t-14 fra visualizing physics behind the trace lachman

© 2011 Doble Engineering Company -78th

Annual International Doble Client Conference

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FREQUENCY RESPONSE ANALYSIS OF TRANSFORMERS:

VISUALIZING PHYSICS BEHIND THE TRACE

Mark F. Lachman, Vadim Fomichev, Vadim Rashkovski and AbdulMajid Shaikh

Delta Star Inc.

ABSTRACT

Many diagnostic measurements on transformers allow for an easy grasp of processes unfolding during the

test. The frequency response analysis, though, is a challenge due to a sweep of test frequencies engaging

multiple transformer segments. This paper reviews the physics at several key resonance frequencies. It

further associates each with a simplified equivalent circuit and contributing transformer components. As a

result, a visual insight into the process behind the trace is emerging.

MOTIVATION

Analysis of transformer diagnostic data is often aided by a clear image of physics behind the measurement.

For example, we can examine how during the exciting current and loss test the applied voltage creates

current, which in turn creates magnetic flux. The current has only one objective - to maintain magnetic flux

capable of inducing cemf into the excited winding that balances the applied voltage. This flux is confined to

the core. Therefore, any changes in the core reluctance would affect the amount of current (and loss) needed

to create the required flux. If there is a magnetically-coupled loop – intentional or un-intentional – a “load”

current is created, resulting in a counter flux. This counter flux increases the core reluctance. Whereas the

applied voltage must remain balanced, the measured current increases, making up for reduction in the

magnetic flux. Understanding this simple physics helps to envision potential scenarios creating abnormal

components in current and losses.

A similar process unfolds during the leakage reactance and loss test with exception of the flux path

including the core and the leakage channel. The latter is much larger and dominates the overall reluctance of

the path. The radial deformation, a chief target of this test, changes reluctance of the leakage channel. It

makes it more permeable (for the flux) by shifting a segment of the winding into what otherwise is a unit

permeability space. Less reluctance allows creation of the same flux with less current; thus increase in

leakage reactance.

We can easily follow the above processes as each unfolds at a single frequency. During the frequency

response test, the measurement is performed at a sweep of frequencies (e.g., 10 Hz - 2 MHz). An insight into

that process is a challenge to master. The task, however, can be simplified by recognizing that analyses of

this data rely, for the most part, on several key indicators:

Appearance of a new resonance

Disappearance of an existing resonance

Shift in a resonant frequency (significant above certain frequency).

Having a visual insight into a process associated with the key resonant points can be instructive in reading

the often convoluted narrative behind the frequency response trace.



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PHYSICS BEHIND THE TRACE

The following discussion, rooted in the material presented in Appendices and references [1-3, 6-12],

describes the process behind several characteristic resonance points. These points are found in the frequency

ranges shown to be associated with the key transformer components [4, 5]. Although the exact boundaries of

each range depend on the transformer design, for each unit the frequency span can be sub-divided (from low

to high frequency) into four regions, each correspondingly dominated by core, interaction between windings,

winding structure and leads (including the measuring leads). The first three regions carry the most diagnostic

significance and are the subject of this paper. The data for discussion (Figures 1-4) was obtained on the

middle phase of the low-voltage (LV) winding. The tested unit was a 20 MVA, 138-12.47Y/7.2 kV

transformer with an LTC on the neutral end of the LV side.

Core

As the magnitude trace is examined in the direction of frequency increase, the first point of interest is the

maximum impedance, typically found below 1 kHz (Fig. 1). Its corresponding phase angle (not shown) is

zero, indicating a resistive circuit. Therefore, even though the transformer consists of inductive and

capacitive elements, at this frequency, the source (i.e., the instrument) does not see them. The “ignorance” of

the source, however, does not mean that these elements do not participate in the process: inductive elements

still exchange energy with capacitive elements. However, all the energy supplied by the source is dissipated

in the resistive elements only.

Energy Oscillation between Electric Field (red) in LV Winding Bulk Insulation and Magnetic Field

(purple) in the Core

FIGURE 1



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(The oscillating energy was provided by the source during the initial transient process. Then, the inductive

and the capacitive elements “realized” that this energy is delivered at their natural frequency, and they can

be swapping it with each other for as long as the source “is paying the bills”, i.e., it is compensating for

losses. The focus of our discussion is the steady-state process only, one that follows the transient period).

