r.w.erickson - university of colorado...

R. W. Erickson Department of Electrical, Computer, and Energy Engineering

University of Colorado, Boulder

Introduction to Inverters

Single-Phase Inverter ApproachesThe Solar ApplicationSingle-Phase Solar InvertersMicroinverters

Robert W. Erickson Department of Electrical, Computer, and Energy Engineering

University of Colorado, Boulder

A Basic Single-Phase Inverter Circuit

Power Electronics Lab� 6�

Two ways to generate a PWM sinusoid�

t

vac(t)

(a)� Operate left and right sides with same (complementary) gate drive signals�� v(t) = (2d(t) – 1) Vg�

(b) �PWM one side, while other side switches at 60 Hz�� v(t) = ± d(t) Vg�

Two-level waveform�

Three-level waveform�

DC power source

AC load

H-bridge inverter circuit

•  L-C filter may or may not be present•  Simple resistive load illustrated; actual AC loads are more complex•  Even in the single-phase case, there are multiple ways to control the

switches•  Applications:

•  Uninterruptable power supply (UPS)•  AC motor drive•  Fluorescent lamp driver•  Solar power inverter•  Automobile AC power inverter

Some three-phase applications:•  AC motor drives•  Inverters for wind and solar•  Electric vehicles•  DC transmission line stations

The “Modified Sine Wave” Inverter


“Modified Sine-Wave” Inverter�

vac(t) has a rectangular waveform�

Inverter transistors switch at 60 Hz, T = 8.33 msec�

T/2

DT/2+ V

HVDC

– VHVDC

vac

(t)

RMS value of vac(t) is:�

��

��

�

�

� � ��

•� Choose VHVDC larger than desired Vac,RMS�

•� Can regulate value of Vac,RMS by variation of D�

•� Waveform is highly nonsinusoidal, with significant harmonics�

H-bridge switches at the output frequency•  Waveform is highly

nonsinusoidal, with significant harmonics

•  Some ac loads can tolerate this waveform, others cannot

•  Inexpensive, efficient

Control of ac rms voltage by control of duty cycle D:


“Modified Sine-Wave” Inverter�

vac(t) has a rectangular waveform�

Inverter transistors switch at 60 Hz, T = 8.33 msec�

T/2

DT/2+ V

HVDC

– VHVDC

vac

(t)

RMS value of vac(t) is:�

��

��

�

�

� � ��

•� Choose VHVDC larger than desired Vac,RMS�

•� Can regulate value of Vac,RMS by variation of D�

•� Waveform is highly nonsinusoidal, with significant harmonics�

Standalone inverter: inverter drives a passive load, and regulates the voltage supplied to the load

The “True Sine Wave” Inverter



t

vac(t)




Three-level waveform�Power Electronics Lab� 6�


t

vac(t)






t

vac(t)






t

vac(t)






t

vac(t)




Three-level waveform�

•  Switches 1a and 1b are driven by the same gate drive signal, and conduct during d interval

•  Switches 2a and 2b are driven by the complement and conduct during d’ interval

•  For positive half cycle, switch 1b is on. Switches 1a and 2a operate with PWM

•  For negative half cycle, switch 2a is on. Switches 2b and 1b operate with PWM

Alternate: b switches switch at the line frequency, and a switches operate with PWM•  Three-level exhibits reduced switching loss

Use of PWM with high frequency switching to produce a sinusoidal ac voltage having low harmonic content. Typically an L-C filter is included to meet EMI requirements

Standalone vs. grid-tied applications

Inverter

vac(t)

dinv

×

iref(t)

Phase-locked loop

vac(t)

PWM

+ –

Sinusoidunit amplitude

phase-locked to vac

H3

iac(t)

Gc3(s)

Vcontrol

+–Vdc

Inverter

dinv

vref(t)

vac(t)

PWM

+ –

H3Gc3(s)

+–Vdc AC

load

Sinusoidalreference

Standalone inverter:Inverter regulates ac voltage

Grid-tied inverter:Inverter regulates its ac current

Reactive power

For a standalone application, the inverter must be capable of supplying whatever current waveform is demanded by the ac load

•  Reactive load, in which current is phase-shifted relative to voltage•  Distorted current

In most grid-tied applications, the inverter supplies a low-THD current waveform to the grid, with power factor very close to unity.

