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Phys. Status Solidi C 8, No. 9, 2804–2809 (2010) / DOI 10.1002/pssc.201084057 p s scurrent topics in solid state physics

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Investigations of the complex impedance of photovoltaic cells under illumination Erika Kancsar, Markus Drapalik, Julian Schmid, and Viktor Schlosser *

University of Vienna, Faculty of Physics, Dept. of Electronic Materials Properties, Vienna

Received 3 October 2010, accepted 4 February 2011 Published online 8 September 2011

Keywords photovoltaic, solar cell, impedance, bandwidth * Corresponding author: e-mail viktor.schlosser@univie.ac.at, Phone: +43 4277 51428, Fax: +43 4277 51429

The low pass characteristic of a photovoltaic device in the power converting voltage range was investigated by impedance spectroscopy and signal response measure-ments. For a c-Si solar cell it was found that the band-width varies with bias voltage, illumination intensity and circuit termination. At low intensities or voltages below

+0.15 V the bandwidth becomes less than 100 Hz when the cell was terminated with a high resistive load. As a consequence reliable data acquisition of the open circuit voltage transient for a fast intensity variation (≥ 100mWcm-2/s) is limited to voltages above +0.35 V.

© 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction Although a photovoltaic power gen-erator in principle is a d.c. current source the determination of its a.c. parameters is essential for (i) basic device char-acterisation, (ii) process control procedures and (iii) out-door operation as illustrated by the following examples: Doping and impurity distribution of solar cells are com-monly deduced from impedance spectroscopy of the re-verse biased junction [1-3] The quality control during solar cell and module production prefers fast techniques to ac-cess electrical parameters. So called quasi-steady-state, QSS conditions during transient recording are widely ap-plied [4,5]. In the electrical circuit formed by the photo-voltaic array and the connected power conditioning unit the modules act as a low pass filter for the transmitted noise [6].

Two experimental approaches to access the a.c. pa-rameters of a photovoltaic device are in use: The first method frequently is referred to as the small signal or dif-ferential impedance determination. The device under test is operated at a well defined electrical working point ensur-ing a steady state. A small sinusoidally modulated signal is either electrically or optically introduced into the circuit. The current response by means of amplitude and phase is acquired and used to directly the derive real and imaginary part of the impedance, Z or admittance, Y=Z-1 described by the dynamic conductance, G and the dynamic capacitance,

C. The decay method records the current or voltage tran-sients of a solar cell after an instantaneous change of the working point. Since the observable transient is determined by the product of the real and the imaginary part the com-plete complex impedance can solely be reconstructed by applying model assumptions [4,7]. For a photovoltaic cell C as well as G depend individually on voltage and fre-quency. Therefore assumptions most often apply only for a limited range. Nevertheless the short acquisition time makes this method attractive for quality control applica-tions [10]. Only little effort was made so far to determine experimentally Z in the operating regime of solar cells from small signal experiments [9,11].

In this work we investigated the correlation between results from small signal and transient experiments. Both approaches are analysed with respect of the device’s equivalent electronic circuit model [9] avoiding a physical interpretation based on possibly unjustified assumption.

2 Experimental Three sets of experiments were car-ried out on a c-Si solar cell. The sample we investigated was delivered by Solartek (Czech Republic) with an rec-tangular area, A=8.8 cm2 and a thickness of 320 μm. It was made from nominally 1 Ωcm p-Si with a planar, diffused pn-junction.