The observed peak in impedance can be described by a circuit with the parallel RLC elements, where

inductance L represents storage of energy in the magnetic field and capacitance C in the electric field. Then,

with C = 1/L the source sees L and C branches as an open circuit, and the remaining R as impedance

maximum. This condition is variously referred to as anti-resonance, parallel resonance or current resonance

(the latter due to current through L being equal but opposite of current through C).

To identify components inside the transformer that make-up L and C in this oscillation, we recall that in the

low-frequency range under the open-circuit condition, the magnetic flux is confined to the transformer core.

Hence, magnetic energy is stored in the core as well. Therefore, here, L represents the ability of current to

create magnetic flux in the core and is referred to as the magnetizing inductance. This ability can be

influenced by magnetic viscosity [6, 7], presence of residual magnetism, and any closed circuit that acts as

load on the instrument.

The electric energy in the low-frequency range is confined, for the most part, to the bulk segments of

insulation, i.e., those with large capacitance. They include various capacitive paths to ground, i.e., through

the winding-to-ground insulation of the winding under test, through the winding-to-winding insulation and

then to ground, and further out to ground through the phase-to-phase insulation. The latter is not fully shown

on Fig. 1 to avoid details of the outer phases. All in all, we are dealing with a comprehensive capacitive

network that couples the winding under test to ground. Given the above, the energy supplied by the source,

for the most part, covers losses in the core laminations due to hysteresis and eddy currents.

Even though various capacitive elements contribute differently to the bulk capacitance involved in this

oscillation, knowing each contribution is not terribly important. Although both L and C play a role, it is

convenient to think about this particular point as one influenced by L only. The rationale for this is as

follows: when C is involved, the other frequency ranges are affected as well. However, if the shift in

frequency is limited to the low-frequency range only, then the change is caused only by L, since, with

increase in frequency, the impact of L (as magnetizing inductance) becomes negligible.

Summary: This resonant point can be visualized as a parallel RLC circuit where the oscillation between L

and C is associated with magnetic energy stored in the core and electric energy stored in the winding bulk

insulation of the tested phase.

Windings Interaction

As we travel past 2 kHz, the next point of interest is the minimum impedance found (on this trace) around 7

kHz (Fig. 2). Its corresponding phase angle is zero, once again indicating a resistive circuit, i.e., the reactive

components cancel each other, leaving resistance as minimum impedance. This can happen when, with 1/C

= L, the RLC elements are connected in series. Then, the L and C branches create a short-circuited

segment, leaving R as minimum impedance. This condition is variously referred to as resonance, series

resonance or voltage resonance (the latter due to voltage across L being equal but opposite of voltage across

C).



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Energy Oscillation between Electric Field in Delta-Connected HV Winding and Magnetic Field in the

Core and the Leakage Channel

FIGURE 2

To identify components influencing L, we recall that increase in frequency intensifies the eddy currents in

the core laminations. These currents create magnetic flux of their own, which opposes the flux creating them

in the first place. The higher is the frequency, the stronger is the demagnetizing effect pushing the flux out of

the core and into the space between and within the windings, i.e., the transformer inductance decreases [8].

To illustrate that, Figure 2 shows that the magnetic field penetration into the core is reduced as compared

with that on Figure 1. Therefore, at this resonance frequency, magnetic energy is stored both in the core and

in the leakage channel, and L is a combination of both the magnetizing inductance and the leakage

inductance.

To identify components influencing C in this oscillation, we rely on results first reported in [1]. That study

showed that for an open-circuit test on a wye-connected winding, the first main resonance in the 2-20 kHz

range is dominated by the other delta-connected winding (Fig. 2). This phenomenon can be rationalized as

follows. The tested winding through coupling creates voltage in the delta-connected winding. This voltage

generates currents circulating through conductors and flowing through the winding-to-ground insulation of

all phases. As a result, the changes in the conductors’ continuity and winding-to-ground insulation of all

phases in the delta-connected winding would influence the current flow through the winding under test.

Given the above, the source energy covers the dielectric losses in insulation, losses in the core and losses in

conductors created by eddy currents induced by the leakage flux.

This phenomenon can be demonstrated when the delta-connected winding has a de-energized tap changer.

Placing the latter between the taps, opens up the delta and interrupts the current flow. The impact of this is

easily observed in Figure 3.



