AN OVERVIEW OF TECHNICAL CHALLENGES IN THE DESIGNOF CURRENT TRANSFORMERS

Nisha Das and Marian K. KazimierczukWright State University

Abstract: This paper presents approximate equationsthat may be used to obtain the lower cutoff frequencyfL and higher cutoff frequency f,, and thereby thebandwidth of a current transformer. The equationsare validated by a PSpice simulation. The Bode plotsfor both low-frequency range and high-frequencyrange for a selected design are presented. Also anoverview of the various problems that should beconsidered while designing a current transformer isgiven, including leakage inductance, stray capaci-tance, ringing, parasitic resonance, saturation, andmechanical clamping.

Key Words: Current Transformer, Band width.

I. INTRODUCTION

Devices for measuring current waveforms or values arecalled current probes, current transformers, current-measuring transformers, and current-monitoring trans-formers. They are simple, inexpensive, and also provideconductive electrical isolation from the rest of the circuit,especially for high voltage applications. This paper dealswith ac current transformers, which are based onmagnetic coupling principles. A typical current trans-former arrangement is shown in Figure 1.

Figure 1 .Current Probe.

A Current transformer [CT] is desired to have adequatebandwidth so that it can regenerate the measured currentapart from the uncertainties caused by the non-sinusoidalwaveforms along with the harmonics. A significantamount of literature concentrating on various issues andtheir solutions is available.Current transformers are usually designed for perfectsinusoidal conditions. But in real networks, owing to the

presence of distorted loads, current transformers have tomeasure distorted waveforms. These non-sinusoidalwaveforms, along with the significant amount ofharmonics, introduce large uncertainties resulting ingross measurement errors or malfunctioning of thedevices connected to it. Therefore, one of the mostimportant performance parameters of the currenttransformer is its bandwidth.Section II describes the construction of a current probe.The principle of operation of the current transformer isgiven in Section III. The important probe parameters and

The design challenges are given in Sections IV and V. InSection VI, the approximate equations for fL and fH arederived and bode plots for low-frequency range and high-frequency range are shown. The results obtained usingthe approximate equations are validated using the PSpiceand Matlab simulation results in Section VII. In SectionVIII, some of the important applications of the currentsensing transformers are listed.

II. CONSTRUCTION OF CURRENT TRANSFORMERS

Current probes can be classified into dc current probesand ac current probes. The dc current probes are basedon Hall effect and the ac current probes are based on themagnetic coupling principle. A typical ac current probe(current transformer) consists of a toroidal ferromagneticcore, on which a copper wire of n turns is wound. Thiswinding is terminated with a low-inductance senseresistor or burden resistor R. Thin-film chip resistorsexhibit low inductive and capacitive reactances. Theconductor carrying the measured time-varying currentacts as the primary of the current transformer. The toroidcan be clamped around the current carrying conductor.The winding wound on the toroid acts as the secondaryof the current transformer. The probe normally isconnected through a 50-Q coaxial cable to a high-impedance oscilloscope to observe the measured currentwaveform.

For better performance, current transformer cores aredesired to have high permeability, high resistivity, lowhysteresis and eddy current losses, and wide bandwidth.The most commonly used is MPP (Molybdenumpermalloy powder) core whose permeability ranges from14p to 550p. But the bandwidth reduces as thepermeability increases. For example, a current probe may

0-7803-91 45-4/05/$20.00©2005IEEE 369

have the following construction parameters: the relativepermeability of the toroidal ferromagnetic core pr = 200,the toroid mean radius r = 2 cm, the core cross-sectionalarea A = 4cm2, R = 50 Q, and number of turns, n = 500.