•  Improved efficiency•  This opens the possibility of simpler converter topologies using single-

quadrant switches

The grid-tied solar inverter application

•  AC voltage is determined by the utility system (“infinite bus”)•  Power is determined by the solar array•  Inverter produces ac current synchronized to the utility, with amplitude dependent on

array power

A residential solar array system

Functions performed by the inverter

Maximum power point tracking: operate the solar array at the voltage that maximizes generated power

120 Hz energy storage: the difference between the constant power supplied by the array and the 120 Hz pulsating power flowing into the utility is supplied/stored in the capacitor

AC current control: ac line current must meet harmonic requirements (THD < 5%), with unity power factor

•  Array voltage, capacitor voltage, and ac line voltage are independent

•  System resembles PWM rectifier system, but with power flow reversed

Ppv

pac(t)

t

vc(t)

An Inverter System

DC-DCvbus(t)vpv(t) Inverter EMI

vac(t)

dinv

×

iref(t)

PV

d

H2

H1 Phase-locked loop

vac(t)

PWM

+ –

Sinusoidunit amplitude

phase-locked to vac

H3

iac(t)

+–

Gc3(s)

Gc2(s)Vref-bus

ibus

PWM

Gc3(s)

+–

MPPT

Vref-pv

vbus(t)

vpv(t)

Standards

IEEE 1547: standard for connecting a renewable energy source to the utility grid•  Current harmonic limit (THD < 5%)•  Anti-islanding (detect loss of grid, shut down within 1 sec)•  Disconnection when grid frequency or grid voltage is out of bounds

National Electric CodeUL 1741

Weighted Efficiency standards: California Energy Commission (CEC)

Power level, % of rated

Weight

100% 0.05

75% 0.53

50% 0.21

30% 0.12

20% 0.05

10% 0.04

•  Provides a way to compare products of different companies

•  Weightings reflect typical distribution of array power experienced in California

Microinverters

One inverter per panel •  Mounted on or near the panel—on roof •  MPPT on per-panel basis •  Conventional AC wiring reduces

Balance-of-system cost •  Straightforward expandability •  Reliability? Efficiency?

Rated temperature?

Enphase microinverter,

2008

Ascension Tech. microinverter, 1998

Elements of a Microinverter System

ACutility

Transientprotection

ACdisconnect

Communications

Smart gridComputer

Central box

RoofAC

Microinverter

Power stage

MPPT Currentcontrol

Anti-islanding

Microinverter

Power stage

MPPT Currentcontrol

Anti-islanding

DC-ACInverter

DC-DCConverter

v(t)

i(t)

PVCells

Energystorage

+

vac

–

iac

PacPdc

iac(t) t

vac(t)

+

v

–

i

i

v

PacPdc

+

vcap

–

Microinverter power train: • DC-DC converter (high boost ratio) • Energy storage capacitor • Inverter

Rooftop system • Microinverters include most or all of grid interface control • Central box

Microinverter Approaches

enables efficient design of the inductor. We also introduce a fast and stable cycle-by-cycle controller that does not

require sensing of the average output current, and thus avoids an expensive current sensor. Overall, the proposed

method enables a low-cost design that operates with a small inductor and achieves a peak efficiency of 99.5 % and a

weighted efficiency of 99.15 %.

II. VARIABLE FREQUENCY PEAK CURRENT CONTROLLER

In this section we present the constant peak current switching scheme. This new scheme is derived through an

analysis of weighted losses in BCM, an analysis that demonstrates that the dominant loss for BCM is switching loss.

To improve the weighted efficiency, we explore which peak current is optimal at each output power, and show that in

DCM the optimal peak current is constant.

