IEEE Power Electronics Magazine - March 2016 - 18

Analysis of Magnetically Coupled
Resonant Circuits
Magnetically coupled circuits were a
hot research topic in the early 20th century as high-frequency radio circuits
were being investigated. The mathematical analysis of magnetically coupled oscillatory systems based on LC
resonators was reviewed in [10]. E.L.
Chaffee studied the general case of two
coupled circuits with different resonance frequencies. The effects, such as
the amplitude and phase relationships
of the primary and secondary current
amplitudes and their phase shift of
using the frequency below, at, and
above the resonance frequency, were
examined.

power transfer for an artificial heart
and used a magnetic resonance technique to transfer power wirelessly
through a closed chest wall of an animal. A series capacitor was used to
compensate the leakage inductance in
the secondary winding, and the operating frequency was set at the resonance
frequency of the LC resonator. In their
analysis, the significance of the product of the mutual coupling coefficient
k and the Q factor of the winding was
related to the power losses and, thus,
energy efficiency of the system. It was
found that a power loss is inversely
proportional to the kQ product, indicating that the energy efficiency ^h h is
proportional to the kQ product
h \ kQ.

Energy Efficiency and kQ Product
in Magnetically Coupled Resonant
Circuits

The higher the kQ product is, the
higher the energy efficiency of the
WPT system becomes.
■ For loosely coupled systems in
which the coupling coefficient k is
low, windings with high Q factor
should be used to increase the energy efficiency of the WPT systems.
This feature applies to both shortand midrange applications.
These two points have been known for
several decades in the electrical engineering community. They are also in
agreement with Tesla's suggestions of
using high frequency and low-resistance windings.
■

Nonradiative WPT For Short- and
Midrange Applications

(3a)

Short-Range Applications Based on
Magnetic Resonance

For magnetically coupled windings
with two different Q factors

A biological heart uses typically
20-35 W depending on whether it is in
the resting or heavy exercise state.
John Schuder et al. [11] studied the

h \ k Q1 Q2 .

The substantial research on nonradiative WPT by Tesla and other scientists
did not lead to widespread applications of WPT in the early half of the

(3b)

Equation (3) has two important
meanings.

tlec-2015_power-poster-final-cmyk-HIRES.pdf

1

11/9/15

2:54 PM

Line Voltage, Current, and Power - The Basics
LINE VOLTAGE

LINE CURRENT

SINGLE-PHASE

THREE-PHASE

Single-phase line voltage consists of one voltage
vector with:

Three-phase line voltage consists of three voltage vectors.
* By deﬁnition, the system is "balanced"
* Vectors are separated by 120°
* Vectors are of equal magnitude
* Sum of all three voltages = 0 V at Neutral

Line

At any given moment in time, the voltage magnitude is V * sin(α)
* V = magnitude of voltage vector
* α = angle of rotation, in radians

Neutral

120°

C

THREE-PHASE

Voltage value = VX*sin(α)
* VX = magnitude of phase voltage vector
* α = angle of rotation, in radians

* 50 Hz in Europe
* 60 Hz in US
* Either 50 or 60 Hz in Asia
* Other frequencies are sometimes used in non-utility
supplied power, e.g.
* 400 Hz
* 25 Hz

Important to Know
* Voltage is stated as "VAC", but this is really VRMS
* Rated Voltage is Line-Neutral
* VPEAK = 2 * VAC (or 2 * VRMS )
* 169.7 V in the example below
* VPK-PK = 2 * VPEAK
* If rectiﬁed and ﬁltered
* VDC = 2 * VAC = VPEAK
AC Single-Phase "Utility" Voltage

120 VAC Example

Volts (Peak), Line-Neutral

200

120VAC

150
100
50
0

800

Important to Know
* Voltage is stated as "VAC", but this is really VRMS
* Rated Three-phase voltage is always Line-Line (VL-L)
* Line-Line is A-B (VA-B), B-C (VB-C), and C-A (VC-A)
* Line-Line is sometimes referred to as Phase-Phase
* VPEAK(L-L) = 2 * VL-L
* 679 V in the example to the right
* VPK-PK(L-L) = 2 * VPEAK(L-L)

-150
-200

"True" RMS

600
400

0

-800

800

If a neutral wire is present, three-phase voltages
may also be measured Line-Neutral
* VL-N = VL-L/ 3
* 277 VAC (VRMS) in this example
* VPEAK = 2 * VL-N
* 392 V in the example to the right
* VPK-PK = 2 * VPEAK

Time

AC Three-Phase "Utility" Voltage
480VAC , Measured Line-Neutral

600
400
200
0
-200

B

-400

-800

VRMS =

1 V
PK-PK
2 2

For one power cycle

VRMS = VAC2
For one power cycle

B

IB

-600

Time

A-N Voltage
B-N Voltage
C-N Voltage
Three-phase Rectiﬁed DC

IC

N

IA

C

12
9
6
3

I

-6

Time
A Current
B Current
C Current

A

Reactive Power
* Q, in Volt-Amperes reactive, or VAr
* Q = S2 - P2
* Does not "transfer" to load during a power cycle,
just "moves around" in the circuit

10 ARMS Example

IA

Period 1
Mi = 18 points

Period 2
Mi = 18 points

A

mi = point 7

mi = point 25

V B-N

IB

IA

N

V C-N

IB

N
V A-N

I

P≠V*I

φ

N

V

Capacitive load

* The digital samples are grouped
into measurement cycles (periods)
* For a given cycle index i....
* The digitally sampled voltage
waveform is represented as having a
set of sample points j in cycle index i
* For a given cycle index i, there are
Mi sample points beginning at mi
and continuing through mi + Mi -1.
* Voltage, current, power, etc.
values are calculated on each
cycle index i from 1 to N cycles.