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Influence of HV Winding Open Delta on Resonance Observed on LV Winding

FIGURE 3

In auto-transformers, the 2-20 kHz frequency range, typically, has a second major resonance preceded by an

anti-resonance, which carry much diagnostic significance [1]. It is created by the energy oscillation between

C and L associated with windings in the same phase. Specifically, C includes the interwinding and winding-

to-ground insulation and L includes both the mutual inductance and the leakage inductance. This resonance

depends on the strength of capacitive and inductive coupling between the two windings, and, depending on

the proximity and the turns ratio of two windings, can be observed in transformers as well.

Summary: The first main resonance in the 2-20 kHz range can be visualized as a series RLC circuit and is the

result of the oscillation between magnetic energy stored in the core and the leakage channel and electric

energy stored in the inter-winding insulation of the tested phase and the winding-to-ground insulation of the

not tested delta-connected winding.

Winding Structure

As we travel past 20 kHz, many resonance and anti-resonance points are encountered, with different types of

windings exhibiting a variety of characteristic patterns. These patterns to a large degree are governed by the

relationship between the winding series and shunt capacitances [2]. A resonance point chosen in Figure 4 is

presumably associated with an oscillation where the electric energy stored in the winding turn-to-turn (or

layer-to-layer) insulation is exchanged with the magnetic energy stored in the field created by the leakage

flux linked to the corresponding turns. Both, the capacitance and the flux path shown on Figure 4 are selected

to illustrate the local character of this particular resonance and do not reflect the actual segments involved.

We describe this condition by a series ZLC network, with Z including resistor that dissipates energy covering

losses in conductors due to eddy currents induced by the leakage flux and dielectric losses in the turn-to-turn

insulation. It is noted that the phase angle associated with this resonance is not zero (note Z instead of R on

Figure 4). This observation is discussed further in Appendix B.



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Energy Oscillation between Electric Field in Turn-to-Turn Insulation and Local Magnetic Field in the

Leakage Channel

FIGURE 4

As the frequency changes, it is instructive to recognize the processes taking place inside the conductors.

There are two phenomena at play here: skin and proximity effects [9, 10]. The skin effect depends solely on

magnetic field created by current in the conductor itself. This field exists inside and outside the conductor

(Fig. 5) and depends on current, how fast it changes and on the distance from the center of the conductor.

The inside field induces eddy currents in the conductor. These currents produce magnetic field of their own

that tends to cancel the field that created them. This, effectively, reduces the flux penetration into the

conductor. The eddy currents are directed so that they add to the main current on the surface of the conductor

and subtract from it in the interior region. The result is the magnetic field (H) and current (i) distribution as

shown in Figure 5. The net current in the conductor is unchanged by the eddy currents; it is the radial

distribution over the cross-section that changes. As the intensity of eddy currents is directly proportional to

frequency, a non-uniform current distribution becomes more pronounced at higher frequencies, i.e.,

conductor resistance increases. The outside magnetic field depends only on the net current and is unchanged

by the inside current redistribution caused by the skin effect.

The proximity effect depends solely on magnetic field generated by currents in other conductors;

specifically, by the field component normal to the axis of the conductor. Figure 6a shows a conductor,

carrying no current, in an external magnetic field Hext. This field is assumed uniform and is increasing in

magnitude in the direction shown.



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Skin Effect: Eddy Currents Induced by Inside Magnetic Field Change Current Radial Distribution

and Demagnetizing Effect of Eddy Currents Reduces Magnetic Flux Penetration into Conductor

FIGURE 5

Proximity Effect: a) Conductor in External Magnetic Field, b) Eddy Currents Induced by External

Magnetic Field and Resulting Circulating Current, c) Demagnetizing Magnetic Field Produced by

Circulating Current

FIGURE 6

R r

H

Outside magnetic field

Inside magnetic field

Demagnetizing field due to eddy currents

R

r

R r

iConductor

Eddy current Eddy current



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The field lines passing through the conductor induce eddy currents (Fig. 6b). In the shared path, the

neighboring eddy currents have opposite directions thus cancelling each other. The net effect of that is a

circulating current that flows into the right side of the cross-section and out of the left side producing an

opposing magnetic field (Fig. 6c). The superposition of this demagnetizing field and the external field results

in reduction of the net magnetic flux that penetrates the conductor, reducing the amount of magnetic energy

stored in the conductor.

