=\sqrt{1000^2+269.444^2}=1035.664\,\mathrm A.$$

\[

\mathrm{THD}_I=26.944\%,\qquad
I_{\mathrm{rms}}=\sqrt{1000^2+269.444^2}=1035.664\,\mathrm A.

\]

Total RMS current is only approximately 3.57% above the fundamental
current. This does not mean that all losses increase by
3.57%. Resistive losses are quadratic in current, AC resistance is
frequency-dependent, and winding eddy-current losses require additional
frequency weighting. Chapter 4 separates these effects.

1.3. Why
half-wave antisymmetry eliminates even harmonics

The decisive condition is sign reversal after half a
fundamental period:

$$f(t+T/2)=-f(t).$$

\[

f(t+T/2)=-f(t).

\]

This must not be confused with odd-function symmetry
$f(-t)=-f(t)$.
Odd-function symmetry eliminates cosine coefficients for the selected
time origin, but it can still allow
$\sin(2\omega_1t)$.
Elimination of even orders requires half-wave antisymmetry of the
complete waveform.

Using complex Fourier coefficients,

$$C_h=\frac1T\int_0^T f(t)e^{-jh\omega_1t}\,dt.$$

\[

C_h=\frac1T\int_0^T f(t)e^{-jh\omega_1t}\,dt.

\]

Split the integral and substitute
$u=t-T/2$
in its second half:

$$C_h=\frac1T\int_0^{T/2}f(t)e^{-jh\omega_1t}\,dt+ \frac1T\int_0^{T/2}f(u+T/2)e^{-jh\omega_1(u+T/2)}\,du.$$

\[

C_h=\frac1T\int_0^{T/2}f(t)e^{-jh\omega_1t}\,dt+
\frac1T\int_0^{T/2}f(u+T/2)e^{-jh\omega_1(u+T/2)}\,du.

\]

Since
$e^{-jh\pi}=(-1)^h$,

$$C_h=\frac{1-(-1)^h}{T}\int_0^{T/2}f(t)e^{-jh\omega_1t}\,dt.$$

\[

C_h=\frac{1-(-1)^h}{T}\int_0^{T/2}f(t)e^{-jh\omega_1t}\,dt.

\]

For
$h=2k$,
the prefactor is zero, so
$C_{2k}=0$.
For
$h=2k+1$,
the prefactor is two: odd harmonics are permitted, not
guaranteed to exist at every odd order. Taking
$h=0$
also gives zero DC content.

This is a condition on the entire waveform. Unequal half-cycle
conduction caused by asymmetric control, a failed component or biased
core flux invalidates the cancellation. Conversely, a sharply peaked
current or a saturated core does not by itself prove the presence of
even harmonics.

1.4. Physical
mechanisms that produce even orders

A simple algebraic example makes the distinction clear. Let

$$i=\alpha v+\beta v^2+\gamma v^3,\qquad v=V_m\sin(\omega_1t).$$

\[

i=\alpha v+\beta v^2+\gamma v^3,\qquad v=V_m\sin(\omega_1t).

\]

The identities

$$\sin^2x=\frac{1-\cos2x}{2},\qquad \sin^3x=\frac{3\sin x-\sin3x}{4}$$

\[

\sin^2x=\frac{1-\cos2x}{2},\qquad
\sin^3x=\frac{3\sin x-\sin3x}{4}

\]

show that the quadratic term creates DC and a 2nd harmonic, whereas
the cubic term creates fundamental and 3rd-harmonic components. A
nonlinear characteristic symmetric about the origin may contain odd
powers only and retain half-wave antisymmetry. Nonlinearity alone is not
sufficient to establish even-harmonic emission.

For a transformer winding,

$$v(t)\simeq N\frac{d\Phi(t)}{dt}.$$

\[

v(t)\simeq N\frac{d\Phi(t)}{dt}.

\]

Symmetrical flux variation with a symmetrical magnetizing
characteristic can produce a distorted exciting current dominated by odd
orders. DC bias, residual flux combined with the energization instant,
or unequal excitation can drive one half cycle more deeply into
saturation. Transformer inrush is often examined through its
2nd-harmonic content, but it is a transient phenomenon: windowed Fourier
coefficients change with time. A steady-state THD number cannot fully
describe the event.

An open diode, a nonconducting rectifier leg or unequal firing angles
between half cycles can likewise create asymmetry, DC and
noncharacteristic harmonics. However, a measured 2nd harmonic is not a
unique diagnosis of diode failure. Phase waveforms, DC voltage,
converter condition, operating state and measurement integrity must be
checked. A biased sensor or a saturated CT can itself distort the
recorded secondary waveform.

Even harmonics should therefore be treated as evidence motivating a
search for broken symmetry, not as a unique signature of one failure
mode. The investigation needs time-correlated operating data rather than
a single THD screenshot.

Chapter
2. Symmetrical components and harmonic sequence

2.1.
Frequency decomposition and phase decomposition are different
operations

Fourier analysis separates a waveform by frequency. The Fortescue
transformation separates three phasors at the same
frequency into balanced sets with different spatial phase
structures. For each order
$h$,
first obtain
$\underline X_{Ah},\underline X_{Bh},\underline X_{Ch}$
and then perform the sequence transformation. A phase-A 5th harmonic and
a phase-B 7th harmonic do not belong in one Fortescue calculation. The
underlying method and sequence-network connections are covered by the
MIT and SEL references. [R2, R3]

Define

$$a=e^{j2\pi/3},\qquad a^2=e^{j4\pi/3},\qquad 1+a+a^2=0,\qquad a^3=1.$$

\[

a=e^{j2\pi/3},\qquad a^2=e^{j4\pi/3},\qquad
1+a+a^2=0,\qquad a^3=1.

\]

For output order positive, negative, zero,

$$\begin{bmatrix}\underline X_+\\\underline X_-\\\underline X_0\end{bmatrix} =\frac13 \begin{bmatrix}1&a&a^2\\1&a^2&a\\1&1&1\end{bmatrix} \begin{bmatrix}\underline X_A\\\underline X_B\\\underline X_C\end{bmatrix}.$$

\[

\begin{bmatrix}\underline X_+\\\underline X_-\\\underline X_0\end{bmatrix}
=\frac13
\begin{bmatrix}1&a&a^2\\1&a^2&a\\1&1&1\end{bmatrix}
\begin{bmatrix}\underline X_A\\\underline X_B\\\underline X_C\end{bmatrix}.

\]

The inverse is

$$\begin{bmatrix}\underline X_A\\\underline X_B\\\underline X_C\end{bmatrix} = \begin{bmatrix}1&1&1\\a^2&a&1\\a&a^2&1\end{bmatrix} \begin{bmatrix}\underline X_+\\\underline X_-\\\underline X_0\end{bmatrix}.$$

\[

\begin{bmatrix}\underline X_A\\\underline X_B\\\underline X_C\end{bmatrix}
=
\begin{bmatrix}1&1&1\\a^2&a&1\\a&a^2&1\end{bmatrix}
\begin{bmatrix}\underline X_+\\\underline X_-\\\underline X_0\end{bmatrix}.

\]

Positive sequence has structure
$[1,a^2,a]^{\mathsf T}$,
negative sequence
$[1,a,a^2]^{\mathsf T}$
and zero sequence
$[1,1,1]^{\mathsf T}$.
References using zero–positive–negative order simply permute the rows
and corresponding inverse columns. The convention must remain consistent
throughout the calculation.

