A novel measurement method for extraction of the base resistance in the BJT

Microelectronics Journal Microelectronics Journal 30 (1999) 41–44

A novel measurement method for extraction of the base resistance in the BJT Ali Toker* ˙Istanbul Technical University, Electrical and Electronics Faculty, Electronics and Communications Department, 80626 Maslak, I˙stanbul, Turkey Accepted 29 June 1998

Abstract A novel high frequency measurement method for determining the base resistance of the bipolar transistor is presented. The new method is based on the short-circuit input impedance measurements. The measurement is performed at lower frequencies compared to the circlediagram method and therefore can be applied very easily. Furthermore, it is also possible to determine some other BJT parameters from these measurements. The new method is given in detail and correction factors to parasitics are derived. It was shown that the measurement results are in good agreement with results of conventional methods. 䉷 1998 Elsevier Science Ltd. All rights reserved. Keywords: Measurement methods; Base resistance; BJT

1. Introduction Computer simulation with SPICE is one of the most important and indispensable steps in both analog and digital IC design. The accuracy of computer simulation is limited primarily by the accuracy of the device model, because computer analysis is only as accurate as the model used in the simulation program. On the other hand, accurate circuit simulation is possible only if accurate meaningful parameters are specified for each device model. Therefore, parameter extraction from measurement data plays also an important role in IC design, since it directly influences the precision of computer analysis. There are several algorithms for extracting model parameters of the BJT manually and with computer programs [13–15]. The base resistance of bipolar transistors has a strong influence on device performance in several areas. The base resistance has a dominant effect on the high frequency signal performance, on the noise performance and on the distortion performance of the device. For this reason and for true device modeling, accurate methods of estimating the value of bipolar transistor base resistance are necessary. Although a large amount of papers have been published on this subject, various methods of estimating base resistance ¨ . Elektrik–Elektronik Faku¨ltesi, Ayazagˇa Kampu¨su¨, 80626 * I˙.T.U Maslak-I˙stanbul Tu¨rkI˙ye. Fax: +90-0212-2853679; E-mail: alitoker@ ehb.itu.edu.tr

have different disadvantages. Some of these methods show difficulties in application such as noise measurement based methods and some of them yield unreliable results. Therefore the spreading base resistance is one of the hardest to estimate model parameters of the BJT [1–5] On the other hand, because of the influence on the high frequency signal performance of the base resistance, many high frequency based methods such as the circle-diagram method are investigated. In these methods, the measurement frequencies are relatively high and close to the transition frequency (f T). At these frequencies the distributed character of the base resistance and the parasitic effects influence adversely the measurement results [6–8]. In this work, a new method is proposed which is based on the input impedance measurement at relatively lower frequencies compared to the transition frequency. Application of this method is easy and the results are satisfactory. In addition, some other parameters can also be simultaneously extracted [9,10]. The effects of the second-order parameters on the results are investigated.

2. The new method It is a well-known behavior that the module of the input impedance of the BJT in common emitter configuration is inversely proportional to frequency. The plot of the module of the input impedance as a function of frequency has a

0026-2692/99/$ - see front matter 䉷 1998 Elsevier Science Ltd. All rights reserved. PII: S0 02 6 -2 6 92 ( 98 ) 00 0 82 - 2

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A. Toker / Microelectronics Journal 30 (1999) 41–44

Fig. 1. The hybrid-p equivalent circuit of the BJT.

semicircular shape. The intersection of this curve with the frequency axis in the increasing frequency direction gives the value of the base resistance. The measurements must be performed in the high frequency region very close to the transition frequency for estimating the intersection point. On the other hand, the distributed character of the base region raises difficulties for the estimation of this point [6,7]. In this work, instead of the module of the input impedance, its real part is examined, which is inversely proportional to the square of the frequency for high frequencies. It is well known, that the hybrid-p high-frequency small signal model of the BJT gives satisfactory results at frequencies smaller than f T/2 and yields exact results when f ⬍ f T/3 where f T is transition frequency [9,11,12]. The real part of the input impedance, which is derived from Fig. 1 with the short-circuited output, is given by: Re{Zi } ¼ rbb⬘ þ

hfeo re 1 þ (f =fhfe )2

(1)

where r bb⬘ denotes the base resistance, g m ¼ 1/r e ¼ I Cq/kT denotes the transconductance, h feo denotes low frequency current gain and f hfe denotes the cut-off frequency of the current gain (h fe) of the BJT respectively. Since Re{Z i} can be measured and the second term of Eq. (1) can be calculated, let k ¼ f/f hfe and solve Eq. (1) for r bb⬘: rbb⬘ ¼ Re{Zi } ¹

hfeo re 1 þ k2

(2)

from which the value of the base resistance can be easily calculated. It is a well known fact of general error analysis that if some value of any quantity is calculated as the difference of two other measured values, the relative error of the result becomes inadmissibly high if the two measured values are much greater than the calculated difference. If the second term of Eq. (1) is carefully observed, it is easily seen that for frequencies which are greater than only several times f hfe, the value of this term quickly decreases with f 2 and becomes comparable to r bb⬘. This is very helpful since the relative error will now be much smaller when calculating r bb⬘ with Eq. (2). For k ¼ 3, the second term of Eq. (1) decreases to one-tenth and for k ¼ 5, to one-twenty-sixth of

its low frequency value. Therefore, the measurements are performed at relatively lower frequencies compared to the transition frequency, which is given by: fT ¼ hfeo fhfe