111. PRINCIPLE OF OPERATION OF AC CURRENTTRANSFORMERS

The principle of operation of ac current transformers isbased on combination of Ampere's and Faraday's laws[21. According to Ampere's law, the measured current ithrough the primary conductor produces a circular time-varying magnetic field Hp. The purpose of theferromagnetic toroidal core is to contain the majority ofthe magnetic field within it. The magnetic field passesthrough the secondary winding and, according toFaraday's law, induces a voltage v, across the terminals ofthe secondary winding. The voltage v, causes the currentis through the sense resistor R. The measured current is

i = n(v,/R) (1)

The current probe can be calibrated by passing a knownvalue of current through the primary conductor andmeasuring the voltage across the sense resistor. Thesensitivity of the current probe is expressed by

v RS = (V/A). (2)p n1 fl

For example, if n = 500 and R = 50, then Sp = Rln =50/500 = 0.IV/A. The current flowing through the senseresistor is much lower than the measured cufrent i.

IV. CURRENT TRANSFORMER PARAMETERS

The major parameters of the current probes are asfollows:

1. Lower cutoff 3-dB frequencyfL.2. Upper cutoff 3-dB frequencyfH.3. Bandwidth BW = fH -fL.4. Zero phase shift.5. Rise time tr and fall time tf (describing high-

frequency response).6. Droop (describing the low-frequency response).7. Sensitivity Sp8. Accuracy (%), usually 1% of the nominal

sensitivity for mid-frequencies.9. I-t capability (i.e., the ability to withstand core

saturation effects).10. Maximum peak measured primary current 4p,.11. Maximum peak secondary current I.sm12. Insulation breakdown voltage.13. Operating temperature.

V. CHALLENGES IN THE DESIGN

The technical challenges in designing high-qualitycurrent transformers are related to following issues:

1. High magnetizing inductance.2. Low leakage inductance.3. Low stray capacitance.4. Core saturation.5. Ringing.6. Parasitic resonance.7. Dependence of core permittivity $\mu$ on

frequency.8. Extra inductance created by the loop carrying the

measured current.9. Mechanical clamping.

VI. BANDWIDTHA. Model

As already mentioned, most often the current involvedwill be non-sinusoidal and may contain a wide range ofharmonics. In order to reproduce the waveform at thesecondary side with a proportional magnitude and sameshape as in the primary, the device must have adequatebandwidth. The physical circuit as well as the equivalentcircuit of the current transformer are shown in Fig. 2,where L,, is the magnetizing inductance of the secondarywinding, R, is the core parallel equivalent resistance, LIis the leakage inductance of the secondary, R, is thesecondary winding resistance, R is the burden or senseresistance, and LR is the inductance of the sense resistor.

I-

aIR

R

LR

T -

LI RsRtet to th

Sict-2C R,>sP5. cK

C

Fiur 2. Curen Trnsore. (a)......Phy..sica Cici.

(c Trnfre Moe wit Curn Source

Relete to th Seoday

370

In order to determine the bandwidth, low-frequency andhigh-frequency equivalent circuits are analyzed separ-ately. The models shown in Figs. 2 (b) and (c) are validfor a wide frequency range, i.e., low-frequency range,mid-frequency range, and high-frequency range.

The magnetizing inductance of the secondary winding isgiven by

Lm = (1 pon2A)/27r (3)

where A is the core cross-sectional area and I is the coremean length. For the given example withPr=200, r =2 cm, A =4 cm2, and n = 500,

Lm =4Pon2 A)/2tr

= [(4)(I 07)(500)2 (4 x 10-4)]/2it(2)( 1 -2)

= 200 mH. (4)

B. Low-frequency Response

The lower cutoff frequency is determined by the relativevalues of the transformer secondary winding impedanceand its sense resistance. The low-frequency equivalentcircuit [6] is shown in Fig. 3 (a).

so that the current through the sense resistor R isapproximately equal to i/n. However, at low-frequencies,the reactance of the magnetizing inductance iscomparable to the resistance RL,, = RCII(R + Rs) seen bythe magnetizing inductance Lm. As a result more currentwill flow through Lm as the frequency decreases. Hence,the current transformer behaves like a first-order high-pass filter in the low-frequency range. From Fig. 3 (a),the voltage across the sense resistance R is