Multiplication of the two matrices verifies the inverse: off-diagonal
terms vanish because
$1+a+a^2=0$,
while the diagonal terms become one. The decomposition is linear and
unique for any three phasors, including a completely unbalanced set.

2.2. Deriving the
harmonic-order rule

Assume that the three phase waveforms are identical shapes displaced
by one third of the fundamental period:

$$x_B(t)=x_A(t-T/3),\qquad x_C(t)=x_A(t+T/3).$$

\[

x_B(t)=x_A(t-T/3),\qquad x_C(t)=x_A(t+T/3).

\]

For a phase-A component

$$x_{Ah}(t)=\sqrt2X_h\cos(h\omega_1t+\varphi_h),$$

\[

x_{Ah}(t)=\sqrt2X_h\cos(h\omega_1t+\varphi_h),

\]

the three-phase phasor vector is

$$\underline{\boldsymbol X}_h= \underline X_h \begin{bmatrix}1\\e^{-j2\pi h/3}\\e^{j2\pi h/3}\end{bmatrix}.$$

\[

\underline{\boldsymbol X}_h=
\underline X_h
\begin{bmatrix}1\\e^{-j2\pi h/3}\\e^{j2\pi h/3}\end{bmatrix}.

\]

When
$h=3k+1$,
the second and third factors are
$a^2$
and
$a$,
giving positive sequence. When
$h=3k+2$,
they are
$a$
and
$a^2$,
giving negative sequence. When
$h=3k$,
both are one, giving zero sequence. Thus

$$h\bmod3=\begin{cases} 1&\Rightarrow\text{positive sequence},\\ 2&\Rightarrow\text{negative sequence},\\ 0&\Rightarrow\text{zero sequence}. \end{cases}$$

\[

h\bmod3=\begin{cases}
1&\Rightarrow\text{positive sequence},\\
2&\Rightarrow\text{negative sequence},\\
0&\Rightarrow\text{zero sequence}.
\end{cases}

\]

This is a rule for a system balanced by cyclic time
displacement. It is not an unconditional property of any
arbitrary signal labelled “5th harmonic” or “3rd harmonic”.

2.3. Triplen
harmonics: odd and even multiples of three

For
$h=3k$,
the harmonic phase displacement is

$$\Delta\varphi_{AB,h}=h\frac{2\pi}{3}=2k\pi\equiv0\pmod{2\pi}.$$

\[

\Delta\varphi_{AB,h}=h\frac{2\pi}{3}=2k\pi\equiv0\pmod{2\pi}.

\]

For odd multiples
$h=6m+3$,
the result is
$(4m+2)\pi$,
equivalent to zero. For even multiples
$h=6m$,
it is
$4m\pi$,
also equivalent to zero. Consequently, there is no valid proof
that the 6th and 12th must be positive or negative sequence under these
balanced conditions.

The common misconception results from learning only the
characteristic spectrum of half-wave-symmetric three-phase loads. Even
orders vanish in that source model, so zero-sequence lists often show
only 3, 9, 15 and 21. Absence from a particular source spectrum does not
change the mathematical classification of an even triplen component when
one is present.

2.4. Complete
classification through the 25th order

For all even orders,
$h=6k$
is zero sequence,
$h=6k+2$
negative sequence and
$h=6k+4$
positive sequence. The same rule extends beyond the table.

2.5.
Unbalance: any order can contain all three sequences

Consider a 3rd-harmonic current present only in phase A:

$$\begin{bmatrix}\underline I_{A3}\\\underline I_{B3}\\\underline I_{C3}\end{bmatrix} =\begin{bmatrix}\underline I\\0\\0\end{bmatrix}.$$

\[

\begin{bmatrix}\underline I_{A3}\\\underline I_{B3}\\\underline I_{C3}\end{bmatrix}
=\begin{bmatrix}\underline I\\0\\0\end{bmatrix}.

\]

Applying Fortescue gives

$$\underline I_{3+}=\underline I_{3-}=\underline I_{30}=\frac{\underline I}{3}.$$

\[

\underline I_{3+}=\underline I_{3-}=\underline I_{30}=\frac{\underline I}{3}.

\]

A 3rd-harmonic current is therefore not necessarily pure zero
sequence in a system containing unequal single-phase loads. The actual
zero-sequence component, rather than the order label alone, determines
the neutral return current and the blocking behaviour of a delta’s
external line terminals.

For consistently defined current directions,

$$\underline I_{Nh}=\underline I_{Ah}+\underline I_{Bh}+\underline I_{Ch}=3\underline I_{0h}.$$

\[

\underline I_{Nh}=\underline I_{Ah}+\underline I_{Bh}+\underline I_{Ch}=3\underline I_{0h}.

\]

If each phase carries 30 A of equal, in-phase 3rd harmonic, the
neutral carries 90 A at that frequency. With several orders present,
phasors are added within each frequency first; the resulting RMS
contributions at different frequencies are then combined by
root-sum-square.

For pure zero-sequence voltage,

$$\underline V_{AB,h}=\underline V_{Ah}-\underline V_{Bh}=0.$$

\[

\underline V_{AB,h}=\underline V_{Ah}-\underline V_{Bh}=0.

\]

Line-to-line voltage measurements can therefore miss this common
phase-to-neutral component. “There is no 3rd-harmonic voltage” is not a
justified conclusion from a measurement scheme observing only line
voltages.

Chapter
3. Propagation from the 500 kV grid to end-use loads

3.1.
Harmonic current follows network impedance, not the fundamental-power
arrow

The following is a functional supply chain, not a mandatory national
substation configuration:

500 kV sources and network <----> 220/110 kV network
                                         |
                                  HV/MV transformer
                                         |
                                    35/22 kV bus
                                         |
                                  MV/LV transformer
                                         |
                                PCC / factory 0.4 kV MSB
                                   |               |
                              Drives, arcs,    Capacitors,
                              rectifiers, UPS  cables, linear loads

Fundamental active power may flow predominantly from utility to load,
while load-generated harmonic currents divide among paths according to
their impedance at each harmonic frequency. There is no physical rule
requiring every harmonic ampere to travel upstream. Some can enter a
nearby capacitor, another parallel transformer or a filter at a
neighbouring installation. [R8]

In a linearized frequency-domain model, represent nonlinear loads by
Norton harmonic-current injections:

$$\boldsymbol Y_h\boldsymbol V_h=\boldsymbol I_{\mathrm{inj},h},\qquad \boldsymbol V_h=\boldsymbol Z_h\boldsymbol I_{\mathrm{inj},h}.$$

\[

\boldsymbol Y_h\boldsymbol V_h=\boldsymbol I_{\mathrm{inj},h},\qquad
\boldsymbol V_h=\boldsymbol Z_h\boldsymbol I_{\mathrm{inj},h}.

\]

Background harmonic voltage sources can be incorporated through
Norton equivalents. Alternatively, a consistently separated background
representation is

$$\boldsymbol V_h=\boldsymbol V_{\mathrm{bg},h}+\boldsymbol Z_h\boldsymbol I_{\mathrm{inj},h}.$$

\[

\boldsymbol V_h=\boldsymbol V_{\mathrm{bg},h}+\boldsymbol Z_h\boldsymbol I_{\mathrm{inj},h}.

