The Fast Hartley Transform Used in the Analysis of

Electrical Transients i n Power Systems

G. T. Heydt

Electrical and Communications Systems

National Science Foundation

Washington, DC 20550

Abstract

This paper concerns the use of the fast Hartley

transform in the analysis of electrical transients in power sys-

tems. Both periodic and nonperiodic signals are studied.

The fast Hartley transform (FHT) is a real transform which

is very similar to the fast Fourier transform (FPT). The

FHT has many of the useful properties of the FFT, however

the FIJT is a purely real transformation and computational

advantages are associated with this fact. This paper

addresses several of the properties of the FHT as used in

power systems including representation of initial conditions,

convolution properties and symmetries, filtering and window-

ing, aliasing errors in both the Hartley frequency and time

domains, and certain properties related to time scaling and

zero padding. Practical methodologies are presented for the

calculation of electrical transients; these signals may occur in

electric power systems due to nonperiodic phenomena such as

switching surges and transformer inrush or periodic

phenomena such as those found in electronic power process-

ing devices. The methodology presented is useful for power

quality assessment. The basis of the method is Ohm’s law

which, in the time domain, is the convolution

v(t) = z * i(t).

This convolution is rendered to be a sum of products in Hart-

ley space, and under some conditions, these products collapse

into a single product which is identical to the FFT case. The

usual parameters found in a power system result in frequency

band limits which translate as especially rapid calculation

possibilities in Hartley space. A discussion of the potential

pitfalls cf the method are also presented.

I. The Hartley Transform

The Hartley transform [l] of time function x(t) is X(Y)

00

X(Y) = x(t) cas(t) d t (1) -03

where cas(.) is the cosine - and - sine function

cas(t) = cos(t) + sin(t) .

The Hartley transform enjoys many of the familiar properties

of the Fourier transform, but the Hartley transform is purely

real. Bracewell has written a definative text on the Hartley

transform and its properties [2] (in fact, the transform is

sometimes termed the Hartley-Bracewell transform).

Marly of the most useful properties of the Hartley

transform are also useful properties in Fourier transform

methodo!ogy. For example, convolution in time is rendered

to be a product,

oc,

Z ( t ) = x * y(t) = J x(t -.)Y(T)dT --M

1 Z ( V ) = - [X(U)Y(V) 2 + X(-V)Y(Y) + X(Y)Y(-Y) - X(-U).(-@].

Also, there are analogs to both the discrete Fourier transform

and the fast Fourier transform algorithm. The discrete Hart-

ley transform (DHT) is

N - 1

i = O X(kAv) = x(iAT) cas(ikATAv) (3)

where x(iAT) is periodic with period NAT and resolution A u

is 27r/(NAT). Evidently, (3) is identical to the discrete

Fourier transform with cas(.) replacing the complex exponen-

tial. The notation v is used to distinguish the Hartley

transform from the Fourier transform. In in the latter, w or

f is usually used as the transform variable.

Equation (3) has been evaluated in a fast algorithm

known as the'fast Hartley transform (FHT) [3,4]. The FHD

is an exact calculation of the DHT (in the same way that the

fast Fourier transform is an exact calculation of the discrete

Fourier transform).

Before leaving preliminaries related to the Hartley

transform, note that the Hartley and Fourier transform pairs

and the discrete Hartley and Fourier transform pairs

X(kAv) ++ x(iAT)

X(kAR) ++ x(iAT)

are related by

X(kAv) = L Re(X(kAn) -I- Im(X(kAR))]An = A y

X(w) = [ Ev(X(v)) - jOd(X(v))]v=L,

X(kAR) = [Ev(X(kAv)) - jOd(X(k i \u ) ) ]~ ,=~n

where Re, Im, ET, and Od denote the real part, imaginary

part, even part, and odd part respectively. The latter two

parts are

1 Ev(X(v)) = 5 ( X ( Y ) + X(-Y))

1 2

Od(X(v)) = - (X(Y) + X(-U)).

Using these relationships, the familiar Fourier transform pro-

1814

perties (e.g., time scaling, modulation, etc.) can readily be

transcribed to Hartley notation. Bracewell has tabulated the

most important of these relations in 121.