RcsLmi RcR + sLmVR =R-CR n R sL,n +R (R+RR + sL,"

iR Rr s

n RC + R + Rs + Rc || (R + Rs5)Lin

iR RC sn R +R+R. 1

c S s +-

iR RC s

n RC+R+Rs S+OL(5)

Rs

(,N~~Litz <3R.)I I., K '*I JI^¢[{ gL.w2 gR;~~~~~~~~~

Hence, the transfer function of the current transformer forlow-frequencies is given by a transresistance. 4)

* 0,ai

Rs

..C,

--

I b7

R() VR -R RC: si n RC + R +RSs +a.L

whereLm =_____=LmRD=l Rcll(R+Rs)

and

=1 R|RII(R+Rs) (8)T Ls~~~~n

Therefore, the lower 3-dB cutoff frequency of the currenttransformer is given by

Figure 3. Equivalent Circuit for Determining the LowerCutoff Frequency. (a) Low-frequency Equivalent

Circuit.(b) Low-frequency Dead Circuit

This equivalent circuit is derived from the general modelshown in Fig. 2 (c) by neglecting the leakage inductanceLI, the sense resistor inductance LR, and the straycapacitance CS since L1 and LR present very lowreactances and CS presents very high reactance at lowfrequencies. The reactance of the magnetizing inductanceLm and the core parallel resistance R, must be very large

1 R, 1II(R+Rs)fL = =r2xT 2/-7,t

(9)

If RC >> R + RS , then the transresistance of the currenttransformer can be approximated by

R(s)=- VR = R s .(OR ~~~~~~~~(10)ii n S+W

The time constant T can be derived directly from Fig. 3(b), which is obtained from Fig. 3 (a) by reducing thecurrent source iln to zero.

371

(6)

(7)

For example, with Lm = 200 mH, R = 50 Q, Rs = I Q, andRC = I0 kQ, we obtain

R, (R+Rs)RLDn=-=R R R =50.74Q (11)

and

fL =RL,n 50.74 3 =40.38 Hz. (12)21Itn 2zx 200 xl 0

Setting s = jw, the transresistance of the currenttransformer becomes

Rm = (R/n)[R/(Rc+R+ Rs)][(l + j(fL/f)]05 (13)or

IRm (R/n)[R/(R,+R+ RS)][( 1+ (fL/f)2]0-5 (14)

andR,Rm = arctan(fL/f) (15)

Figs. 4 and 5 shows the Bode plots for the designedcurrent transformer for low frequencies with Lm = 200mH, RC = 10 kQ, R, = I Q, and R =S50 .

0.0

0.090 08

007

0.06

005 /

-t.D- 004!

0.03 1

0 02

0,01

10 10 10If(Hzl

Figure 4. Magnitude Bode Plot

0.09

0 08

o 07

0.06

005

0,03

002 '

0.01

1o 10 10

f (Hz)

Let us consider the step response of the currenttransformer for low frequencies. The step change in themeasured current is given by

(16)

which in the s-domain becomes

Ii(s) =-

s(17)

Hence, the voltage across the sense resistor in the s-domain is expressed by

VR(S)=Rtn(S) i(S) In R.+R+RS 1'c s 5+-

which gives the step response in the time domain as

where

VR0) R RC In Rc + R + R R

(18)

(19)

(20)

Fig. 6 illustrates the step response of current transformerfor low frequencies.