Summary: The discussed resonance in the > 20 kHz range can be visualized as a series ZLC circuit and is the

result of the oscillation between the electric energy stored in the winding turn-to-turn insulation and the

magnetic energy stored in the space occupied by the leakage flux linked to the corresponding turns. As

frequency increases, the impact of skin and proximity effects becomes more prominent; it increases

conductor resistance and reduces magnetic flux penetration into conductors.

Mechanical Analogy of LC Oscillations

At times, a physical phenomenon can be more visual by examining its analog in another discipline. An LC

oscillation resembles in many ways an oscillation in a mass-spring system depicted on Figure 7. Here, a

vibrator V, imposing an external alternating force F, corresponds to a generator G, imposing an external

alternating electromagnetic force e. If the driving frequency equals the natural frequency of the system, the

resonance occurs. The friction of the mechanical system is depicted as a viscous friction encountered by the

dashpot moving through liquid.

Mechanical Analog to Forced Oscillations in Electromagnetic System with Losses

FIGURE 7

When a mass-spring system performs a simple harmonic motion, just as in LC oscillation, two kinds of

energy occur. One is potential energy (UP) of the compressed or extended spring, corresponding to electric

energy (UC) in C; the other - kinetic energy (UK) of the moving mass m, corresponding to magnetic energy

(UL) in L. The well-known equations relating both types are presented in Table I [11].

TABLE I

ENERGY IN MASS-SPRING AND LC OSCILLATING SYSTEMS

Mechanical Electromagnetic

Spring UP = ½kx2 Capacitor UC = ½(1/C)q

2

Mass UK = ½mv

2

v = dx/dt Inductor

UL = ½Li2

i = dq/dt



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The table suggests that capacitor “mathematically” behaves like a spring and inductor like a mass (we

assume the spring has no mass); moreover, certain electromagnetic quantities “correspond” to certain

mechanical ones. Namely,

charge q to linear deflection x

current i to velocity v

capacitance C to spring constant in 1/k

inductance L to mass m

Appendix C demonstrates the validity of this comparison by describing a lossless oscillation in each system.

It illustrates how the energy alternates between two forms: magnetic and electric in the LC system and

kinetic and potential in the mass-spring system.

The practical outcome of this analogy is realization that an oscillation between electric and magnetic fields at

a resonance on the frequency response trace can be visualized as a mass oscillating on a spring. When an

external force operates at the resonance frequency, the mass reaches the highest deflection from the

equilibrium position. If we change geometry of the spring (similar to changing distance between plates in a

capacitor) or the spring material becomes fatigue (similar to dielectric material changing its dielectric

properties), the mass-spring system will change its natural frequency. This will require changing the driving

frequency to produce a resonance, i.e., a highest deflection of the mass. Likewise, changing the mass

attached to a spring changes the amount of kinetic energy the system can carry. This is similar to changes in

inductance affecting the amount of magnetic energy inductor can store; in both cases, the point of resonance

is shifting. Finally, changing viscosity of the liquid or geometry of the dashpot is similar to changes in circuit

resistance; both change the loss of energy during the oscillation. Also, the faster the dashpot moves, the

greater is liquid resistance; this is analogous to increase in conductor resistance with increase in frequency.

CONCLUSION

Review of basic physics behind the transformer frequency response trace demonstrated how the process at

key resonance frequencies can be visualized using the simplified equivalent circuits. This process is the

electromagnetic energy oscillation between capacitive and inductive elements. The use of mechanical mass-

spring system to further the insight is suggested.

ACKNOWLEDGMENT

The comments by Simon Rider of Doble PowerTest Ltd. and Professor L. Satish and Mr. Saurav from the

Indian Institute of Science in Bangalore are greatly appreciated.

REFERENCES

1. Sofian, D. M. “Transformer FRA Interpretation for Detection of Winding Movement”, PhD thesis,

University of Manchester, July 2007.

2. Wang, Z., Li, J. and Sofian, D. M. “Interpretation of Transformer FRA Responses - Part I: Influence of

Winding Structure”, IEEE Transactions on Power Delivery, Vol. 24, No. 2, April 2009, pages 703-710.



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3. Sofian, D. M., Wang, Z. and Li, J. “Interpretation of Transformer FRA Responses - Part II: Influence of

Transformer Structure”, IEEE Transactions on Power Delivery, Vol. 25, No. 4, October 2010, pages 2582-

2589.

4. “Draft Trial-Use Guide for the Application and Interpretation of Frequency Response Analysis for Oil

Immersed Transformers”, IEEE PC57.149™/D8, November 2009.