\]

These are alternative bookkeeping choices for a consistently defined
network. Background voltage must not be added again after its Norton
source has already been included. Rectifier current also depends on the
distorted terminal voltage; an independent current source is then an
approximation. Iterative harmonic load flow or time-domain simulation
may be required.

3.2. What a closed delta
actually traps

Let the delta branch currents be
$\underline I_{ab},\underline I_{bc},\underline I_{ca}$.
External line currents are

$$\underline I_A=\underline I_{ab}-\underline I_{ca},\quad \underline I_B=\underline I_{bc}-\underline I_{ab},\quad \underline I_C=\underline I_{ca}-\underline I_{bc}.$$

\[

\underline I_A=\underline I_{ab}-\underline I_{ca},\quad
\underline I_B=\underline I_{bc}-\underline I_{ab},\quad
\underline I_C=\underline I_{ca}-\underline I_{bc}.

\]

If the three branch currents circulate equally,

$$\underline I_{ab}=\underline I_{bc}=\underline I_{ca}=\underline I_{\Delta0},$$

\[

\underline I_{ab}=\underline I_{bc}=\underline I_{ca}=\underline I_{\Delta0},

\]

all three external line currents are zero although winding current
remains nonzero. This is the precise meaning of internal trapping. The
delta does not destroy the current: it prevents that circulating
component from appearing as zero-sequence current at the three external
line terminals. Internal loss and heating remain. [R2, R3]

Equal-phase branch EMFs can add around the closed loop. A conceptual
relation is

$$\underline I_{\Delta0,h}\simeq \frac{\underline E_{ab,h}+\underline E_{bc,h}+\underline E_{ca,h}} {Z_{ab,h}+Z_{bc,h}+Z_{ca,h}}.$$

\[

\underline I_{\Delta0,h}\simeq
\frac{\underline E_{ab,h}+\underline E_{bc,h}+\underline E_{ca,h}}
{Z_{ab,h}+Z_{bc,h}+Z_{ca,h}}.

\]

This is not a replacement for transformer design data. The actual
EMFs and impedances depend on magnetic coupling, core behaviour and
loading. It simply explains how a closed loop supports circulation
without an external neutral conductor.

3.3.
Zero-sequence equivalent of a three-winding Yn/Yn/Δ transformer

Consider a hypothetical isolated three-winding
500/220/110 kV transformer connected Yn/Yn/Δ. The 110 kV delta is
retained here to address the stated example; it is not a claim that all
practical 500/220 kV interconnecting transformers have a 110 kV delta
tertiary. Actual nameplate information, full vector group, individual
winding ratings and earthing arrangements must be obtained before
modelling a real unit.

The following conceptual circuit is referred to one common base and
uses star-connected leakage branches with a closed delta:

H0 port: 500 kV grounded wye              M0 port: 220 kV grounded wye
       o--[ ZH0 ]--[ 3ZnH ]--o m--[ 3ZnM ]--[ ZM0 ]--o
                              |
                       +------+------+
                       |             |
                     [ ZD0 ]       [ Zmag,0 ]
                       |             |
                       +------+------+
                              |
                       Zero-sequence reference

D0 port: external line terminals of the 110 kV delta
       o---------------- OPEN for external zero-sequence current

ZD0 represents internal delta circulation, NOT a physical delta earth.
Zmag,0 is retained unless neglecting excitation is justified.

A neutral impedance appears as
$3Z_n$
because neutral current is
$3I_0$,
producing a neutral drop
$3I_0Z_n$.
Connecting the delta branch to the reference of the transformed
equivalent circuit does not mean that the physical
delta is earthed. This is a sequence-network diagram, not an
installation wiring diagram. [R3]

If both grounded-wye windings have effective neutral return paths, H0
and M0 are connected through the leakage network. Zero-sequence current
can transfer between them and divide into the delta branch.
Consequently, “the delta ensures that all 3rd-harmonic current on the
220 kV side cannot affect the 500 kV side” is not a valid general
conclusion. An ungrounded wye port can be open to zero-sequence line
current in the low-frequency idealization, but capacitance and other
earth connections may need to be retained at higher frequencies. [R2,
R3]

Where compatible zero-sequence test data support a star-branch
representation,

$$Z_{H0}=\frac{Z_{HM,0}+Z_{HD,0}-Z_{MD,0}}{2},$$

\[

Z_{H0}=\frac{Z_{HM,0}+Z_{HD,0}-Z_{MD,0}}{2},

\]

$$Z_{M0}=\frac{Z_{HM,0}+Z_{MD,0}-Z_{HD,0}}{2},\qquad Z_{D0}=\frac{Z_{HD,0}+Z_{MD,0}-Z_{HM,0}}{2}.$$

\[

Z_{M0}=\frac{Z_{HM,0}+Z_{MD,0}-Z_{HD,0}}{2},\qquad
Z_{D0}=\frac{Z_{HD,0}+Z_{MD,0}-Z_{HM,0}}{2}.

\]

All values must use the same base and compatible test conditions. The
three positive-sequence nameplate short-circuit impedances cannot simply
be relabelled as zero-sequence data. If magnetic excitation and core
construction have a substantial effect, the simple star model may be
insufficient. A negative component in an algebraically derived star arm
should not automatically be interpreted as an independent physical
negative inductor.

3.4. Why
an autotransformer requires a different treatment

An autotransformer links voltage levels not only magnetically but
also through a common winding and a direct electrical connection.
Common-winding current, series-winding current, neutral current and
tertiary circulation satisfy coupled constraints. Replacing it with
three isolated windings by changing the labels on a diagram does not
preserve those constraints.

A generic linearized port representation is

$$\begin{bmatrix}\underline V_{H0}\\\underline V_{M0}\end{bmatrix} = \begin{bmatrix}Z_{HH,0}&Z_{HM,0}\\Z_{MH,0}&Z_{MM,0}\end{bmatrix} \begin{bmatrix}\underline I_{H0}\\\underline I_{M0}\end{bmatrix}.$$

\[

\begin{bmatrix}\underline V_{H0}\\\underline V_{M0}\end{bmatrix}
=
\begin{bmatrix}Z_{HH,0}&Z_{HM,0}\\Z_{MH,0}&Z_{MM,0}\end{bmatrix}
\begin{bmatrix}\underline I_{H0}\\\underline I_{M0}\end{bmatrix}.

\]

Its parameters must include the common winding, delta state, neutral
return and consistent current referral. Where there is one physical
common neutral, two independent
$3Z_n$
branches must not be added as if two separate neutrals existed.
Three-limb cores, five-limb cores and banks of single-phase units also
provide different zero-sequence flux paths.

A detailed study therefore requires manufacturer models, test
information and tertiary thermal limits. A conceptual sequence circuit
is useful for understanding current paths, but it is not enough to
approve protection settings or loading capability for an EHV
autotransformer.

3.5. Upstream
propagation of the 5th and 7th harmonics

An ideal balanced six-pulse rectifier has characteristic AC-side
harmonic orders

$$h=6k\pm1,\qquad k=1,2,\ldots$$

\[

h=6k\pm1,\qquad k=1,2,\ldots

\]

Under the assumptions of Chapter 2, the 5th is negative sequence and
the 7th positive sequence. Neither needs a neutral conductor. An
ordinary transformer transmits these components through magnetic
coupling, with current-magnitude conversion according to ratio and phase
conversion according to vector group. A delta winding does not
automatically eliminate them. [R7]

For an ideal transformer with line-voltage ratio
$n=U_H/U_L$,

$$I_{H,h}=\frac{I_{L,h}}{n}.$$

\[

I_{H,h}=\frac{I_{L,h}}{n}.