PTOTAL = VA-N * IA + VB-N * IB + VC-N * IC

IA
PTOTAL ≠ VA- N * IA + VB- N * IB + VC- N * IC

V C-N

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IRMS

QB

Real Power for each Phase
* P, in Watts
* = instantaneous V * I for a given
power cycle

VRMSi =

IRMSi =

Reactive Power for each Phase
* Q, in Volt-Amperes reactive, or VAr
* Q = S2 - P2

PB

φ

SB
PA

SC

* PTOTAL = PA + PB + PC
* STOTAL = SA + SB + SC
* QTOTAL = QA + QB + QC

φ
φ

QA

SA

PC
QC

Line-Line Voltage Sensing Case

B
IB
VB-C

Current is measured L-N

N

IC

L-L voltages must be transformed to L-N reference:

B

VB-N

VA-B

IA

IB

A

N

IC

VA-N
IA

A

VC-A

C

Calculations are straightforward, as described above:
* PTOTAL= PA + PB + PC
* STOTAL = SA + SB + SC
* QTOTAL = QA + QB + QC

C

VC-N

Two Wattmeter Method - 2 Voltages, 2 Currents with Wye (Y or Star) or Delta (∆) Winding

S

Q

φ
P

Real Power

Note: Any distortion present on the Line voltage
and current waveforms will result in power
measurement errors if real power (P) is
calculated as |S|*cos(φ). To avoid measurement
errors, a digital sampling technique for power
calculations should be used, and this technique is
also valid for pure sinusoidal waveforms.

Voltage is measured L-L on two phases
* Note that the both voltages are measured with
reference to C phase

mi + Mi - 1
1
V j2
Mi
j=mi

Σ

mi + Mi - 1
1
I j2
Mi
j=mi

Σ

Real Power
(P, in Watts)

Mathematical assumptions:
* Σ(IA + IB + IC) = 0
* Σ(VA-B + VB-C + VC-A) = 0
This is a widely used and valid method
for a balanced three-phase system

Pi =

Apparent Power
(S, in VA)

Reactive Power
(Q, in VAR)

PTOTAL = VA-C * IA + VB-C * IB
STOTAL= VRMSA-C * IRMSA + VRMSB-C * IRMSB
QTOTAL = STOTAL2 - PTOTAL2

Current is measured on two phases
* The two that flow into the C phase

Formulas Used for Per-cycle Digitally Sampled Calculations

VRMS

V A- N

φ

IC

IC

Voltage is measured L-L
* Neutral point may not be accessible, or
* L-L voltage sensing may be preferred

Inductive load

-9

Delta (∆) 3-phase Connection
* Neutral is not present in the
winding (in most cases)

C

Real Power
* P, in Watts
* = instantaneous V * I for a given power cycle

0
-3

-15

Digital Sampling Technique for Power Calculations�

C

Apparent Power
* |S|, in Volt-Amperes, or VA
* = VRMS * IRMS for a given power cycle

-12

IB

A
IC

B
A

V

φ

Single-phase Real, Apparent and Reactive Power
AC Three-Phase "Line" Currents
15

C

Three-Phase Winding Connections

N

N

* For inductive loads
* The current vector "lags" the voltage
vector angle φ
* Purely inductive load has angle φ = 90°
* Capacitive Loads
* The current vector "leads" the voltage
vector by angle φ
* Purely capacitive load has angle φ = 90°

Important to Know
* Current is stated as "lAC", but this is really IRMS
* Line currents can represent either current
through a coil, or current into a terminal
(see image below) depending on the three-phase
winding connection
* IPEAK = 2 * IRMS
* 14.14A for a 10 ARMS current in the example
to the right
* IPK-PK = 2 * IPEAK

A-B Voltage
B-C Voltage
C-A Voltage

480 VAC Example

Wye (Y) 3-phase Connection
* Neutral is present in the winding
* But often is not accessible
* Most common conﬁguration

P≠V*I

Line Current Measurements

-600

As with the single-phase case, Power is not the simple
multiplication of voltage and current magnitudes, and
subsequent summation for all three phases.