A common topology for micro-inverters is the two-stage topology [22], [23], which includes a boost stage and an

inverter stage, as shown in Fig. 1. Typically the boost stage tracks the maximum power point of the PV source, and

boosts the low PV input voltage to a higher voltage. The inverter stage generates the AC current that is injected to the

AC line. Despite various new topologies that have been demonstrated in recent literature [5], the typical low-cost

micro-inverter is still designed either as a full-bridge stage, or as a buck stage with an unfolder stage. The unfolder

stage, if present, switches at the zero-crossings of the line voltage to convert the rectified sinusoid at the buck output

to a full sinusoid on the AC line.

Fig. 1. Common micro-inverter power stages. (a) Full bridge. (b) Buck stage with an unfolder stage.

An illustration of the Boundary Conduction Mode (BCM) waveform is shown in Fig. 2. Although it is soft-

switching, and operates with low RMS current, a disadvantage of BCM is its high average switching frequency, which

causes high switching losses. As demonstrated by equation (1), the BCM waveform has the highest switching

frequency among all DCM waveforms. This is because the peak current of BCM is equal to ipk(t)=2iout(t), which is the

H-bridge inverter

Buck converter plus unfolder

Unfolder: similar to bridge rectifier, but power flows in reverse direction. Implemented using transistors that switch at ac line frequency

Inverter sinewave synthesis approaches

We can employ any of the approaches we have already discussed for PWM rectifier systems:

•  Average current control•  Peak current control•  Boundary conduction mode•  Hysteretic control•  Discontinuous conduction mode control•  Cycle-by-cycle control

(and there are a few we didn’t discuss, most notably harmonic elimination, that could be employed for either rectifiers or inverters)

Synthesizing a Sinusoidal Current: Boundary Conduction Mode (BCM)

lowest possible peak current in DCM, and as a result the switching frequency in BCM is maximal. Equation (1) also

predicts that the switching frequency of BCM increases at low output powers, creating a switching frequency profile

that causes disproportional switching losses at low powers. This is demonstrated by the last expression in equation (1)

for which the power-factor is unity and the switching frequency is proportional to Rout. A lower output voltage vout(t)

results in a higher switching frequency, so the switching frequency and switching losses in BCM substantially increase

at low voltages and low powers.

Fig. 2. Illustration of the inductor current in BCM, showing soft switching transitions (ZVS, ZCS) and the variations in switching

frequency over the line cycle.

The switching frequencies in DCM and BCM are given by:

� � � ��

� ��

� ��

� � � ��

� ��

� � � ��

� ��

22

DCM: 12

BCM: 12

BCM with unity power factor :

1 ,where2

out out outs

out dc pk

out outs

out dc

out outouts out

dc out

v t v t i tf t

L i t v t i t

v t v tf t

L i t v t

v t v tRf t RL v t i t

§ ·§ · � ¨ ¸¨ ¸¨ ¸¨ ¸� © ¹© ¹

§ · �¨ ¸¨ ¸� © ¹

§ · � ¨ ¸¨ ¸

© ¹

(1)

Fig. 3 shows how the total loss in BCM distributes at various output powers. The losses in this figure are averaged

over a line cycle, and are shown in percent relative to the cycle averaged output power. The total loss is composed of

four types of loss: conduction losses, switching losses, proximity loss in the inductor, and core loss in the inductor.

The losses are computed according to the calibrated loss model presented in Section IV. The conditions for the test

are AC voltage of 220 Vrms @ 60 Hz, average AC power of 300 W, bus voltage of 425 V, and an inductor of 300 µH

built on a PQ 26/20 core. The results demonstrate that the dominant loss mechanism in BCM at low powers is

switching loss.

Inductor current waveform, BCM

Fig. 3. Distribution of losses in BCM. The vertical bars represent average losses over an AC line cycle. The losses are shown in percent

relative to the average AC output power. Switching losses dominate at low powers.