\]

If 20 A of 5th-harmonic line current at 0.4 kV flows entirely through
an ideal 22/0.4 kV transformer, the corresponding 22 kV current is
approximately 0.364 A. The ampere value decreases as voltage is raised,
but its voltage impact or significance on a consistent power base does
not thereby become zero. In a real network, the fraction flowing through
each branch must be calculated before applying the ratio.

Two different phase operations must be distinguished. A time
displacement of a waveform multiplies its phase displacement by
$h$.
In contrast, a transformer’s vector relationship at a given frequency
has opposite phase-shift signs for positive and negative sequence. For
line voltage,

$$\underline V_{AB,+}=\sqrt3e^{j\pi/6}\underline V_{A,+},\qquad \underline V_{AB,-}=\sqrt3e^{-j\pi/6}\underline V_{A,-}.$$

\[

\underline V_{AB,+}=\sqrt3e^{j\pi/6}\underline V_{A,+},\qquad
\underline V_{AB,-}=\sqrt3e^{-j\pi/6}\underline V_{A,-}.

\]

It is therefore incorrect to represent all transformer harmonic
transfer simply by multiplying its fundamental clock angle by harmonic
order. Multipulse cancellation combines the phase at which each
secondary produces its harmonic with the vector conversion back to a
common primary bus. Chapter 5 uses this distinction explicitly.

3.6. Multi-bus modelling at 500
kV

A high EHV short-circuit level usually reduces the harmonic voltage
produced by one particular injected current, but network impedance is
not uniformly low at all frequencies. Long lines, cables, capacitor
banks, shunt reactors, switching configurations and large power
converters can create response peaks or change impedance phase. A lumped
RL approximation should not be extrapolated indiscriminately to every
500 kV line.

For a linear distributed line at the frequency being studied,

$$\begin{bmatrix}\underline V_s\\\underline I_s\end{bmatrix} = \begin{bmatrix} \cosh(\gamma\ell)&Z_c\sinh(\gamma\ell)\\ Z_c^{-1}\sinh(\gamma\ell)&\cosh(\gamma\ell) \end{bmatrix} \begin{bmatrix}\underline V_r\\\underline I_r\end{bmatrix},$$

\[

\begin{bmatrix}\underline V_s\\\underline I_s\end{bmatrix}
=
\begin{bmatrix}
\cosh(\gamma\ell)&Z_c\sinh(\gamma\ell)\\
Z_c^{-1}\sinh(\gamma\ell)&\cosh(\gamma\ell)
\end{bmatrix}
\begin{bmatrix}\underline V_r\\\underline I_r\end{bmatrix},

\]

$$\gamma=\sqrt{zy},\qquad Z_c=\sqrt{z/y}.$$

\[

\gamma=\sqrt{zy},\qquad Z_c=\sqrt{z/y}.

\]

Here
$z$
and
$y$
are frequency-dependent series impedance and shunt admittance per unit
length. Terminal-current conventions must match the implementation.
Unbalance, earth return and shield-wire coupling may require
phase-domain or appropriate sequence-domain matrices rather than a
single scalar pair.

A meaningful study examines strong and weak sources, capacitor
switching combinations, LV standby generation, parallel transformers and
loss-of-element configurations. Aggregating factory emissions requires
phasor information or a justified statistical method. Adding all
magnitudes provides one conservative scenario; it is not proof that this
scenario occurs continuously.

Chapter
4. Technical damage and techno-economic assessment

4.1. Skin effect and proximity
effect

For a good conductor, a characteristic electromagnetic penetration
depth is

$$\delta=\sqrt{\frac{2\rho}{\omega\mu}}=\sqrt{\frac{\rho}{\pi f\mu}}.$$

\[

\delta=\sqrt{\frac{2\rho}{\omega\mu}}=\sqrt{\frac{\rho}{\pi f\mu}}.

\]

Using illustrative copper parameters near 20 degrees Celsius,
$\rho\simeq1.72\times10^{-8}\,\Omega\mathrm m$
and
$\mu\simeq\mu_0$,
the calculated depth is approximately 9.3 mm at 50 Hz, 4.2 mm at 250 Hz
and 3.5 mm at 350 Hz. These are results of the good-conductor model, not
cable-sizing tables. Temperature, conductor geometry, stranding,
metallic enclosures and phase arrangement affect actual AC
resistance.

Skin effect produces nonuniform current density
within a conductor. Proximity effect is redistribution
caused by the magnetic fields of nearby conductors. In busbars and
windings, spacing, phase stacking, parallel-bar arrangement and field
direction can be as important as cross-section. The rule
$R_{\mathrm{ac},h}=\sqrt h\,R_{\mathrm{ac},1}$
must not be applied to every conductor as a universal design law.

Balanced three-phase conductor loss is

$$P_{\mathrm{cond}}=3\sum_{h=1}^{H}I_h^2R_{\mathrm{ac},h}.$$

\[

P_{\mathrm{cond}}=3\sum_{h=1}^{H}I_h^2R_{\mathrm{ac},h}.

\]

Where neutral current exists, add

$$P_N=\sum_{h=1}^{H}I_{Nh}^2R_{N,\mathrm{ac},h}.$$

\[

P_N=\sum_{h=1}^{H}I_{Nh}^2R_{N,\mathrm{ac},h}.

\]

A single neutral conductor’s loss is not multiplied by three. Under
unbalanced loading, compute each phase separately and then sum over
phases and frequencies.

4.2. Transformer
heating is not just copper loss

Transformer heating includes no-load loss, resistive loss, winding
eddy-current loss and stray loss in structural metal. Within a commonly
used approximation for winding eddy-current loss, the contribution at
order
$h$
receives an
$h^2$
weighting. An illustrative harmonic loss factor is

$$F_{\mathrm{HL}}=\frac{\sum_{h=1}^{H}h^2I_h^2}{\sum_{h=1}^{H}I_h^2}.$$

\[

F_{\mathrm{HL}}=\frac{\sum_{h=1}^{H}h^2I_h^2}{\sum_{h=1}^{H}I_h^2}.

\]

The example spectrum gives
$F_{\mathrm{HL}}\simeq4.446$.
This does not mean total transformer losses become
4.446 times larger. The factor weights the relevant winding eddy-current
term; current level and the reference share of that loss still matter.
No-load loss, DC resistive loss and other stray losses must not be
folded indiscriminately into the same multiplier.

IEEE C57.110-2018 addresses transformer capability under
nonsinusoidal load currents. Application requires checking scope, test
reports and loss data. A one-factor calculation is not a thermal
certification of a 500 kV autotransformer or a three-winding unit.
[R6]

Internal delta circulation may heat a tertiary without appearing in
external line CT measurements. Similarly, low HV-side THD does not prove
that an LV winding or neutral is cool. A derating assessment needs
hot-spot behaviour, oil or air temperatures, load cycles, ambient
temperature and design information.

4.3. Motors, protection
and auxiliary systems

In the balanced induction-machine model, negative sequence produces a
field rotating opposite to the fundamental field. Interaction between
each harmonic field and the moving rotor can increase losses and produce
torque pulsations. Merely reducing fundamental current to keep total
stator RMS constant does not necessarily preserve rotor heating.
Rotor-current frequency follows the relative speed of the rotor and each
harmonic field; it is not simply identical to the stator harmonic
frequency.