(3)

Since, in most situations, the condition k Ⰶ h feo is satisfied, the measuring frequency will be always much smaller than f T. An additional advantage of the new method is in the possibility of estimating several other parameters of the BJT using the same measurement set-up. As an example, the measurement of the real part of the input impedance yields at low frequencies Re{Zi } ¼ rbb⬘ þ hfeo re ⬵ hfeo re

(4)

The value of the current gain at low frequencies can be determined from Eq. (4). Furthermore, the cut-off frequency (f hfe) can be approximately determined from the frequency value, where the module of the input impedance decreases to 0.707 times the low frequency value. So all the parameters in Eq. (3) can be simultaneously determined using the same test set-up. 3. Second-order effects There are several second-order effects, which influence the measurement method discussed above. These are caused by the other body resistances, the lead inductances and also by the parasitic capacitances between the pins if the transistor is a discrete device. Their effects on the measurement results are investigated. 3.1. The effect of the emitter resistance By adding the emitter resistance (r ee⬘) into the hybrid-p small signal model, the effect of this parameter can be determined from the input impedance calculation. If the condition C b⬘e q C cb⬘ is satisfied and, taking into account that the measurements are made at relatively lower frequencies compared to the transition frequency, then: rbb⬘ ⬵ Re{Zi } ¹

hfeo (re þ ree⬘ ) k2 ¹ ree⬘ 2 1 þ k2 1þk

(5)

where f ¼ kf hfe is the measuring frequency. It can be seen

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which has the greatest influence on the measurement results, is the emitter inductance (L e). Taking into account the effect of the emitter inductance, the base resistance can be calculated as follows:

Table 1 Comparison of experimental results Transistor

Base resistance measured with other methods (Q)

Base resistance measured with the new method (Q)

2N3866 (NPN) BFT66 (NPN) BF967 (PNP) BF509S (PNP)

⬍7 ⬍ 10 10 35

6 9.5 9.7 35

rbb⬘ ⬇ [Re{Zi }]f ¼ kfhfe ¹

3.2. The effect of the collector resistance The effect of the collector resistance (r cc⬘) can also be investigated. At measurement frequencies the following conditions are almost satisfied: (6)

Taking into account the conditions given above, the real part of the input impedance of the BJT can be expressed from the hybrid-p small signal model including r cc⬘ with short circuited output: Re{Zi } ⬵ rbb⬘

hfeo re 1 þ (f =fhfe þ f =fc )2

(11)

It can be seen from Eq. (11) that L e is more effective on measurement results at high collector currents.

easily from this equation that the value of the base resistance calculated from Eq. (2) is greater than the real value.

q2 Cb⬘e Ccb⬘ hfeo re rcc⬘ p 1 and qCcb⬘ rcc⬘ p 1

hfeo re k2 Le =re ¹ 2 2C 1þk 1 þ k b⬘e þ Ccb⬘

(7)

3.4. The effect of the parasitic capacitances The sum of the parasitic capacitances (C K) between B–E and C–B is shunting the input port of the measured transistor because the output is short-circuited. Let Z⬘i be the input impedance when parasitic capacitances are present. Taking the total effect of the parasitic capacitances into account, Re{Z i} can be determined in terms of the measurement value Re{Z⬘i } and the parasitic capacitance C K with the following equation: Re{Zi } ¼

Re{Zi ⬘} (1 þ qCK Im{Z⬘i })2 þ (qCK Re{Z⬘i })2

(12)

If this correction is not made, the base resistance value calculated from Eq. (2) will be greater than the actual value because Re{Z i} is smaller than Re{Z⬘i }.

where f c is determined as follows: 4. Experimental results

1 fc ¼ 2pCcb⬘ rcc⬘ (1 þ hfeo )

(8)

When the influence of the collector resistance on the measurement results is not negligible, k ¼ f/f hfe must be replaced by: k ¼ f (1=fhfe þ 1=fc ) ¼ f =f ⬘hfe

(9)

In this case, the measurement is performed at f ¼ kf ⬘h f e instead of f ¼ kf hfe, and the base resistance can be determined from Eq. (7), which now becomes: rbb⬘ ¼ [Re{Zi }]f ¼ kf ⬘hfe ¹

hfeo re 1 þ k2

To verify the method presented above, many measurements were made on various transistors and compared to the results obtained with other known methods. Consistent results have been obtained even for the samples with low values of base resistance. Some examples are shown in Table 1. From the comparative results given in Table 1, it is obvious that the results of the proposed methods are in good agreement with the results of conventional methods even for the overlay transistor 2N3866 which has a small base resistance.

(10)

If the cut-off frequency of the short circuit current gain (f hfe) is determined from the input impedance measurements data, then the correction given above is not necessary, because the value of the cut-off frequency which is determined in this way is actually f ⬘hfe and not f hfe. 3.3. The effect of the emitter inductance In practice, the results are influenced by the lead inductances of the transistors with very high transition frequency. Because the value of the base resistance is determined from the real part of the input impedance, the lead inductance of the base has no effect on the results. The lead inductance,

5. Conclusions The measurement method presented above for the extraction of the base resistance has been examined in detail and correction factors due to the parasitics have been derived. Some other parameters of the BJT can be simultaneously obtained from the same measuring set-up. Since the measurements are performed at relatively low frequencies, the effects of the parasitics are small and the measurements are realized easily. Furthermore, the corrections due to the second-order effects can be made easily. Finally, the results are in good agreement with the results of the other methods.

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