5. CIGRE Working Group A2.26, “Mechanical-Condition Assessment of Transformer Windings Using

Frequency Response Analysis (FRA)”, Brochure 342, 2007.

6. Abeywickrama, N., Serdyuk, Y. V. and Gubanski, S. M. “Effect of Core Magnetization on Frequency

Response Analysis (FRA) of Power Transformers”, IEEE Transactions on Power Delivery, Vol. 23, No. 3,

July 2008, pages 1432-1438.

7. Lachman, M. F., Fomichev, V., Rashkovsky, V., and Shaikh, A. “Frequency response analysis of

transformers and influence of magnetic viscosity”, Proceedings of the Seventy-Seventh Annual International

Conference of Doble Clients, 2010, Sec. TX-11.

8. Abeywickrama, N., Podoltsev, A. D., Serdyuk, Y. V. and Gubanski, S. M. “Computation of Parameters of

Power Transformer Windings for Use in Frequency Response Analysis”, IEEE Transactions on Magnetics,

Vol. 43, No. 5, May 2007, pages 1983-1990.

9. Urling, A. M., Niemela, V. A., Skutt, G. R. and Wilson, T. G. “Characterizing High-Frequency Effects in

Transformer Windings – A Guide to Several Significant Articles”, 1989, IEEE.

10. Podoltsev, A. D., Abeywickrama, N., Serdyuk, Y. V. and Gubanski, S. M. “Multiscale Computations of

Parameters of Power Transformer Windings at High Frequencies. Part I: Small-Scale Level”, IEEE

Transactions on Magnetics, Vol. 43, No. 11, November 2007, pages 3991-3998.

11. Feynman, R. P., Leighton, R. B. and Sands, M. The Feynman Lectures on Physics, Definitive Edition,

Vol. I, Reading, MA, Addison-Wesley, 1977.

12. Ragavan, K. and Satish, L. “Localization of Changes in a Model Winding Based on Terminal

Measurements: Experimental Study”, IEEE Transactions on Power Delivery, Vol. 22, No. 3, July 2007,

pages 1557-1565.

BIOGRAPHIES

Mark F. Lachman, Ph.D., P.E., has been with the power industry for over 30 years. In 2005, he joined Delta

Star in San Carlos, CA, where, as Test Manager, he was responsible for the test department operation. In

2011, he returned to Doble Engineering Company as Director of Diagnostic Analyses.

Vadim Fomichev has been with the power industry for 13 years. He joined Delta Star in San Carlos, CA in

2006, where he is a Lead Tester in the test department, responsible for the swing shift testing of power

transformers and mobile substations.

Vadim Rashkovski has been with the power industry for over 35 years. He joined Delta Star in San Carlos,

CA in 2000, where he is a Lead Tester in the test department, responsible for the day shift testing of power

transformers and mobile substations.

AbdulMajid Shaikh has been with the power industry for 9 years. He joined Delta Star in San Carlos, CA in

2004, where he is presently a Test Supervisor responsible for the test department operation.



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APPENDIX A

ENERGY FLOW IN BASIC RLC CIRCUIT

In electrical circuits, capacitors and inductors are often treated as “throw-away” items - if fails, replace, no

need to understand the failure mechanism. In other words, each element is viewed as a black box, and the

need to consider physics inside the element is generally optional. In power transformers, C and L are

parameters of enormous importance. Hence, the process is worth considering. Although we will discuss a

series RLC circuit, similar considerations apply to a parallel circuit as well. The following review describes

the energy oscillations without the use of vector diagrams and impedance relationships; these can be found in

any text book. The objective here is to facilitate a more intuitive look into the process behind the frequency

response trace; one that can be easily recalled when data is analyzed.

We begin with the energy flow through a simple series RLC circuit (Fig. 1A), where L > 1/C is assumed

and the driving frequency is constant.

Series RLC Circuit

FIGURE 1A

Fig. 2A shows parameters associated with the resistor. We observe that at any moment during the cycle the

instantaneous power, pR = i2R, is positive, i.e., the energy always flows into resistor. (The shown time scale is

for 60 Hz).