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2.1 Determination of the differential imped-ance We used a conventional set up for the determination of the dynamic impedance. The source for C(V), G(V) and drive level capacitance-profiling, DLCP [3] measurements was a 20 MHz function generator (Agilent, Model 33220A) which supports both, the d.c. voltage and an sinu-soidal a.c. signal, VAC which was fed into a voltage con-trolled power amplifier (±12V, ±3A, 1MHz bandwidth). The electronics of the self made amplifier was capable to act as voltage source as well as a voltage controlled current sink. The photovoltaic device was connected in series with an induction-less thick film 5W resistor. The resistor serves as current shunt and was varied between 1 Ω and 1 kΩ. For data acquisition either a 60 MHz oscilloscope, DSO (Agilent, Model 3062) or a dual phase lock in ampli-fier, LIA (Stanford Research, model 830) was used. The DSO was used because of its higher bandwidth, and re-duced sensitivity to noise overload during measurements of the illuminated sample. Measurements at lower frequencies (below 100 kHz), in the dark were preferentially carried out with the LIA. With the oscilloscope the total voltage and the current could be acquired simultaneously. Both re-cords in the time domain was numerically Fourier trans-formed in the PC and amplitude and phase of the a.c. volt-age and current were evaluated together with the d.c. volt-age from the DSO signals. The d.c. current, the cell tem-perature and light intensity were measured with a Keithley multimeter (model 199) with an integrated input source scanner unit sequentially. When we used the LIA for the experiment its input was wired to the scanner output of the multimeter and the total voltage signal was connected to one spare scanner channel input. a.c. voltage and current signals were alternately detected and registered by the LIA. In order to derive real and imaginary part of the impedance as a function of the applied d.c. voltage the set up was ei-ther operated at a constant frequency (referred further on as CF measurement) or at a constant phase between G and ωC (CP condition) as suggested by Suresh [8]. Here ω is the angular frequency (ω=2πf).

We started our experiments with conventional C(V) and G(V) measurements in the dark for the reverse biased solar cell between –1.9 V and 0 V for two reasons: (i) to proof the reliability of our set up and measurement condi-tions. (ii) to access the depletion capacitance which is still present at forward voltages however superimposed by the exponentially growing diffusion capacitance.

According to the classical space charge model of an abrupt pn-junction [1] our data were used to determine the acceptor concentration of the sample. In Fig. 1 the evalua-tion of the acceptor concentrations as a function of the space charge region width, W is shown. Results from a C(V) measurement at f=3.8 kHz, VAC=20mVrms and from a DLCP configured experiment in the frequency range 3.6 kHz±1 Oct. are plotted. All experiments were carried out in the dark at T =(306.6±0.1) K using the CF method. The boron doping concentration of the substrate remains independent from the distance and the measurement fre-

quency at NSub≈1×1016cm-3 which corresponds to the speci-fied resistivity of 1 Ωcm. Following the theory of DLCP [3] an additional impurity concentration, NImp from Eq. (1) was obtained.

30

Imp 212

CN

q A Ce∫ - (1)

Here q is the magnitude of the elementary charge, ε the material specific permittivity and C0 and C1 are the first 2 coefficients of a series expansion for the exact solution of C from dQ/dV. C0 corresponds to the capacitance arising from the background acceptor concentration only and equals C found from the simple C(V) measurements. NImp is about 1/5th of the doping level. At frequencies below 1.7 kHz the dependence on W began to deviate from the almost constant level found at higher frequencies indicat-ing a lower experimental frequency limit for useful data evaluation.

Figure 1 The solar cell’s base doping concentration evaluated from the differential capacitance of a reverse biased c-Si solar cell.

With the same set of experimental parameters C(V)

measurements of the forward biased cell in the dark were performed. In addition capacitance and conductance were evaluated from measurements in CP mode with a phase, ϕ of π/4. Experimentally, at a constant forward voltage the excitation frequency was scanned from 20 Hz up to 100 kHz or 1 MHz and Z was determined for f where the magnitude of Im(Z) equals Re(Z) (|X|=R). These experi-ments were carried out in the dark and repeated under il-lumination at a low intensity level. For one measurement set in the dark the parameters which match the CP criterion are summarised in Table 1. Table 1 Parameters used to determine the impedance

DC Voltage [V] Re(Z) [Ω] Frequency [Hz] +0.100 510 480 +0.200 152 1429 +0.294 32 5500 +0.404 1.5 70000

2.2 Signal response measurements In order to determine the signal response as a function of the excita-tion frequency the previous set up was slightly modified