II. Calculation and Symmet ry Considerations

In this paper, the bus voltage in a power system, v(t) is

calculated. This is a time function and the voltage itself is

usually measured with respect to ground. Let the bus vol-

tage be the result of some nonsinusoidal injection current,

i(t). If z(t) denotes the inverse transform of the transfer

impedance Z(.d) between the busses at which v(t) is measured

and at which i(t) is injected, then

v(t) = z * i(t)

1 2

V(U) = - (Z(v)I(v) - Z(-U)I(Y) - Z(U)I(-U) - Z ( - l ~ ) I ( - L ~ ) ~ )

Equation (4) is purely real, a fact which suggests computa-

tional convenience over the Fourier transform. However, the

several terms in (4) are clearly less convenient than the

Fourier analog,

For this reason, special cases and symmetries are studied to

simplify (4). Heydt, Olejniczak, Sparks, and Viscito first sug-

gested the use of (4) for the calculation of bus voltages due to

nonsinusoidal injection currents in [5]. The main application

is in the area of assessment of the impact of power electronic

loads. These loads typically have load currents i(t) which are

band limited, periodic, and nonsinusoidal. For example,

most rectifier or inverter devices are generally considered to

be band limited to below the 49th harmonic of the funda-

mental frequency. Power modulated devices rarely involve

frequencies above 20 K H z . Harmonic impact studies are

often limited to the 25th harmonic of the power frequency.

The symmetry properties of I(kAv) are readily found

noting that if i(kAT) is an even periodic function, I(kAn) is

purely real, and X(u) is purely an even function,

x(nAT) = x(-nAT) -+ X(n4v) = X(-nAL/).

Similarly, if i(kAT) is odd, X(nAc/) is also odd

x(nAT) = -x(-nAT) -+ X(nAu) = X(-nAu). III. Errors and the FHT

The essence of the proposed algorithm is the following These results are also shown using (3) directly along with the

periodicity property c a d = cas(8+2~). Under even and odd

symmetry of x(nAT), the convolution property becomes, for

the case ii. Evaluate the FHT of i(t).

steps:

i(t) at = nAT.

z(nAT) = x * y(nAT) iii. Calculate the FHT of z(t) of sampling Z(lir) at

w = nAn. when either x ( n A T ) and/or y ( n A T ) is even

iv. Convolve in Hartley space.

v.

There are errors associated with each step and they are

Z(nAu) = X(nAu) Y(nAu), (5) Calculate v(t) using the inverse FHT.

when e i ther x (nAT) and/or (ynAT) is odd

Z(nAv) = - X( -n A u) Y( -n A U). (6) briefly discussed in this section. Note that the desired end

The odd half wave symmetric case, result is v(t) a t times t = nAT,

x(nAT) = x(n + (N/2)),

inay be analyzed by resolving x into a purely even and purely

w

v(nAT) = Z(w)I(w) eJwnAT dlir * (7)

odd function In connection with step (i), enumerated above, the sam-

pling of i(t) results in frequency domain aliasing as evidenced

by the overlapping versions of I(w) which make up the

discrete time Fourier transform I(eJwAT) [5]. The aliasing in

Hartley space is identical since I(u) is readily obtained at

Then E(nAu) and O(nAu) are purely even and odd respec- each value of nAR from the discrete Fourier transform.

t,ivcly. Convolutions with x(nAT) are therefore analyzed by Taking the FHT of the samples i(nAT) yields a set of

superimposing signals e and o and using the special results discrete Hartley transform values which are samples of

(5) and (6) respectively. Figure 1 shows an example of an

1 2 1 2

e(nAT) = - (x(nAT) + x(-nAT))

o(nAT) = - (x(nAT) - x(-nAT))

x(nAT) = e(nAT) + o(nAT).

I(cas nAv) at spacing

odd half wave symmetric x(nAT) (a continuous depiction is 2n NAT

Au=-. shown in the figure for illustrative convenience; however,

tficse signals are sampled at time resolution AT). The even Taking the discrete (or fast) Hartley transform results in no

arid odd parts of x are e and 0. The discrete Harley further loss of information since the samples i(nAT) may be

transforms are E and 0 respectively. exactly recovered from the FHT provided that the length of

the FHT, N, is larger than the number of the time domain

samples.