Parameters Describing Process in Resistor R

FIGURE 2A

Fig. 3A depicts parameters describing the process in the inductor. Here, the instantaneous power depends on

the current rate-of-change, qL = d(Li2/2)/dt. (Note that the extrema of qL correspond in time to the highest

R L

C

i

vR vL

vCv

-6

-4

-2

0

2

4

6

8

10

12

14

0.0 4.2 8.3 12.5 16.7

i t[ms]

Rp

v

t1

t3

t2

flow into R

R



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slope in the current sinewave.) Hence, the energy flows into inductor (qL > 0) when the absolute value of

current increases (positive rate-of-change) and out of inductor (qL < 0) when the absolute value of current

decreases (negative rate-of-change). With the former the energy accumulates in the magnetic field and with

the latter the inductor returns the energy and the field collapses.

Parameters Describing Process in Inductor L

FIGURE 3A

Fig. 4A presents parameters describing the process in the capacitor. Here, the instantaneous power depends

on the voltage rate-of-change, qc = d(Cvc2/2)/dt. (Note that the extrema of qc correspond in time to the highest

slope in the voltage sinewave.) Therefore, the energy flows into capacitor (qc > 0) when the absolute value of

voltage increases (positive rate-of-change) and out of capacitor (qc < 0) when the absolute value of voltage

decreases (negative rate-of-change). With the former the energy accumulates in the electric field and with the

latter the capacitor returns the energy and the field collapses.

Parameters Describing Process in Capacitor C

FIGURE 4A

Fig. 5A presents parameters as viewed from the terminals of the series circuit. As expected, the current lags

the voltage (by an angle ). We also observe that during the period 0 - t2, the instantaneous total power s is

positive; hence, the energy flows from the source into the circuit. During the period t2 – t3, s < 0, and the

energy returns to the source. Furthermore, given 0 < < 90, the source only “knows” that the circuit is

inductive and has a resistance. Moreover, if L is determined from the terminal measurements, its value is

-10

-8

-6

-4

-2

0

2

4

6

8

10

0.0 4.2 8.3 12.5 16.7

L

q

v

i

t[ms]

L

t3

t1 t2

flow into L

flow out of L



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reduced by the presence of capacitance, i.e., Lterm = L - 1/C. In other words, from the terminal

measurements at a given frequency, we don’t learn that the circuit also includes a capacitor.

Parameters at Terminals of Series RLC Circuit

FIGURE 5A

To probe further, we consider the energy exchange between components in the circuit. During the interval

0 – t1, the current is increasing (Fig. 3A), and the voltage across the capacitor is dropping (Fig. 4A).

Therefore, the energy flows to the inductor and from the capacitor (Fig. 6A, a). Since L > 1/C, the

capacitor cannot satisfy all of inductor’s energy needs. We re-state this as follows: at a given frequency, the

rate-of-change of energy (qC) flowing out of C is less than the rate-of-change of energy (qL) demanded by L.

Hence, the source has to make up the difference. In addition, the source also provides for the energy loss in

the resistor (Fig. 6A, a). All in all, s > 0 (Fig. 5A).

During the interval t1 – t2, the current is decreasing (Fig. 3A), and the voltage across the capacitor is

increasing (Fig. 4A). Therefore, the energy flows from the inductor and to the capacitor, and also to the

resistor (Fig. 6A, b). We re-state this as follows: at a given frequency, the rate-of-change of energy (qL)

flowing out of L is higher than the rate-of-change of energy (qC) demanded by C. Therefore, the balance of

energy is available for R. During this interval, the level of current is such that losses in the resistor (i2R) are

too high for the inductor to fully cover them. Once again, the source has to come in to make up the

difference. At t2, the losses in the resistor drop to a level when the energy flowing out of inductor finally

covers the needs of both, the capacitor and the resistor. Hence, s = 0 (Fig. 5A).

-7

-5

-3

-1

1

3

5

7

9

11

13

15

17

0.0 4.2 8.3 12.5 16.7

v

s

i

t1

t2

t3

t[ms]

flow into

source

flow out of source



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During the interval t2 – t3, the current is decreasing from its level at t2 to 0 (Fig. 3A), and the voltage across

capacitor is increasing but at a slower rate (Fig. 4A). The demands of the capacitor and the resistor are such

that the inductor, after meeting them, still has some energy left. This balance of energy returns to the source

(Fig. 6A, c) and s < 0 (Fig. 5A).

If 1/C > L, then L and C would trade their respective roles, i.e., the source would then “think” that it

exchanges energy with C and have no knowledge that L exists in the circuit. The 0 t t3 interval covers a

Energy Flow in Series RLC Circuit

FIGURE 6A

complete cycle of energy oscillation. It repeats itself during the next half cycle of the source frequency;

however, the directions of rms current and voltages will be opposite.