Peaks, crest factor, zero crossings and frequency components outside
an instrument’s intended range can also affect protection, controls and
measurement. Nevertheless, a breaker trip, PLC fault or capacitor
failure should not automatically be attributed to harmonics. Overload,
voltage events, temperature, connections, protection coordination and
switching chronology must be checked. Harmonics are a testable
engineering hypothesis, not a default explanation for every electrical
fault.

4.4.
Parallel and series resonance involving capacitors

A source represented by
$R_s,L_s$
and a shunt capacitor
$C$
has approximate driving-point admittance

$$Y(\omega)=\frac{1}{R_s+j\omega L_s}+j\omega C.$$

\[

Y(\omega)=\frac{1}{R_s+j\omega L_s}+j\omega C.

\]

Neglecting resistance and load branches for an initial estimate,

$$\omega_r\simeq\frac{1}{\sqrt{L_sC}},\qquad h_r\simeq\sqrt{\frac{S_{\mathrm{sc}}}{Q_C}}.$$

\[

\omega_r\simeq\frac{1}{\sqrt{L_sC}},\qquad
h_r\simeq\sqrt{\frac{S_{\mathrm{sc}}}{Q_C}}.

\]

The three-phase short-circuit power
$S_{\mathrm{sc}}$
and capacitor reactive rating
$Q_C$
must use the same units and bus. The relation follows from
$X_{s1}\simeq U_{LL}^2/S_{\mathrm{sc}}$,
$X_{C1}=U_{LL}^2/Q_C$
and
$hX_{s1}=X_{C1}/h$.
It is a screening estimate for an inductive network, not a substitute
for a frequency scan.

For
$S_{\mathrm{sc}}=10\,\mathrm{MVA}$
and
$Q_C=0.4\,\mathrm{MVAr}$,
$h_r\simeq5$,
approximately 250 Hz. A six-pulse load can inject substantial
5th-harmonic current into this high-impedance region. Adding capacitance
to improve fundamental displacement factor can consequently worsen
voltage distortion and capacitor duty. Switching capacitor steps also
moves the resonance over time.

At parallel resonance, the impedance seen by a
harmonic-current injection can become large, increasing bus harmonic
voltage. At series resonance, one path can have low
impedance, producing large capacitor/reactor current under
harmonic-voltage excitation. These descriptions must identify the
excitation port and circuit path. A multi-branch network may exhibit
both response peaks and minima.

For a capacitor voltage component,

$$I_{C,h}=h\omega_1C V_{C,h}.$$

\[

I_{C,h}=h\omega_1C V_{C,h}.

\]

A relatively small high-order voltage can therefore make a
significant contribution to capacitor RMS current. Capacitor element
voltage, reactor current, temperature, capacitance tolerance and
protective duty all require checking, not only MSB voltage THD.

4.5.
Detuned capacitor banks are not universal harmonic filters

Define the fundamental-frequency reactor ratio

$$p=\frac{X_{L1}}{X_{C1}},\qquad h_t=\frac1{\sqrt p},\qquad f_t=\frac{f_1}{\sqrt p}.$$

\[

p=\frac{X_{L1}}{X_{C1}},\qquad
h_t=\frac1{\sqrt p},\qquad f_t=\frac{f_1}{\sqrt p}.

\]

For
$p=0.07$,
$h_t\simeq3.780$
and
$f_t\simeq189\,\mathrm{Hz}$.
This is below the 5th harmonic in a 50 Hz system. It does not establish
suitability for every network containing a 3rd harmonic or
interharmonics. Source-side impedance, background distortion and the
current attracted into the branch still need to be assessed.

For the ideal series branch at fundamental frequency,

$$U_{C1}\simeq\frac{U_{\mathrm{bus},1}}{1-p}.$$

\[

U_{C1}\simeq\frac{U_{\mathrm{bus},1}}{1-p}.

\]

Adding a reactor without reconsidering capacitor voltage rating is
therefore unsafe as a design method. The relation uses voltages in the
same per-phase equivalent; the actual star or delta assembly must be
referred correctly.

4.6. THD, TDD and
power factor must remain distinct

Total demand distortion uses a demand-reference current rather than
instantaneous fundamental current:

$$\mathrm{TDD}=\frac{\sqrt{\sum_{h=2}^{H}I_h^2}}{I_L}\,100\%.$$

\[

\mathrm{TDD}=\frac{\sqrt{\sum_{h=2}^{H}I_h^2}}{I_L}\,100\%.

\]

A facility with 20 A harmonic RMS, 100 A fundamental current at the
measurement instant and 500 A reference demand has
$\mathrm{THD}_I=20\%$
but
$\mathrm{TDD}=4\%$.
These numbers are not contradictory; they use different denominators.
The precise definition of
$I_L$
follows the applicable document and must not be replaced arbitrarily by
a breaker rating. [R1, R10]

For a balanced three-phase system with approximately sinusoidal
voltage,

$$\mathrm{PF}=\frac{P}{S}\simeq \frac{\cos\varphi_1}{\sqrt{1+\mathrm{THD}_I^2}},$$

\[

\mathrm{PF}=\frac{P}{S}\simeq
\frac{\cos\varphi_1}{\sqrt{1+\mathrm{THD}_I^2}},

\]

where THD is entered as a decimal. Capacitors can improve
$\cos\varphi_1$
without eliminating current distortion. When voltage is distorted or the
system unbalanced, the expression is not a general identity. For a
balanced periodic system, active power can be calculated as

$$P=3\sum_{h=1}^{H}V_hI_h\cos(\varphi_{Vh}-\varphi_{Ih}),$$

\[

P=3\sum_{h=1}^{H}V_hI_h\cos(\varphi_{Vh}-\varphi_{Ih}),

\]

using consistently defined phase voltage and current. DC power, if
present, must be added separately.

4.7.
Vietnamese requirements and the role of IEEE 519

The Vietnamese reference checked for this monograph is Article 8 of
Circular 05/2025/TT-BCT, as consolidated in 38/VBHN-BCT following
Circular 46/2025/TT-BCT. The amendment generally took effect on 22
September 2025, subject to the specifically stated exceptions. The
following tables are based on that source; each project still needs to
check applicable instruments, connection agreements and requirements at
the contractual date. [R10, R11]

Voltage limits from Table 4a:

“Not specified” in the 500/220 kV row does not mean that any
individual harmonic is harmless or that contractual requirements
disappear. It describes only the corresponding cell of the cited
regulatory table.

Current limits for loads connected to distribution networks,
Table 4c:

For 220/500 kV, Article 8 sets a current TDD limit of 3%. Generating
plants connected to distribution networks have a separate table: 3%
total and 2% individual at 110 kV; 5% total and 4% individual at MV/LV.
The load table must not automatically be applied to generation.
[R10]

Under this Vietnamese instrument,
$I_L$
relates to fundamental current at maximum load or generation, using the
average of the monthly maxima over the previous 12 months; for new
connections or missing history, the maximum over the measurement period
is used as specified. Article 8 also requires 95% of values over at
least one week, recorded at 10-minute intervals, to remain within the
limits. The 10-minute interval is not the ADC sampling rate for
resolving a 250 Hz waveform: the instrument samples internally at a
suitable rate and produces the required reporting values. [R10]

IEEE 519-2022 is a system harmonic-control standard addressing the
point of common coupling, PCC, under steady-state
conditions. It is not an individual-drive product certification. Its
current limits must be read with the relevant voltage class, harmonic
band and
$I_{\mathrm{sc}}/I_L$
ratio. A blanket statement that “THD must be below 5%” is not a
universal application of IEEE 519. IEEE lists the 2022 edition as active
at the source-check date. [R1]

IEEE requirements do not automatically constitute a statutory penalty
mechanism in Vietnam. They may be incorporated into design criteria,
investor specifications or contracts alongside national requirements.
Where criteria differ, the project must identify which obligations
apply, their respective measurement locations and how simultaneous
compliance will be demonstrated. Replacing a national table with an
international table without explanation is not a defensible compliance
method.