This data in Fig. 3 suggests that the weighted efficiency of BCM may be substantially improved if the low power

switching losses are reduced. According to equation (1), this may be achieved by increasing the peak current. To

demonstrate this idea, Fig. 4 compares a BCM waveform and a DCM waveform with a higher peak current. Although

both waveforms provide the same average current (iout), the DCM waveform delivers more energy to the output at

every cycle, and as a result operates with a lower switching frequency that enables lower switching losses.

Thus, to achieve a certain average current, the controller can vary either the peak current or the switching frequency

of the inductor current waveform. A higher peak current reduces the switching frequency, but also raises the RMS

current and causes more conduction losses. At each operating point, there is an optimal peak current that minimizes

the sum of these losses. To discover which peak current is optimal, Fig. 5 shows a plot of efficiency as a function of

peak current (ipk) at various DC operating points. Each curve in this figure corresponds to one DC operating point,

with fixed average current and voltage (iout and vout). The minimal ipk at every curve is 2iout, which corresponds to a

BCM waveform. The DC operating points are selected with constant ratio of voltage and current vout/iout = Rout, and

thus reside on the same output sinusoid. The efficiency is computed according to the calibrated loss model presented

in section IV, with conditions as follows: Rout = 215.1 Ω, average AC power of 225 W, bus voltage of vbus = 425 V,

and an inductor of 300µH built on a PQ 26/20 core. Each curve is label by its output power pout = voutiout, which is

given in percent relative to the maximal instantaneous output power of 450 W.

Fig. 4. By increasing the peak current in DCM, the switching frequency is reduced, while the average inductor current (iout) is unchanged.

Loss components at different solar irradiance levels, BCM(300 W, 240 Vac example)

Discontinuous conduction mode (DCM)

Fig. 3. Distribution of losses in BCM. The vertical bars represent average losses over an AC line cycle. The losses are shown in percent

relative to the average AC output power. Switching losses dominate at low powers.

This data in Fig. 3 suggests that the weighted efficiency of BCM may be substantially improved if the low power

switching losses are reduced. According to equation (1), this may be achieved by increasing the peak current. To

demonstrate this idea, Fig. 4 compares a BCM waveform and a DCM waveform with a higher peak current. Although

both waveforms provide the same average current (iout), the DCM waveform delivers more energy to the output at

every cycle, and as a result operates with a lower switching frequency that enables lower switching losses.

Thus, to achieve a certain average current, the controller can vary either the peak current or the switching frequency

of the inductor current waveform. A higher peak current reduces the switching frequency, but also raises the RMS

current and causes more conduction losses. At each operating point, there is an optimal peak current that minimizes

the sum of these losses. To discover which peak current is optimal, Fig. 5 shows a plot of efficiency as a function of

peak current (ipk) at various DC operating points. Each curve in this figure corresponds to one DC operating point,

with fixed average current and voltage (iout and vout). The minimal ipk at every curve is 2iout, which corresponds to a

BCM waveform. The DC operating points are selected with constant ratio of voltage and current vout/iout = Rout, and

thus reside on the same output sinusoid. The efficiency is computed according to the calibrated loss model presented

in section IV, with conditions as follows: Rout = 215.1 Ω, average AC power of 225 W, bus voltage of vbus = 425 V,

and an inductor of 300µH built on a PQ 26/20 core. Each curve is label by its output power pout = voutiout, which is

given in percent relative to the maximal instantaneous output power of 450 W.

Fig. 4. By increasing the peak current in DCM, the switching frequency is reduced, while the average inductor current (iout) is unchanged.

•  Higher conduction loss•  Lower switching loss•  A net improvement in

CEC efficiency

To find the optimal values of the inductance L and magnetic flux density ΔBmax, we used a computer program that

numerically scanned the weighted efficiency of every combination of these two parameters. The program runs over

each output power level, and over single DC operating points in the output sinusoid, and computes the efficiency at

each operating point, using the calibrated loss model. The resulting efficiency data is averaged according to the CEC

efficiency formula (see section V). Typical results of this simulation are shown in Fig. 10, for a PQ 26/20 magnetic

core, and conditions as detailed in Table I.

Fig. 10. Weighted CEC efficiency with a PQ 26/20 magnetic core, for a BCM controller and the proposed constant peak current

controller.