0.1

0.09

008

10 103

for Low Frequencies0.07

0.06

**cz 0055M0.041

0.031

0.021

0.01

v0 5 10 15

II(.ms)20 25 30

Figure 6. Step Response for Low Frequencies

The step response can be approximated by

10 103

Figure 5. Phase Bode Plot for Low FrequenciesVR(t) VR(01 -

t

for t << X, (21)

372

//

11

I

I 11

-IIII

I

//

/

i(t) = I u (t),

.-

from which

VR (At) = VR ()-AVR = VR(O Q1--jAt

= VROV()VR (°) VR (°)T

Therefore,

Sag= AVR - =At() rvR(O) i

Therefore, the voltage across the sense resistor R is givenby

VR = (Rj||- i,(22) i

(23)

C. High-frequency Response

Unlike the lower cutoff frequency, the higher cutofffrequency is determined by the effects of leakageinductance and stray capacitance. Figure 7 shows thehigh-frequency equivalent circuit for the currenttransformer.

RC

Hence, the transresistance of the current transformer forhigh frequencies is expressed by

Rtn (S ) VRi

RC 1nCL1 s2+ R 1A s+ 1)

\"LI RC) CL, TR )

RcC1-

1

where---el '

2X 2,r CL, KR )and

Figure 7. High-Frequency Equivalent Circuit

This model is obtained from the equivalent circuit shownin Fig. 2 (c) by neglecting Lm, RS, and LR. The reactanceof magnetizing inductance Lm is much higher that theresistance core resistance R, at high frequencies. Thesecondary winding resistance R, is much lower than thereactance of the leakage inductance L,. It is assumed thata high quality sense resistor is used so that its inductanceLR is negligibly small. Thus the equivalent circuit of acurrent transformer resembles a second order low-passfilter.

Referring to Fig. 7, the current through the leakageinductance is

Li =-i Rc

+sL, + C|| RI

i

Rc

Li RC

2 (RC__(28)

Figs. 8 and 9 shows the Bode plots for the designedcurrent transformer for high frequencies with LI = 10 pH,RC= l0klQ,R=5OQ,andC= lOpF.

U.

0.09

0.08

0.07

_ 0.06

- 0.05C- 0.041

0 03-

0.02[RC({s+0.01

(24)

CL( s+ )S+ J+106 10 1t1010

f(Hz)

Figure 8. Magnitude Bode Plot for High Frequencies

373

CLjs+ ;:J+ L J+

(25)

ij <R.wC0

nCL, s' + 2ca0s +a02(26)

(27)