4.8.
Polluter responsibility requires attribution, not assumptions

A harmonic-producing customer cannot reasonably treat the utility’s
delta winding as a free, unlimited filter. Utility equipment has finite
thermal capability; positive- and negative-sequence harmonics can pass
through, and zero-sequence circulation still creates loss. Emission
control at the agreed boundary cannot be dismissed by pointing to a
transformer vector group.

The opposite error is equally important: distorted voltage at the PCC
does not prove that one nearby customer caused all of it. Background
harmonic voltage, a utility capacitor resonance and simultaneous
emissions from several users may contribute. Even the sign of harmonic
active power,

$$P_h=\Re\{\underline V_h\underline I_h^*\},$$

\[

P_h=\Re\{\underline V_h\underline I_h^*\},

\]

should not be used alone to assign liability in a multi-source
network with changing impedance angles.

Article 8(5) of the cited Vietnamese instrument provides for checking
or independent testing in the specified relationship between
transmission-grid users and the transmission utility. Where testing
identifies the party causing the violation, that party bears the
specified checking, verification, damage and remediation costs. This is
not an automatic “harmonic tax” charged for every percentage point of
THD. Nor should the clause’s exact scope be extended without examining
other applicable provisions and contracts. [R10]

A defensible attribution process combines synchronized PCC and branch
measurements, switching states, operating loads, CT/VT verification, an
impedance model and controlled load-change tests. The aim is to separate
emission, network amplification and damage, then select the most
effective remedy rather than blame whichever converter or capacitor is
nearest the meter.

4.9. A reproducible
loss and payback calculation

Continue with the Chapter 1 spectrum. Assume that the AC resistance
of one phase of the particular circuit section being
assessed is
$0.010\,\Omega$
at fundamental frequency. At orders 5, 7, 11 and 13, assume resistance
multipliers of 1.4, 1.6, 2.0 and 2.2. These are exercise inputs, not
standard correction factors for every cable or transformer.

Additional loss from the four harmonic components is

$$\Delta P_h=3(200^2\times0.014+140^2\times0.016 +90^2\times0.020+70^2\times0.022)=3430.2\,\mathrm W.$$

\[

\Delta P_h=3(200^2\times0.014+140^2\times0.016
+90^2\times0.020+70^2\times0.022)=3430.2\,\mathrm W.

\]

Suppose a filter reduces each harmonic amplitude by
80%, leaving 20% residual current. Harmonic resistive loss in that
section is then reduced by 96%, not 80%:

$$\Delta P_{\mathrm{save,gross}}=(1-0.2^2)\times3.4302 =3.292992\,\mathrm{kW}.$$

\[

\Delta P_{\mathrm{save,gross}}=(1-0.2^2)\times3.4302
=3.292992\,\mathrm{kW}.

\]

Assume the filter itself consumes 2.0 kW in the assessed operating
state:

$$\Delta P_{\mathrm{save,net}}=3.292992-2.0=1.292992\,\mathrm{kW}.$$

\[

\Delta P_{\mathrm{save,net}}=3.292992-2.0=1.292992\,\mathrm{kW}.

\]

Use 6000 operating hours/year, an assumed electricity value of VND
2500/kWh, investment of VND 400 million and additional maintenance of
VND 5 million/year:

This case is deliberately unattractive when only the assessed
section’s electricity savings are counted. A generic
“whole-factory energy saving” must not be inserted to manufacture a
short payback. Filter losses must not be omitted, and total transformer
loss must not be treated as fully removable by an AHF.

A broader net-present-value formulation is

$$\mathrm{NPV}=-C_0+\sum_{t=1}^{n} \frac{B_{E,t}+B_{R,t}+B_{K,t}-C_{O,t}}{(1+r)^t} +\frac{S_n}{(1+r)^n}.$$

\[

\mathrm{NPV}=-C_0+\sum_{t=1}^{n}
\frac{B_{E,t}+B_{R,t}+B_{K,t}-C_{O,t}}{(1+r)^t}
+\frac{S_n}{(1+r)^n}.

\]

Here
$B_E$
is energy benefit,
$B_R$
is probability-based reliability benefit,
$B_K$
is avoided or deferred future investment,
$C_O$
is additional operating and maintenance cost, and
$S_n$
is residual value. Avoided replacement cost and the life-extension value
of the same asset must not be counted twice.

If avoided failures are estimated to save VND 80 million/year, their
frequency and consequences need evidence. Adding that explicitly assumed
reliability benefit to this example raises total annual benefit to
approximately VND 94.39 million and reduces simple payback to
approximately 4.24 years. The difference is an economic assumption, not
evidence that the filter suddenly saves more electrical energy.

4.10. Asset life: from hot
spots to risk

A relative Arrhenius-type thermal-ageing model is

$$F_A=\exp\left[\frac{E_a}{R_g} \left(\frac1{T_{\mathrm{ref}}}-\frac1{T_{\mathrm{hot}}}\right)\right].$$

\[

F_A=\exp\left[\frac{E_a}{R_g}
\left(\frac1{T_{\mathrm{ref}}}-\frac1{T_{\mathrm{hot}}}\right)\right].

\]

Temperatures must be in kelvin. Activation energy, reference
temperature and failure mechanism must match the material. A rule such
as “every six-degree increase halves life” cannot be imposed universally
on insulation systems, capacitors, semiconductors and moisture
conditions.

For a varying load cycle, equivalent ageing is

$$L_{\mathrm{eq}}=\int_0^{t_{\mathrm{obs}}}F_A(t)\,dt.$$

\[

L_{\mathrm{eq}}=\int_0^{t_{\mathrm{obs}}}F_A(t)\,dt.

\]

Harmonics can increase hot-spot temperature and accelerate ageing,
but one THD measurement cannot establish exactly how many service years
a transformer has lost. A useful estimate requires a thermal model,
ambient and moisture information, loading history and condition
assessment. Economic evaluations should show scenarios and sensitivities
rather than a precise-looking life prediction with missing inputs.

Chapter
5. Filtering technologies and engineering solutions

5.1. Define the
objective before selecting equipment

Possible objectives include lower PCC current emissions, lower
voltage distortion at a sensitive bus, lower neutral current, avoidance
of capacitor resonance or recovery of thermal capacity. These objectives
overlap but are not identical. An MSB-connected filter can improve
current through the supply transformer without reducing the harmonic
current in the cable between the MSB and a drive. Load-level filtering
can reduce branch losses more effectively, but equipment count,
distributed maintenance and installation costs increase.

Selection should begin with operating-state spectra, phase
information, network impedance, earthing structure and acceptance
limits. Guaranteed results should be tied to a defined operating
envelope. A good full-load measurement does not establish light-load
performance, standby-generator compatibility or acceptable behaviour
with one filter module out of service.