Fig. 10 shows the optimal inductance values (L) for the BCM control method, and for the proposed constant peak

current method. These optimal inductance values achieve the best mix of frequency dependent losses and Ohmic

losses. Notice that a higher inductance value in Fig. 10 does not mean a larger inductor, because the magnetic core is

the same (PQ26/20) at all the data points. Nevertheless, the optimal inductance value in BCM is considerably higher

in comparison to the optimal value of the constant current controller. The reason for this difference is that a higher

inductance is needed in BCM, to reduce the otherwise high switching frequencies that occur with this control method.

The higher inductance requires more turns of wire on the inductor core, which is one reason why BCM is less efficient

in comparison to the proposed constant peak current controller.

97.497.697.8

9898.298.498.698.8

9999.2

250 350 450 550 650 750 850

CEC

effic

ienc

y [%

]

inductance [µH]

constant peak currentBCM

Weighted efficiency vs. inductor size, DCM vs. BCM300 W, 240 Vac example

Measured Results: 300 W Microinverter Prototype

The current sense resistor is located between the FET source and ground, and senses the rising slope of the inductor

current. The only function of this sensed signal is to turn off the FET when the inductor current reaches the designated

peak current Ipk. The voltage sensor is implemented by a comparator that connects to an auxiliary winding (3 turns)

on the inductor. Its purpose is to detect the moment in which the inductor current reaches zero, the moment in which

the cycle-by-cycle integrator voltage starts to discharge (See Fig. 8). As explained in Section IV, at this moment the

switching node voltage starts oscillating, and the voltage across the inductor changes polarity. This change is detected

by the comparator, which resets the controller SR flip-flop. The oscillating output of the auxiliary winding comparator

is shown in Fig. 12 (blue waveform). This graph also shows the voltage across the cycle-by-cycle integration capacitor

(vint, yellow waveform). The inverter waveforms over a 60 Hz line cycle are shown in Fig. 13.

Fig. 12. Inverter waveforms at a DC operating point. Ch1 (yellow) cycle-by-cycle integration capacitor voltage vint(t) , Ch2 (blue) auxiliarywinding voltage sensor, at comparator output, Ch4 (green) inductor current iL(t). Conditions: Vdc= 426.8 V, Idc= 0.98 A, vout= 330.1 V, iout=

1.259 A.

Fig. 13. Inverter waveforms over a line cycle. Ch1 (yellow) AC line voltage sensor, Ch2 (blue) AC current iac(t), Ch3 (magenta) referencesignal iref(t), Ch4 (green) inductor current iL(t). Conditions: Vdc= 425 V, Idc= 0.462 A, Rload=253 Ω

The efficiency of the inverter is measured at static DC operating points. These tests are done with a power supply

at the input and a variable load resistor (Rload) at the output. To increase the accuracy of the measurements, the meters

at the input and output are filtered by large EMI inductors. The efficiency results are shown in Fig. 14. The various

curves in the figure correspond to tests with different average AC powers (Pac). At each such test the load resistor is

set to Rload=Vac,rms2/Pac, and the instantaneous output power is scanned in the range 0 … 2Pac.

Fig. 14. Efficiency measurements at DC operating points, for various average AC powers

At each power level, the AC efficiency is computed by averaging the DC efficiencies over a line cycle. The results

are again averaged by the CEC weighted average formula to obtain the overall CEC efficiency. The AC efficiency is

computed by equation (9), and the results are summarized in Table II.

� �

� ��

� � � �

/

20/

0

AC efficiency where sin

ac

ac

out

AC out ac acout

DC out

p t dtp t P t

p tdt

p t

S Z

S ZK Z

K

³

³(9)

In this equation, Pac is the average AC power, pout(t) is the output power at a DC operating point, and ηDC(pout) is the

efficiency at those operating points, as shown in Fig. 14. The weighted CEC efficiency is found to be 99.15 %.