Thc fundamental concept of calculating v(nAT) using

the fast Hartley transform is to numerically process

The Hartley analog of time domain convolution is multi-

plication, in this case by I(cas nAu). Use of samples of

Z(nAu) for this convolution is preferable over samples of z( t )

because of the natural band limitation of Z(Au) (i.e., Z(w)) in

Z(nAu)I(nAv) + Z(-nAv)I(nAu)

1 + Z(nAu)I(-nAv) - (n-Av)I(-nAv)

power networks. Sampling of Z(Au) results, however, in (or Eq. (5) or ( 6 ) under special cases of symmetry).

time domain aliasing. Reference to the DFT will help illus-

27r trate this point: i t is necessary to sample Z(W) at U=- NAT

1815

iii.

iv. Leakage [SI

Picket fence effect [6, 71

4 I ,-

I

t

I I t

Fig. 1. Illustration of resolution of an odd half wave

function into even and odd parts.

in order to match its frequency domain sampling rate to that

of I(w) implied by the DFT. There will be an odd number of

samples centered at w=O. It is important to note that the

algorithm uses only samples of Z(w) from a low-pass fre-

quency region whose bandwidth is dictated by the sampling

period A T of i(t). Thus, this sampling period must be

selected small enough both to sufficiently control aliasing in

i(t), and to not severely band-limit Z(w). The band-limiting

of Z(w) can be interpreted as introducing time domain

smoothing. Finally, the inverse DFT of the product of sam-

ples is taken to form i(nAT).

It is clear that the potential pitfalls of the proposed

method are identical to those of the use of the DFT for this

purpose. The main pitfalls are

i. Time domain aliasing

W . Windowing and Other Refinements

The use of the Hartley transform for circuit solutions in

band limited applications is, in effect, a variation on the use

of the Fourier and Laplace transforms for circuit solutions

(examples in power engineering are extensive and illustrated

by references [8] and [9]). What is new is the use of a real

transform and a quantitative assessment of error in band lim-

ited cases. These innovations lead one to the following

interesting areas which are suggeshed for future research:

i. A full quantitative assessment and error analysis for

real transform solutions of circuits problems.

ii. A study of novel and fast methods to calculate Z(nAv)

(e.g., [81).

iii. A serious application in which real time solutions are

required.

The question of leakage raised in the previous section

was discussed by Girgis and Ham in [SI for Fourier frequency

space. In brief, this effect is a phenomenon which occurs

when the sample wave, i(t) for example, is not sampled

sufficiently to capture all the time phenomena. Often, raising

N will satisfy this difficulty, but there are practical cases in

which this simple (but computationally ineffective) solution

may not be appropriate. For example, if i(t) is composed of

two or more periodic waves which result in a non-periodic

wave, e.g.,

i(t) = sin f i t + sin V i t, I(Av) wiil exhibit leakage for any value of N. This case

occurs in practice when a power modulator causes the ac load

current waveform to be modulated by an asynchronous

(often high frequency) signal. One alternative solution to the

minimization of leakage is to time-window the wave in ques-

tion. By this methodology, i’(t) is the windowed value ii. Time domain smoothing

1816

i'(t) = w(t) i(t).

The window w(t) should be small near 5 = 0 and t -NAT,

but it should be unity in mid-range values of t.

V. Conclusions

The fast Hartley transform has been proposed for any

electric circuit calculation involving convolution in time.

The method is especially applicable in cases in which fre-

quency band limitations occur. Such is the case in electric

power systems in which power electronic loads cause non-

sinusoidal load currents and bus voltages to occur. Special

symmetry cases have been listed which are applicable when

i(t) is even or odd.

Among the errors associated with the FHT solution of

electric power circuits are time aliasing, the picket fence

effect, time domain smoothing and leakage. The use of time

windows has been proposed to limit leakage.

VI. Acknowledgement

I would like to acknowledge the useful work of Mr.

Kraig J. Olejnicsak in the preparation of this paper.

PI

121

131

141

151

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1817

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[SI

[7]

[8]

[9]