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and the intensity modulated high power LED was used to generate an oscillating photo-current in the externally bi-ased cell. The light source was a single high power IR LED (JET-870-05 from Roithner Lasertechnik). Since the emitted intensity almost linearly varies with the LED cur-rent the function generator was used to control the current up to the bandwidth limit of 1 MHz of the LED supply. The LED emits around 870 nm up to 90 mW in continuous mode. Rise and fall time are specified to be 30 ns and 40 ns respectively. For cw operation the current is limited to 0.55 A, the permissible pulse currents ranges up to 8 A. The device under test, a shunt resistor, the voltage source and sensing instruments remain unchanged. The voltage source however was controlled by a mere d.c. voltage from one DAC output of the LIA in order to define the working point of the device. The LED current was modulated sinu-soidally with a constant amplitude and a current minimum of about 2 mA. The a.c. signal at the shunt resistor was analysed with respect to the fundamental amplitude and phase. For the light generated current the shunt resistor forms an additional path to the internal components in the circuit The low pass then is extended by the external resis-tor. Therefore the shunt resistor, Rext was used as experi-mental parameter and varied between 1 Ω and 1 kΩ in dis-crete steps.

2.3 Transient recording For the last set of ex-periments the voltage source was removed from the circuit and the cell was connected either directly or with a 10 Ω termination to the input of the DSO. The channel imped-ance being 1 MΩ. The first case approximates the open circuit and the second the short circuit condition for the cell. The LED intensity was either pulse or triangle shaped modulated by the function generator. This set-up was used to derive the open circuit voltage and the short circuit cur-rent for (i) steady state (ii) abrupt step or (iii) QSS like conditions. For (i) and (ii) a light pulse with a pulse length of 10 ms was used and the saturated signal as well as the transient were acquired. For (iii) a symmetrical linear cur-rent ramp with varying period was generated.

3 Results and discussion 3.1 Determination of the differential imped-

ance As already mentioned the first approach to deter-mine the admittance for the forward biased device was the extension of CF measurements from the reverse diode re-gime to forward bias voltages. The same set of experi- mental parameters were used which have been found ap-propriate for space charge region characterisation. The re-sult is plotted in Fig. 2. Beside the originally derived C(V) and G(V) values (square symbols) reduced values are shown for capacitance and conductance (circle markers). From the total capacitance the extrapolated depletion ca-pacitance was subtracted. The conductance was corrected for a constant value accounting for the internal ohmic shunt element of the device. This was done in order to

compare our results with the theoretical diffusion capaci-tance, Cdiff and conductance, Gdiff [12].

0 expdiffB B

q qVdIG IdV k T k T

Ê ˆ= = Á ˜Ë ¯ (2)

2diff

diff

GC

t= (3)

Here I0 is the diode’s saturation current, kB the Boltz-mann constant, and τ a time constant. For comparison the numerically differentiated I(V) curve is additionally shown as line in Fig. 2.

Figure 2 Real and imaginary part of the solar cell’s small signal admittance as a function of the forward voltage. The conductance scales with the left axis, the capacitance with the right axis.

As expected from Eqs. (2) and (3) both components

exponentially increase above 0.3 V. However the slope for the capacitance in the semi-logarithmic plot is twice the slope of the conductance indicating an exponential varia-tion of τ with bias voltage. A potential argument could be that the observed conductance arise from the cell’s recom-bination current rather than from the diffusion current. It is well known that recombination currents dominate the I(V) characteristics of c-Si solar cells at low forward voltages [13]. This contribution to the diode current increase solely half as fast with voltage than the diffusion current.

As a consequence of the much faster increase of C(V) the magnitude of X-1=2πfC will exceed G around 0.3 V. That will result in an increasing cut-off frequency with in-creasing voltage up to 0.3 V. At higher voltages the cut-off frequency will decrease again. Neither from signal re-sponse measurements nor from transient measurements the existence of a bandwidth maximum around 0.3 V however could be confirmed.

For CP measurements the derived G(V) dependence remains unchanged compared with the results from the CF experiment. The capacitance however significantly differs as shown in Table 2.

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Table 2 Comparison of normalised capacitance from CF (2nd col.) and CP (3rd col.) measurements.

DC Voltage [V] C at f=3.7 kHz [nFcm-2]

C at ϕ=π/4 [nFcm-2]

+0.100 38.0 73.9 +0.200 46.2 84.3 +0.294 148.9 102.3 +0.404 24.9×103 165.0

In order to find the reason for the large discrepancy be-tween the two measurement procedures the variation of C and G with frequency were plotted. The result is shown in Fig. 3 where the parameter is the voltage. Data points to derive the C(V) and G(V) plots are marked.