5.2. Passive LC filtering

A single-tuned branch has impedance

$$Z_f(\omega)=R_f+j\left(\omega L_f-\frac1{\omega C_f}\right), \qquad f_t=\frac1{2\pi\sqrt{L_fC_f}}.$$

\[

Z_f(\omega)=R_f+j\left(\omega L_f-\frac1{\omega C_f}\right),
\qquad f_t=\frac1{2\pi\sqrt{L_fC_f}}.

\]

Near tuning, its low impedance attracts harmonic current. Energy does
not disappear: current is redirected, reactive energy is exchanged
between L and C, and real losses occur in resistance, core material and
dielectric. The branch also supplies fundamental reactive power,
requiring checks at light load and across capacitor switching
states.

Passive filters offer comparatively simple construction and can
handle large current or power at target frequencies. Their weaknesses
include dependence on source impedance, load spectrum and component
tolerance, together with the possibility of creating new resonances.
Multiple tuned branches or damped high-pass arrangements are possible,
but all branches must be assessed as an interacting network rather than
selected independently from the largest individual spectrum bar.
[R7]

5.3.
Active harmonic filtering: operation and block diagram

A shunt active harmonic filter is a controlled power converter. With
source current
$i_s$
and filter current
$i_c$
defined as entering the load node,

$$i_s+i_c=i_L,\qquad i_c^*=i_L-i_s^*.$$

\[

i_s+i_c=i_L,\qquad i_c^*=i_L-i_s^*.

\]

If the desired source current is balanced fundamental active current,
the controller estimates
$i_s^*$
from its voltage reference and required power and injects the
difference. The source still supplies AHF losses and DC-link energy
regulation. The filter is not a source of free energy.

GRID ---- Source CT ----o---------------------- NONLINEAR LOAD
             is         |                            iL
                        | ic
                  [ Coupling inductor ]
                        |
                    [ PWM inverter ]
                        |
                   [ DC-link capacitor ]
                        ^
                        |
Load/source CT --> Component extraction --> Compensation reference
Voltage sensing -> PLL / fundamental reference -> Current control -> PWM
DC voltage      -> DC-energy control ----------------------^

Implementations may use synchronous-reference-frame extraction,
adaptive methods, resonant controllers or other algorithms; the diagram
does not prescribe a single method. With distorted or unbalanced supply
voltage, fundamental positive-sequence extraction and current-limiting
strategy become important. Bandwidth, delay, switching frequency,
coupling impedance and grid interaction constrain compensation.
“Filtering every harmonic to infinite order” is not a technically
meaningful claim.

The example’s remaining current THD is approximately 5.389% if
fundamental current is unchanged. This alone does not establish PCC
compliance. TDD must be calculated with the proper
$I_L$,
and individual harmonics, voltage background and operating-state
criteria must also be checked.

5.4.
Three-wire, four-wire and neutral-current requirements

A converter without the required neutral-current path cannot
independently inject a nonzero sum of three phase currents at its
connection. A filter intended for positive- and negative-sequence
compensation must therefore be distinguished from one genuinely able to
compensate neutral zero-sequence current. Some equipment can be
installed in three- or four-wire networks but does not provide neutral
correction. The phrase “three or four wires” is not sufficient evidence
of neutral compensation; dedicated neutral-harmonic products state that
function explicitly. [R9]

Selection must consider phase compensation current, neutral rating,
neutral topology, permissible unbalance, voltage range, thermal derating
and the allocation of capacity between harmonic filtering and
fundamental reactive compensation. The nameplate current is not
necessarily available entirely for harmonics when other functions
operate simultaneously.

For orthogonal components, a preliminary current budget is

$$I_{\mathrm{AHF,rms}}^2\simeq \sum_{h=2}^{H}I_{c,h}^2+I_{c,1Q}^2+I_{c,1-}^2+I_{c,10}^2.$$

\[

I_{\mathrm{AHF,rms}}^2\simeq
\sum_{h=2}^{H}I_{c,h}^2+I_{c,1Q}^2+I_{c,1-}^2+I_{c,10}^2.

\]

Individual phase, neutral and peak-current limits still require
separate checks. Reducing the example spectrum by 80% needs
approximately
$0.8\times269.444=215.555\,\mathrm A$
of harmonic RMS compensation per phase in the balanced model. If 150 A
of fundamental reactive compensation is also required, the ideal RMS
budget becomes

$$\sqrt{215.555^2+150^2}\simeq262.62\,\mathrm A,$$

\[

\sqrt{215.555^2+150^2}\simeq262.62\,\mathrm A,

\]

before expansion allowance, derating or crest-factor limits.
Selecting a 200 A filter merely because the largest single harmonic is
200 A is an incorrect way to aggregate the spectrum.

5.5. Six-,
twelve- and twenty-four-pulse rectification

Harmonics can also be reduced at their source by supplying several
rectifier bridges from phase-displaced transformer secondaries and
summing their currents on a common primary. Ideally, a
$p$-pulse
converter has characteristic orders

$$h=pk\pm1.$$

\[

h=pk\pm1.

\]

Six-pulse rectification has 5, 7, 11 and 13 among its low orders.
Twelve-pulse rectification ideally cancels the characteristic 5th and
7th, leaving 11 and 13 as the first pair. Twenty-four-pulse
rectification has 23 and 25 as its first characteristic pair. Balanced
voltages, appropriate current sharing and correct phase arrangements are
necessary. [R7]

For two six-pulse groups with relative fundamental displacement
$\theta$,
let
$s=+1$
for positive sequence and
$s=-1$
for negative sequence. Under a consistent phase convention, the relative
harmonic phase after referral to the common side has the form

$$\Delta\psi_h=(h-s)\theta.$$

\[

\Delta\psi_h=(h-s)\theta.

\]

For
$\theta=30^\circ$,
the 5th has
$(5-(-1))30^\circ=180^\circ$
and the 7th has
$(7-1)30^\circ=180^\circ$.
Equal currents cancel ideally. The 11th and 13th produce a 360-degree
difference and therefore remain.

For four six-pulse groups spaced by 15 degrees,

$$\underline I_{h,\mathrm{sum}}\propto \sum_{m=0}^{3}e^{jm(h-s)15^\circ}.$$

\[

\underline I_{h,\mathrm{sum}}\propto
\sum_{m=0}^{3}e^{jm(h-s)15^\circ}.

\]

The geometric sum vanishes for several lower characteristic orders,
while 23 and 25 become co-phasal. This is phasor cancellation between
converter groups, not universal absorption by a delta winding. Unequal
voltage, impedance, phase displacement or DC loading leaves residual
characteristic harmonics; noncharacteristic harmonics and voltage
background still require assessment.

Multipulse systems are attractive where the supply topology of a
large load can be controlled, especially in new projects. Replacing
transformers and supplies for many small existing drives may be less
economical than group filtering. An active front end is another option,
but it also has switching emissions, filters and grid-compatibility
requirements. It should not be described as producing “no harmonics” at
every frequency.

5.6. Technology and economic
comparison

These cost descriptions are structural comparisons, not October 2026
market quotations. Procurement comparisons must use the same work scope,
guaranteed spectrum, source conditions and acceptance criteria.
Comparing only cabinet prices while omitting transformers, cables,
ventilation or shutdown time creates false economy.