TABLE II CEC Efficiency

CEC power level

weight average AC power Pac

Average loss over AC cycle

average ACefficiency

100 % 0.05 300 W 2.6 W 99.13 % 75 % 0.53 225 W 1.97 W 99.12 % 50 % 0.21 150 W 1.26 W 99.16 % 30 % 0.12 90 W 0.7 W 99.22 % 20 % 0.05 60 W 0.45 W 99.24 % 10 % 0.04 30 W 0.24 W 99.2 %

Overall weighted CEC efficiency = 99.15 %

97

97.5

98

98.5

99

99.5

100

0 200 400 600

effic

ienc

y [%

]

DC output power pout [W]

300 W 225 W150 W 90 w60 W

AC line voltage

Current reference

Filtered inverter current

Instantaneous inductor current

Development of Electrical Modelof the Photovoltaic Cell, slide 1�

Photogeneration�

Semiconductor material absorbs photons and converts into hole-electron pairs if�

�Photon energy h� > Egap � �(*)�

•� Energy in excess of Egap is converted toheat�

•� Photo-generated current I0 is proportional to number of absorbed photons satisfying (*)�

photon

+

–

Charge separation�

Electric field created by diode structure separates holes and electrons�

Open circuit voltage Voc depends on diode characteristic, Voc < Egap/q�


Current source I0 models photo-generated current�

I0 is proportional to the solar irradiance, also called the “insolation”:�

�I0 = k (solar irradiance)�

Solar irradiance is measured in W/m^2�


Diode models p–n junction�

Diode i–v characteristic follows classical exponential diode equation:�

�Id = Idss (e Vd – 1)�

The diode current Id causes the terminal current Ipv to be less than or equal to the photo-generated current I0.�


Modeling nonidealities:�

R1 : defects and other leakage current mechanisms�

R2 : contact resistance and other series resistances�

Cell characteristic�

Cell output power is Ppv = IpvVpv�At the maximum power point (MPP):�

Vpv = Vmp�Ipv = Imp�

At the short circuit point:�Ipv = Isc = I0�Ppv = 0�

At the open circuit point:�Vpv = Voc�Ppv = 0�

Maximum Power Point Tracking�

Automatically operate the PV panel at its maximum power point�

I-V curve with partial shading

Power vs. voltage

Some possible MPPT algorithms:

• Perturb and observe

• Periodic scan

• Newton s method, or related hill-climbing algorithms

• What is the control variable? Where is the power measured?

!

Example MPPT: Perturb and Observe • A well-known approach"• Works well if properly tuned"• When not well tuned, maximum power

point tracker (MPPT) is slow and canget confused by rapid changes inoperating point"

• A common choice: �control� is switchduty cycle"

Basic algorithm!!Measure power"Loop:"• Perturb the operating point in

some direction"

• Wait for system to settle"

• Measure power"

• Did the power increase?"

Yes: retain direction for next perturbation"

N: reverse direction for next perturbation"

Repeat"

Control Issues: MPPT by Perturb-and-Observe

��)!' �

/�&� **-.�

�*'.�#!�!)-!��

�*).,*'�**+��

��

�� ,

��

�� /,,!).��!)-!��

��

�

�0!,�#!��

��*)0!,.!,�1%.$�� -! ��%#%.�'��*).,*'��**+

�� ,

Key elements of digital controller�

Measured power vs. commanded PV voltage�

Magnified view�

• Find PV voltage that maximizes power output• Switching converter is high noise environment• This “noise” is partly correlated to the control,

and hence isn’t entirely random• The highly-filtered dc control characteristic

exhibits many small peaks (“traps”), whereP&O algorithm gets stuck

• More noise makes P&O work better! ��

��

Typical experimental data

�� ( ) α �� ( ) α �� →� ��

• Perturb-and-observe step time of 15 msec• Perturb-and-observe algorithm may take minutes to find max power• Weather can change in seconds• Improved algorithm achieves max power in seconds• Adaptive algorithm finds max power quickly, then reduces jitter size to

improve equilibrium MPPT accuracy

r.w.erickson - university of colorado...

Documents