Figure 3 Frequency dependence of the capacitance (top) and the conductance (bottom). Parameter is the applied voltage. Circular symbols trace values at a constant frequency, square symbols de-fine points of constant phase.

Evidently the region where C becomes independent of

the frequency begins at higher frequencies when the volt-age increases. Below the frequency independent regime C varies with ω-2. In contrast G remains frequency independ-ent from d.c. up to ω≈105 rads-1. At higher frequencies an increase with frequency will presumably begin. Tracing the symbols for CF and CP data evaluation in Fig. 3 it is obvious that the determination of G(V) was independent of the chosen method (CF, PF or ΔI/ΔV from d.c. measure-

ments). For the C(V) dependency however the photovoltaic cell experiences a transition of its operating mode during CF measurements. At low voltages the device’s capaci-tance is frequency independent. Towards higher voltages the imaginary part of the device’s impedance becomes fre-quency dependent. During the CP measurements for ϕ=π/4 the device maintains its operating point close to the fre-quency independent range of the capacitance which ap-pears more favourable for data interpretation. Intuitively the choice of ϕ which shifts the operating point completely towards the frequency independent range of C however could be advantageous. This likely will solve the observed discontinuity between C(-V) determined in the CF ar-rangement and C(+V) from CP measurements.

Further C(V) measurements in CP configuration were performed in the dark and under illumination at a low in-tensity level (<10 mWcm-2). No resolvable changes due to illumination were observed. Although the voltage depend-ence of the capacitance remains exponential its increase was only about 1/10th compared with the results from CF measurements.

In the low pass filter model the measurement fre-quency for ϕ=π/4 defines the cut-off frequency, fLP of the device under open circuit conditions.

3.2 Signal response measurements The results of the light induced current oscillations were analysed in a Bode plot in order to determine fLP. Both, the 3 dB ampli-tude attenuation and the phase change of π/4 were evalu-ated. The sometimes not even resolvable differences in the bandwidth justify our assumptions concerning the circuit. No influence of an additional inductance or series resis-tance were observed. The experimentally derived fLP were compared with calculations based on the previously deter-mined impedance according to Eq. (4)

1

2 extLP LP

G Rf

Cw p

-+= = (4)

A comparison between calculation and experimentally found fLP is shown in Fig. 4. For the two results with re-verse biased device C and G were taken only from CF measurements. In forward direction the calculation was additionally done for the data set from CP measurements.

For the reverse biased cell calculation and experiment are in good agreement (top plot). That again supports the application of the CF mode with the above given parameter set. For the forward biased cell neither the impedance de-rived from CF nor from CP measurements matches well. However the variation of fLP with Rext is better reproduced by the CP results. As mentioned earlier the choice of a phase apart of π/4 could be more appropriate for the de-termination of the impedance thus leading to a better agreement and will be further investigated.

For transient experiments the extreme limits of Rext are of special interest: 0 Ω and ∞, or in other words short cir-cuit and open circuit condition for the solar cell. Since

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these limits in the experiment often are approximated we used a value for 10 Ω for short circuit and 1 MΩ for open circuit condition and calculated the bandwith as a function of the forward voltage for both results of the impedance (Fig. 5). This was done in order to estimate (i) decay time constants from transient experiments and (ii) rates for the change of the working point during QSS data acquisition. The decay time constant, τ simply relates to the bandwidth with τ-1=ωLP.

Figure 4 A comparison between experimentally derived and calculated dependence of the cut-off frequency on the external termination resistance.

Figure 5 Calculated cut-off frequency as a function of the ap-plied voltage.

The data set from the CF measurements predict that the

bandwidth for both, short circuit and open circuit condi-tions decrease rapidly above 0.3 V. Above 0.4 V it will be below 10 Hz which is contradictory to experimental ex-

perience with d.c. measurements. The dependence of the data from CP measurements on the other side appear more realistic exhibiting a continuous increase of fLP with volt-age in the open circuit case. For short circuit conditions fLP remains almost constant. Although the plots in Fig. 5 sug-gest a cross over of the bandwidths at higher voltages this is inhibited by circuit theory. More likely both curves will approach asymptotically a common constant value.