If we now begin to change the frequency of the source, we may arrive to a frequency when the rate at which

L is able to return (and receive) energy is equal to the rate at which C is able to receive (and return) energy,

i.e., qL = qC.

Parameters at Resonance: L - C Energy Oscillation and One-Way Energy Flow from Source to R

FIGURE 7A

Under these conditions, the energy oscillates between C and L, and the only energy provided by the source is

one dissipated in R (Fig. 6A-d shows polarity and directions during the first 4.2 ms). Furthermore, the source

has no knowledge that the circuit contains C and L and perceives it as purely resistive. This condition is

R L

C

i

0 < t < t1

+ - + -

-

+

a)

R L

C

i

t1 < t < t2

+ - - +

+

-

b)

R L

C

i

t2 < t < t3

+ - - +

+

-

c)

R L

C

i

+ - + -

-

+

d)

at resonant frequency

-15

-10

-5

0

5

10

15

0.0 4.2 8.3 12.5 16.7

ps =R

Cq

Lq

flow out of C

flow into L

flow from source into R

t[ms]

flow out of L

flow into C



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referred to as electrical resonance (Fig. 7A depicts this condition at 60 Hz). At resonance in the series circuit,

1/C = L; if the RLC elements were connected in parallel, then, at resonance, C = 1/L.

+

APPENDIX B

ENERGY FLOW IN POWER TRANSFORMER EQUIVALENT CIRCUIT

In a power transformer, we are dealing with an extremely complicated equivalent network of distributed,

non-linear and frequency-dependent R, L, and C elements. As the frequency of the instrument changes,

various combinations of C and L may find themselves in resonance with each other, i.e., the energy returned

by one is fully received by the other and vice versa. Hence, numerous resonant frequencies, each associated

with a different pair of C and L, are observed. In frequency ranges between the resonant points, the circuit

will be predominantly inductive or predominantly capacitive with the energy flow as described earlier.

However, even though the network is seen as inductive, one should recognize that capacitance is present and

makes a contribution, and, conversely, when the network is capacitive, inductance plays a role as well.

Furthermore, as shown above, in a simple series RLC circuit the impedance extrema at a resonant frequency

is achieved by L and C elements canceling each other, resulting in resistive circuit with a zero phase angle. In

a power transformer, the complexity of the equivalent network creates conditions which, on the surface, may

appear counter-intuitive. In this circuit, the capacitive and inductive energies are being continuously

interchanged, and during this transition there could be one or more frequencies at which the two are exactly

equal and cancel each other. For these frequencies the phase angle is exactly zero. Additionally, there are

also frequencies at which this cancellation does not occur, but still an impedance extrema is observable at the

terminals. This results in a non-zero phase angle resonance points [12]. In other words, even though the

impedance of that circuit is at the extremum, the energy will oscillate between the instrument and the

network.

This can be explained further if we define a resonance as an extrema of impedance magnitude seen by the

instrument. For the transformer, the impedance equation would include inductive and capacitive components

along with frequency in both the real and the imaginary parts:

Therefore, at the frequency when imaginary component is zero and impedance, being equal to the real part,

has a zero phase angle, this impedance is not resistive. As a result, on the sfra magnitude trace we may

observe an inductive roll-off or a capacitive climb-up (this description is function-dependent) corresponding

to a zero value on the phase angle trace. Moreover, just because the imaginary component is zero, does not

necessarily imply that impedance is an extrema. Due to frequency being present in both real and imaginary

parts, there could be another frequency at which impedance is more “extreme” than it is at the frequency

when the imaginary part is zero. This is why we may observe a resonance when the phase angle is non-zero.



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16

APPENDIX C

ENERGY OSCILLATION IN MASS-SPRING AND LC SYSTEMS

To appreciate the similarity between the electromagnetic oscillation in LC system and the mechanical

oscillation in mass-spring system it is instructive to examine the energy flow in time. For simplicity, we

consider both systems as having no losses: no resistance in LC and no friction in mass-spring. The oscillation

cycle is depicted in Figure 1C and described in Table I-C. It demonstrates how the energy alternates between

two forms: magnetic and electric in the LC system and kinetic and potential in the mass-spring system.

Review of Table I-C is most useful when performed recognizing the correspondence between parameters of

both systems, i.e., x q, L m, i v, C 1/k.