5.7. Survey,
modelling and commissioning workflow

Define the case. Establish the regulatory PCC and
internal sensitive buses. Obtain the single-line diagram, vector groups,
neutral and earthing arrangements, short-circuit data, cable parameters,
capacitors and reactors, converter inventory, generator modes and
operating schedule. A kW nameplate does not replace a current spectrum,
and transformer kVA does not replace network short-circuit power.

Measure. Prefer Class A methods where
reproducibility and formal comparison are important, with a suitable
instrument and CT/VT chain. IEC 61000-4-30:2025 is the fourth edition
and has a July 2026 corrigendum, COR1:2026. Record the edition and
functions actually implemented by the instrument; do not assume every
handheld meter already meets the newest edition. Transducer behaviour
remains a separate issue, particularly at 110–500 kV. [R4, R5]

Model. Build operating-state harmonic magnitudes and
angles, scan driving-point and transfer impedances, and examine
capacitor switching and weak-grid configurations. Use time-domain models
when controls, saturation or transients invalidate fixed
harmonic-current sources. Preserve model versions, background
assumptions, tolerances and scenarios so another engineer can reproduce
the study.

Select. Compare load-level, group-level and PCC
filtering. Evaluate current rating, thermal duty, reactive power, space,
maintenance and shutdown cost together. A hybrid solution may retain
suitably detuned compensation, use passive filtering for stable
background loads and active filtering for variable components. The
interaction must still be simulated; combining individually good devices
does not automatically produce a stable system.

Accept. Compare before and after under genuinely
comparable operating conditions, not just a favourable low-load
screenshot. Report voltage THD, current TDD, individual spectrum,
neutral current, capacitor states, temperatures and source
configuration. Guarantees should identify background voltage,
source-strength range, active loads and degraded operation with one
module unavailable. The regulatory time-based criterion discussed in
Chapter 4 requires a sufficient record for statistical assessment, not a
few seconds of measurement.

5.8. Engineering conclusion

Harmonic management is the management of interaction between emission
sources and an electrical network. Fourier analysis identifies frequency
components. Fortescue analysis identifies their three-phase structure
and return paths. Impedance models establish where currents flow and
where voltage distortion is amplified. Thermal and economic models
establish which consequences justify intervention.

A 500 kV network is not immune to resonance. A delta winding is not
loss-free. Low THD does not replace PCC TDD assessment, and a large AHF
nameplate does not establish life-cycle value. A sound solution has
transparent assumptions, verifiable measurements, a model appropriate to
the actual equipment and reproducible acceptance criteria.

Appendix A.
Reproducing the figures and arithmetic

The accompanying harmonic_examples.py generates six
figures in PNG and SVG, waveform data in CSV, and
calculated_results.json. Run it with Python 3.10 or later,
NumPy and Matplotlib:

python harmonic_examples.py --output results

The script declares RMS currents, phase angles, resistance values,
capacitor ratings and economic assumptions. It also tests the Fortescue
transformation for every order from 1 to 25. No network access or
proprietary data is required. The figures are calculation illustrations
and must not be relabelled as QuangAnhcons field measurements or
substituted for commissioning evidence.

The resonance model is a balanced lumped RL–C network. It does not
fully represent EHV distributed lines, transformer magnetic circuits,
AHF control loops, interharmonics or switching ripple. Modified inputs
should be interpreted within that scope.

Appendix B.
Self-assessment questions

Question 1. Is odd-function symmetry sufficient to
exclude the 2nd harmonic? No. The counterexample
$\sin(2\omega_1t)$
is an odd function. The complete waveform must satisfy half-wave
antisymmetry.

Question 2. Is a cyclically balanced 6th-harmonic
set negative sequence? No. Its phase displacement is an integer multiple
of 360 degrees, so it is zero sequence. Its actual existence depends on
source symmetry and available circuit paths.

Question 3. Does the delta of a Yn/Yn/Δ transformer
guarantee zero-sequence isolation between grounded-wye sides? No. Solve
the sequence network and current division.

Question 4. If each harmonic amplitude is reduced by
80%, how much does its resistive loss fall with unchanged resistance? By
96%. Filter losses must then be subtracted to obtain net savings.

Question 5. Does a measured 12% current THD
establish violation of IEEE 519 or Vietnamese requirements?
No. Measurement location, TDD denominator, voltage class, connection
type, individual orders and time statistics are still needed.

References and source notes

The explanations and derivations are independently authored. The
standards below are not reproduced in full. Contractual or detailed
design use requires the appropriate authorized edition.

[R1] IEEE. IEEE 519-2022, IEEE Standard for
Harmonic Control in Electric Power Systems. Official scope, PCC
application and status: https://standards.ieee.org/ieee/519/10677/.

[R2] James L. Kirtley Jr., MIT OpenCourseWare.
Introduction to Symmetrical Components, 6.061, Chapter 4.
Transformation and sequence-network foundations: https://ocw.mit.edu/courses/6-061-introduction-to-electric-power-systems-spring-2011/7603169f5d8514b769a46fa900b79a57_MIT6_061S11_ch4.pdf.

[R3] Schweitzer Engineering Laboratories. A
Tutorial on Symmetrical Components. Manufacturer reference on
sequence networks and grounding: https://selinc.com/api/download/100688.

[R4] IEC. IEC 61000-4-30:2025, Power quality
measurement methods, fourth edition, corrected July 2026: https://webstore.iec.ch/en/publication/71611.

[R5] IEC. IEC 61000-4-30:2025/COR1:2026, published
17 July 2026: https://webstore.iec.ch/en/publication/113789.

[R6] IEEE. IEEE C57.110-2018, Transformer
Capability when Supplying Nonsinusoidal Load Currents. Official
information on purpose, edition and application: https://standards.ieee.org/ieee/C57.110/5948/.

[R7] ABB. Technical Guide Book, Guide
No. 6, Guide to harmonics with AC drives. Principles of
multipulse rectification and filtering; historical cost figures are not
used as current prices: https://library.e.abb.com/public/a209d9dc29b24b789e0650b8de9c3f22/TechnicalGuideBook_EN_3AFE64514482_RevI.pdf.

[R8] Schneider Electric, Electrical Installation
Guide. Origine des harmoniques. Harmonic-current
sources and network impedance: https://fr.electrical-installation.org/frwiki/Origine_des_harmoniques.

[R9] Schneider Electric. Neutral-compensation
capability references: PowerLogic AccuSine PCSn Neutral Harmonic
Filters, https://www.se.com/vn/vi/product-range/65657-powerlogic-accusine-pcsn-neutral-harmonic-filters/;
the cited PCS+ configuration lists three-/four-wire network connection
but no neutral correction: https://eshop.se.com/sa/pcsp200d5ip00-active-harmonic-filter-200-a-380-480-v-ac-ip00-enclosure.html.

[R10] Vietnam Ministry of Industry and Trade.
Circular 05/2025/TT-BCT, consolidated instrument 38/VBHN-BCT, Article 8
and Tables 4a–4c. EVN-hosted official text, printed pages 14–15 visually
checked: https://www.evn.com.vn/userfile/User/giangtcdl/files/2025/8/VBHN38-20250814160032345.pdf.
Original instrument: https://chinhphu.vn/?classid=1&docid=212774&orggroupid=4&pageid=27160.

[R11] Vietnam National Legal Database. Circular
46/2025/TT-BCT, effective-date and related-instrument information: https://vbpl.vn/tw/Pages/ivbpq-vanbanlienquan.aspx?ItemID=180486.