3.3 Transient recording Once the photo-current of

a solar cell is voltage independent, the diode’s current voltage dependence can be expressed by data pairs of the short circuit current, Isc and the open circuit voltage, Voc at varying illumination intensity.

10 exp 1oc

sc sh ocB

qVI I R V

k T-Ê Ê ˆ ˆ= - +Á Á ˜ ˜Ë Ë ¯ ¯

. (5)

Here Rsh is the diode’s internal ohmic shunt contribu-tion Eq. (5) expresses essentially the same current-voltage relation as given for the dark I(V) dependency except that the diode’s internal series resistance, Rs has no effect. The results of pulsed and linearly ramped photo-voltage (≈Voc) and photo-current (≈Isc) are summarised as jsc(Voc) plots in Fig. 6. The current density, jsc=Isc/A was taken for conven-ience.

Figure 6 Correlation of short circuit current density and open circuit voltage for varying light intensities. Parameter is the sweep frequency (symbols indicate steady state results).

From the signal response experiment we expected that

the upper frequency where the current can trace the inci-dent light intensity will be around 100 kHz. For Voc track-ing the limit was expected around 100 Hz. Therefore we started the light intensity sweep at 83 kHz with a symmet-rical up and down ramp. Here however both current and voltage are distorted. At 8.3 kHz the current follows al-ready the intensity variation without noticeable delay. However the voltage remains distorted. For a frequency as low as 8.3 Hz the voltage still remains delayed at low light intensities. Around the transition from the linear increase of intensity to the linear decrease (highest intensity) no de-lay of the voltage is visible up to 830 Hz. That confirms the assumed continuous increase of bandwidth with volt-

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age and definitely rejects the existence of a maximum in the band-width around 0.3 V.

Qualitatively the results from transient experiments confirm the variation of the bandwidth that was derived from the solar cell’s impedance at a constant phase condi-tion. The magnitude however seems to be over estimated.

Assuming the sweep rate at 8.3 Hz to be slow enough to derive the QSS I(V) record with sufficient accuracy the maximum permissible rate for the short circuit current was calculated to be ±9.3 mAcm-2/s. Taking into account the standard solar cell testing procedure at AM 1.5 condition (100mWcm-2) the initial current density for c-Si solar cells is typically 35 mAcm-2. Hence a linear reduction of the in-tensity down to zero then must last 4 s to continuously maintain a QSS condition for accurate Voc recording. In-stead of a controlled linear intensity reduction the uncon-trolled intensity decay of a flash light tube is practically used. Although here the intensity decays in a much shorter time the decrease is roughly exponential. As a result ini-tially the bandwidth of the solar cell at a high Voc is large when the intensity decreases fast. When the bandwidth of the solar cell narrows the reduction of the intensity slows down. Though a non-linear intensity reduction is the fa-vourable option for solar cell testing procedures the uncer-tainty about the introduced error remains as long as the bandwidth of the device under test is unknown.

4 Conclusions Deriving the d.c. current-voltage characteristic of a c-Si solar cell from transient measure-ments can lead to large errors. The permissible rate with which the operating point of the solar cell can be varied with acceptable deviations from the steady state case de-pend on the bandwidth of the low pass circuitry. Beside the terminating external impedance the bias voltage determine this bandwidth. In particular a high external impedance along with a low operating voltage can reduce the band-width significantly. In our investigations we observed a re-duction of the bandwidth from 10 kHz to about 100 Hz in the open circuit for a voltage decrease from 0.4 V down to 0 V. This variation complicates the definition of a quasi-static-state.

From our measurements we have estimated that the linear reduction of the irradiation intensity has to be lower than 25mWcm-2/s for an accurate reproduction of the com-plete static jsc(Voc) curve. With respect of the interesting operation at maximal power output for 100 mWcm-2 the working point of a high efficiency c-Si solar cell will be around 0.5 V. Restricting the transient recording to this range allows an at least 100 times faster intensity variation.

Acknowledgements The active participation in this work of T. Kitzler and H. Ebe during the measurements is highly ac-knowledged.

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