Comparison of Energy Oscillation in Lossless Mass-Spring and LC Systems

FIGURE 1C

Once started, the lossless oscillations continue indefinitely, with energy being shuttled back and forth from

capacitor to inductor and from spring to mass. In actual systems, there is always some damping present. It

drains energy from electric and magnetic fields through heating the resistance and decreases the amplitude of

mechanical oscillation due to friction. The oscillations described in Table I-C are referred to as “free”

oscillations. They become free oscillations with damping when brought from the abstract into reality. The

presence of a source covering the losses would allow the oscillations to continue indefinitely, as long as the

source is available. These oscillations are referred to as “forced” and they are the oscillations observed at the

resonance point in the measured frequency response data.



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17

TABLE I-C

DESCRIPTION OF LOSSLESS OSCILLATION CYCLE IN MASS-SPRING AND LC SYSTEMS

Time

segment Mechanical oscillation Electromagnetic oscillation

t = 0

The spring is extended to its maximum length x

and the mass is at rest with v = 0. Therefore, the

potential energy UP stored in the spring is at

maximum and the kinetic energy UK acquired by

the mass is zero.

The capacitor carries maximum charge q and the

current i = 0. Therefore, the electric energy UC stored

in the capacitor is at maximum and the magnetic

energy UL stored in the inductor is zero.

t = T/4

As the spring begins to contract, the mass begins

to move up, i.e., the spring is losing potential

energy and the mass is gaining velocity

increasing its kinetic energy. Finally, the spring

is neither elongated nor compressed; hence, with

no stored energy, UP = 0. At the same time,

velocity of mass is at maximum and all energy is

in UK.

As the capacitor begins to discharge, the current

begins to flow, i.e., the capacitor is losing electric

energy and, as current grows, magnetic field is

building up in the inductor, increasing stored magnetic

energy. Finally, there is no charge left in the capacitor;

hence, with no stored energy, UC = 0. At the same

time, current is at maximum and all energy is in UL.

t = T/2

The mass, due to inertia, does not want to stop at

the equilibrium point. Continuous movement of

the spring advances it towards compression and

the energy is transferred from being contained in

the movement of the mass to that contained in the

compressed spring. Eventually, all energy is

transferred back to the spring and the situation is

as it was at t = 0 except that the spring is now

compressed. The deflection x is now at maximum

and the mass once again is at rest with v = 0.

Therefore, the potential energy UP stored in the

spring is at maximum and the kinetic energy UK

of the mass is zero.

Since there is no charge forthcoming from the

capacitor to sustain current through the inductor,

magnetic field begins collapsing. Changing magnetic

field induces current in the direction that sustains the

field, i.e., the same direction as during capacitor

discharge. In other words, current once it starts

flowing through inductor does not want to stop. The

current continues transporting charge from one plate

of the capacitor to the other, and the energy flows

from inductor back to capacitor. Eventually, all energy

is transferred back to the capacitor and the situation is

as it was at t = 0 except that the capacitor is charged

oppositely. The capacitor carries maximum charge q

and the current i = 0. Therefore, the electric energy UC

stored in the capacitor is at maximum and the

magnetic energy UL in the inductor is zero.

t = 3T/4

As the spring begins to stretch, the mass begins

to move down, i.e., the spring is losing potential

energy and the mass is gaining velocity and thus

kinetic energy. Finally, the spring once again is

neither elongated nor compressed and UP = 0. At

the same time, velocity of the mass is at

maximum and all energy is in UK.

As the capacitor begins to discharge, the current

begins to flow in the opposite direction, i.e., the

capacitor is losing electric energy and, as current

grows, magnetic field is building up in the inductor,

increasing stored magnetic energy. Finally, there is no

charge left in the capacitor and UC = 0. At the same

time, the current is at maximum and all energy is in

UL.

t = T

We are back to the initial state at t = 0: the spring

is extended to its maximum length and the mass

is at rest with v = 0. Therefore, the potential

energy UP stored in the spring is at maximum and

the kinetic energy UK of the mass is zero.

We are back to the initial state at t = 0: the capacitor

carries maximum charge q and the current i = 0.

Therefore, the electric energy UC stored in the

capacitor is at maximum and the magnetic energy UL

in the inductor is zero.

t-14 fra visualizing physics behind the trace lachman

Documents

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counter flux

frequency response test

required flux

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loss test

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