---------------0-- -~~~~~

Iv

~~~~~~~~~~~~~~~~~~~~~~~~~~~k.

-LIrI,WW,-I

10o-

-20

-40

-60

-80

-100

-120

-140

-1B0

10 10f ( z)

and

J p2 IRp2 2r 27d?C

Hence, the upper 3-dB frequencytransformer is given by

(35)

of the current

IfH 1

f 2 + 2fp1 p

10e 10(36)

Figure 9. Phase Bode Plot for High Frequencies

For the imaginary poles (; < 1), the upper 3-dBfrequencyfHz kf0, where k depends on the value of 4.

For the real poles (; >> 1), the denominator of thetransresistance Rm can be represented as

s2+ r +__S+ c+1t LI RC) CL, R )

=(soiP,Xs+ p2)= s2+(iPI +cop2)+WP1p2.(29)

fHHence,

and

if

R ICpl + p2

= c +-

RCI+w, I RC

d)pl)p2 RCL, CL, RCLI

I RCCL, RCL,

(30)

(31)

I.

2RC )

For example, for R = 50 , R, = 1O kQ, LI = 1OpH, andC= lOpF,

I , 2 X ~=01003 159.15MHz. (37)-p 27zL, 2ffx10X10-6

Ip2 = =- 1 12 =318.3 MHz2nTRC 2i x50XI0X10(38)and ~~~~~~~~~~~

I(9ISX12 (31861

l(59.15x1o86)2= 142.25 MHz. (39)

Fig. 10 illustrates the step response of current transformerfor low frequencies. The rise time tr = 2.62 ps can beobserved from the plot.O.1

0.09

0.08(32)0071

0.06?which simplifies

~F0.05_r

Rc >> 1.R

0.04?(33)

0031

0021This inequality is well satisfied in practice. Thus, thefrequencies of the real poles can be approximated by

= _IT2} 2;zL

0.01

(34)

0 2 3 4t (iPs)

5 6 7

Figure 10. Step Response for High Frequencies

374

//

/

v0

I I

D. Leakage Inductance

Even though the toroidal core is designed to containmajority of the magnetic flux, there might be someleakage flux. This leakage flux behaves as an inductorand stores energy [3]. The increase in leakage inductanceresults in decrease in bandwidth, as the upper cutofffrequency fH is inversely proportional to Ll. Also, thestored energy causes voltage spikes at the leading edge ofthe voltage waveform or it produces a slope at theleading edge of the current waveform. In most cases, theleakage inductance can be minimized by proper selectionof the magnetic core material, increasing the number ofturns, and reducing the insulation of the windings.

E. Stray Capacitance

Stray capacitance consists of turn-to-turn capacitance,layer-to-layer capacitance, winding-to-winding capac-itance, capacitance between winding and the core, andcapacitance between outer winding and surroundingcircuitry [3]. The effects of stray capacitance includereduced bandwidth, large current wave spikes whenoperated at high frequency, premature resonance of thetransformer, and electrostatic coupling with thesurrounding circuits. The stray capacitance can bereduced by reducing the number of turns in the windingor by increasing the winding insulation.

Therefore, decrease in stray capacitance results inincrease in leakage inductance and vice versa. That is themajor trade-off encountered while designing a currenttransformer.

F. Ringing

The main purpose of toroidal core is to contain most ofthe magnetic flux. For a very high current in the primary,the magnetic field intensity Ho produced will be so largethat the magnetic field density B reaches its saturationlevel and will remain the same for further increase in themeasured current. Even though Hp is reduced to zeroafter saturation, B does not become zero resulting insome remnant flux in the core. Thus core saturationresults in a higher current value than the originalmeasured current value.

I. Mechanical Clamping

The mechanical clampings, used to clamp the probearound the current carrying conductor, determines theposition of the conductor which in turn affects the fluxlinkage between the primary winding and the secondarywinding. Also, the mechanical probe body provides adiscontinuity to the magnetic field produced by thecurrent in the primary winding. As the accuracy of themeasurement depends on the flux linkage, mechanicalclampings play an important role in the accuracy ofcurrent sensing transformers.

VII. VALIDATION

The example design has the following specifications:i = I A, n = 500, Lm = 200 mH, R<. = 10 kQ, LI = 10 pH,RS = 1Q, C = 10 pF, LR = 0.01pH, and R = 50 Q. Thefrequency responses of the circuit obtained using PSpicesimulation as well as Matlab simulation are illustrated inthe Fig. 1 1.

At high frequencies, because of the increasing leakageinductance and the stray capacitance, the currenttransformer may exhibit ringing or roll-off effects.Ringing may be caused by overshoot as well. The effectsof ringing may be removed by reducing the bandwidth ofthe receiver below that of the high-frequency responsecutoff frequency.

G. Parasitic Resonance

The overshoot oscillation causing ringing in thewaveforms occurs at a resonant frequency completelydependent on the parasitic components, i.e., the leakageinductance L and the stray capacitance C [3].