Apparent Power for each Phase
* |S|, in Volt-Amperes, or VA
* = VRMS * IRMS for a given power cycle

Single-phase, Non-resistive Loads
For capacitive and inductive loads
* P ≠ V * I since voltage and current are
not in phase

-400

B
VPK-PK

A

120°

Current value = IX*sin(α)
* IX = magnitude of line current vector
* α = angle of rotation, in radians

200

-200

Line-Neutral Voltage Measurements

Time

"Not True" RMS

Neutral

Like voltage, the resulting time-varying "rotating"
current vectors appear as three sinusoidal waveforms
* Separated by 120°
* Of equal peak amplitude for a balanced load

AC Three-Phase "Utility" Voltage
480VAC , Measured Line-Line

480 VAC Example

If all three phases are rectiﬁed and ﬁltered
* VDC = 2 * VL-N * 3 = VPEAK * 3 = 679 V
in the example to the right

-50
-100

Power Factor (PF, or λ)
* cos(φ) for purely sinusoidal waveforms
* Unitless, 0 to 1,
* 1 = V and I in phase, purely resistive load
* 0 = 90° out of phase, purely capacitive
or purely inductive load
* Not typically "signed" - current either
leads (capacitive load) or lags
(inductive load) the voltage

C

VC

Three-phase, Non-resistive Loads

For purely resistive loads
* PA = VA-N * IA
* PB = VB-N * IB
* PC = VC-N * IC
* PTOTAL = PA + PB + PC
V B-N

Voltage
Current

120°

120°

VA-N

Line-Line Voltage Measurements

V

Resistive load

Three-phase, Resistive Loads

Three-phase, Any Load

ω (rad/s) or
freq (Hz)

Like voltage, three-phase current has three different line
current vectors that rotate at a given frequency
* Typically, 50 or 60 Hz for utility-supplied voltage

VA

I

N

B

By deﬁnition, the system is "balanced"
* Vectors are separated by 120˚
* Vectors are of equal magnitude
* Sum of all three currents = O A at neutral (provided
there is no leakage of current to ground)

VA-B

Neutral

120°

Volts (Peak), Line-Line

The resulting time-varying "rotating" voltage vector appears as
a sinusoidal waveform with a ﬁxed frequency

Voltages can be measured two ways:
* Line-Line (L-L)
* Also referred to as Phase-Phase
* e.g. from VA to VB, or VA-B
* Line-Neutral (L-N)
* Neutral must be present and accessible
* e.g. from VA to Neutral, or VA-N
* VL-L conversion to VL-N
* Magnitude: VL-N * 3 = VL-L
* Phase: VL-N - 30° = VL-L

P=V * I

Power Factor

Phase Angle (φ)
* Indicates the angular difference between the
current and voltage vectors
* Degrees: - 90° to +90°
* Or radians: -π/2 to + π/2

Line

Neutral

The resulting time-varying "rotating" voltage vectors appear
as three sinusoidal waveforms
* Separated by 120°
* Of equal peak amplitude

Volts (Peak), Line-Neutral

Line

Neutral

For purely resistive loads
* P = I2R = V2/R = V * I
* The current vector and voltage vector are in perfect phase

Phase Angle
ω (rad/s) or
freq (Hz)

VB
ω (rad/s) or
freq (Hz)

The resulting time-varying "rotating"
current vector appears as a sinusoidal
waveform

At any given moment in time, the current magnitude
is I*sin(α)
* I = magnitude of current vector
* α = angle of rotation, in radians

A

120°

The three voltage vectors rotate at a given frequency
* Typically, 50 or 60 Hz for utility-supplied voltage
The single-phase voltage vector rotates at a given frequency
* Typically, 50 or 60 Hz for utility-supplied voltage

120°

ω (rad/s) or
freq (Hz)

Typically, the three phases are referred to as A, B, and C,
but other conventions are also used:
* 1, 2, and 3
* L1, L2, and L3
* R, S, and T

THREE-PHASE

Electric Power
* "The rate at which energy is transferred to a circuit"
* Units = Watts (one Joule/second)

Imaginary Power

Neutral

MDA800 Series
Motor Drive Analyzers
8 channels, 12-bits, 1 GHz

SINGLE-PHASE

Like voltage, the single-phase current vector rotates
at a given frequency
* Typically, 50 or 60 Hz

Line Current (Peak)

* Magnitude (voltage)
* Angle (phase)
Typically, the single-phase is referred to as "Line" voltage, and
is referenced to neutral.

LINE POWER

SINGLE-PHASE
B

mi + Mi - 1
1
Vj * Ij
Mi
j=mi

Σ

B

B

IB
VB-C

C

Power Factor (λ)

N

IA
VA-C

A

IB

VB-C

C

λi =

VA-C

A

IA

Pi
Si

Si = VRMSi * IRMSi

magnitude Qi =

S i2 - P i2

Sign of Qi is positive if the fundamental voltage
vector leads the fundamental current vector

Phase Angle (φ)

magnitude Φi = cos-1λi
Sign of Φi is positive if the fundamental voltage
vector leads the fundamental current vector

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18

IEEE POWER ELECTRONICS MAGAZINE

 March 2016
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Table of Contents for the Digital Edition of IEEE Power Electronics Magazine - March 2016