0.05

I

/

I..

I, tO id' ItO1 In10 t 10 10

Figure 11. Frequency Response of CT using Pspice andMatlab Simulation

375

H. Saturation

009

0Q6

on,7

The approximate values of fL and fH are calculated asgiven in (12) and (39). The values of fL and fH are alsoobtained from PSpice and Matlab simulation results.These values as well as the percentage errors obtained as

_ Spicevalue -ApproxvalueI1 -- Sicand Spicevalue

and

Spicevalue - Matlabvalue2 SpicevauaeieiTbevalu.

are given in Table I.

T.ABLE ICOMPAfRSOnT OF fL MUD f1 V-ALJES OBTAUrEO USnIG

APPRoxuIAJ.ATE EQu.knioIs, MATLAB1, AIt' PSPICE

Apprcox Ma.tlab PSpice El.%) fM%fLi.H:) 40.38 40.7 40. 7 0.786 0fH(1',f~ i42.255 132.13 155. 6 18.S 15. 08

The effect of high parasitic inductance LR associated withthe sense resistor in the bandwidth is shown in the Fig. 12.

0 1B

0 14

0.12

0. 1

0.o0

o

0041

002

102 l04 10' 1010I lHz)

Figure 12. The Effect of Parasitic Inductance of SenseResistor in the Bandwidth

From the percentage errors obtained, it can be observedthat the approximate equations for fL and fH are quitereliable for all practical purposes. As the value of L_Rincreases, the peaking of the waveform increases. So, it isdesirable to have low inductive sense resistance for betterperformance.

VIII. APPLICATIONS

Examples of applications of current transformers are asfollows:

1. Observing pulse current waveforms and amplitudes.

2. Current sensors used for current-mode control ofPWM dc-dc converters.

3. Observing transient and steady-state current wave-forms of various active and passive components inelectronic circuits.

4. Current transients and harmonics in power systems.

5. Current waveforms of fluorescent lamps.

6. Current waveforms in automobile electronics.

7. Observing the instantaneous power.

8. Self-oscillatory power inverters to provide negativefeedback.

IX. CONCLUSIONS

Approximate expressions for lower cutoff frequency andhigher cutoff frequency are derived. The values obtainedwith approximated equations are validated using thefrequency response of the current transformer for wideband range obtained by Matlab and PSpice simulations.The effects of various design parameters in the currenttransformer frequency response can be concluded as

1. As the magnetizing inductance L.. is increased, thelower cutoff frequencyfL is decreased.

2. As the leakage inductance LI is increased, the uppercutoff frequencyfH is decreased.

3. As the stray capacitance C is increased, the uppercutoff frequency fHis decreased.

So, for a broad-band current transformer, magnetizinginductance L4, should be very high and leakageinductance LI and stray capacitance C should be verylow. As leakage inductance and stray capacitance areinversely related to each other, that is the major trade-offencountered in the design of a broad band currenttransformer.

376

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I~~~~~~~~~~~I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~/ 1I

II

cr-

v _

100

REFERENCES

1. A. Wright, Current Transformers, Their Transientand Steady State Performance. London: Chapmanand Hall Ltd., 1968.

2. R. Paul, Electromagnetics for Engineers withApplications. New York: John Wiley \& Sons, pp.186-188.

3. Colonel Wm. T. McLyman, Transformer andInductor Design Handbook, 3rd Ed. New York:Marcel Dekker, Inc., 2004.

4. N. Locci and C. Muscas, "Hysteresis and eddycurrent compensation in current transformers,"IEEE Trans. Power Delivery), vol. 16, no. 2, pp.154-159, Apr. 2001.

5. G. Cerri, R. De Leo, V. M. Primiani, S. Pennesi,and P. Russo, "Wide band characterization ofcurrent probes," IEEE Trans. ElectromagneticCompatibility, vol. 45, no. 4, pp. 616-625, Nov.2003.

6. B. V. Cordingley and D. J. Chamund, "Someobservations on the performance of modernwideband current transformers in pulse currentmeasurement applications."

Nisha Das received her B.Tech. degree in electricalengineering from Mahatma Gandhi University, Kerala,India and MS degree in electrical engineering fromWright State University, Dayton, Ohio, where she iscurrently pursuing the PhD. Degree. Her areas ofresearch interest are power electronics and magnetics.

Marian K. Kazimierzcuk is a Professor of electricalengineering at Wright State University, Dayton, OH. Hisareas of research are electronic circuit analysis, high-frequency tuned power amplifiers, and power electronics.He is a Fellow of IEEE and has published more than 230papers. He is the co-author of the book "Resonant PowerConverters," Wiley, 1995. His email address ismkazim@cs.wright.edu.

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