- Widlar current source
A Widlar current source is a modification of the basic two-
transistor current mirror that incorporates an emitter degenerationresistor for only the output transistor, enabling the current source to generate low currents using only moderate resistor values.cite book |title=Analysis and design of analog integrated circuits |author=PR Gray, PJ Hurst, SH Lewis & RG Meyer |year=2001 |edition=4rth Edition |publisher=John Wiley and Sons |isbn=0-471-32168-0 |url=http://www.worldcat.org/search?q=0471321680&qt=owc_search |page= §4.4.1.1 pp. 299–303] cite book |title=Microelectronic circuits |author=AS Sedra and KC Smith |edition=5th Edition |page=Example 6.14, pp. 654-655 |isbn=0-19-514251-9 |year=2004 |publisher=Oxford University Press |url=http://www.worldcat.org/search?q=0195142519&qt=owc_search ] cite book |title=Microelectronic circuits: analysis and design |author=MH Rashid |page=pp. 661-665 |year=1999 |publisher=PWS Publishing Co. |isbn=0-534-95174-0 |url=http://www.worldcat.org/search?q=0534951740&qt=owc_search]The Widlar circuit may be used with
bipolar transistor s, MOS transistors, and evenvacuum tube s. An example application is the 741 operational amplifier,cite book |title=§9.4.2, p. 899 |author=AS Sedra and KC Smith |edition=5th Edition |isbn=0-19-514251-9 |year=2004 |url=http://www.worldcat.org/search?q=0195142519&qt=owc_search ] and Widlar used the circuit as a part in many designs.See, for example, Figure 2 in [https://www.utdallas.edu/~hellums/docs/JournalPapers/WidlarBandgap.pdf "IC voltage regulators"] .]This circuit is named after its inventor,
Bob Widlar , and was patented in 1967.RJ Widlar: US Patent Number 03320439; Filed May 26, 1965; Granted May 16, 1967: [http://patimg1.uspto.gov/.piw?Docid=03320439&homeurl=http%3A%2F%2Fpatft.uspto.gov%2Fnetacgi%2Fnph-Parser%3FSect1%3DPTO2%2526Sect2%3DHITOFF%2526p%3D1%2526u%3D%25252Fnetahtml%25252FPTO%25252Fsearch-bool.html%2526r%3D16%2526f%3DG%2526l%3D50%2526co1%3DAND%2526d%3DPALL%2526s1%3DWidlar.INNM.%2526OS%3DIN%2FWidlar%2526RS%3DIN%2FWidlar&PageNum=&Rtype=&SectionNum=&idkey=NONE&Input=View+first+page" Low-value current source for integrated circuits"] ] See Widlar: [http://scholar.google.com/scholar?as_q=&num=10&btnG=Search+Scholar&as_epq=emitter+degeneration+resistor+&as_oq=&as_eq=&as_occt=any&as_sauthors=Widlar&as_publication=&as_ylo=1965&as_yhi=1975&as_allsubj=all&hl=en&lr= "Some circuit design techniques for linear integrated circuits"] and [http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=1049994 "Design techniques for monolithic operational amplifiers"] ]Analysis
Figure 1 is an example Widlar current source using bipolar transistors, where the emitter resistor R2 is connected to the output transistor Q2, and has the effect of reducing the current in Q2 relative to Q1. The key to this circuit is that the voltage drop across the resistor "R"2 subtracts from the base-emitter voltage of transistor "Q"2, thereby turning this transistor off compared to transistor "Q"1. This observation is expressed by equating the base voltage expressions found on either side of the circuit in Figure 1 as:
:
where β2 is the beta-value of the output transistor, which is not the same as that of the input transistor, in part because the currents in the two transistors are very different.cite book |title=Figure 2.38, p. 115 |author=PR Gray, PJ Hurst, SH Lewis & RG Meyer |year=2001 |isbn=0-471-32168-0 |url=http://www.worldcat.org/search?q=0471321680&qt=owc_search ] The variable "I"B2 is the base current of the output transistor, "V"BE refers to base-emitter voltage. This equation implies (using the Shockley diode law):
"Eq. 1"anchor|Eq1:where "V"T is the .
This equation makes the approximation that the currents are both much larger than the "scale currents" "I"S1, "I"S2, an approximation valid except for current levels near cut off. In the following the distinction between the two scale currents is dropped, although the difference can be important, for example, if the two transistors are chosen with different areas.
Design procedure with specified currents
To design the mirror, the output current must be related to the two resistor values "R"1 and "R"2. A basic observation is that the output transistor is in active mode only so long as its collector-base voltage is non-zero. Thus, the simplest bias condition for design of the mirror sets the applied voltage "V"A to equal the base voltage "V"B. This minimum useful value of "V"A is called the "compliance voltage" of the current source. With that bias condition, the
Early effect plays no role in the design.Of course, one might imagine a design where the output resistance of the mirror is a major consideration. Then a different approach is necessary.]These considerations suggest the following design procedure:
* Select the desired output current, "I"O = "I"C2.
* Select the reference current, "I"R1, assumed to be larger than the output current, probably considerably larger. (That is the purpose of the circuit.)
* Determine the input collector current of "Q"1, "I"C1:::
* Determine the base voltage "V"BE1 using the Shockley diode law:::where "I"S is a device parameter sometimes called the "scale current".:The value of base votlage also sets the compliance voltage "V"A = "V"BE1. This voltage is the lowest voltage for which the mirror works properly.
* Determine "R"1:::anchor|R2* Determine the emitter leg resistance "R"2 using "Eq. 1" (to reduce clutter, the scale currents are chosen equal):::Finding the current with given resistor values
The inverse of the design problem is finding the current when the resistor values are known. An iterative method is described next. Assume the current source is biased so the collector-base voltage of the output transistor "Q"2 is zero. The current through "R"1 is the input or reference current given as,
:
:: ::Rearranging, "I"C1 is found as:
"Eq. 2"anchor|Eq2:
The diode equation provides:
anchor|Eq3"Eq. 3":
"Eq.1" provides::
These three relations are a nonlinear, implicit determination for the currents that can be solved by iteration.
* We guess starting values for "I"C1 and "I"C2.
* We find a value for "V"BE1:::
* We find a new value for "I"C1:::
*We find a new value for "I"C2:::This procedure is repeated to convergence, and is set up conveniently in a spreadsheet. One simply uses a macro to copy the new values into the spreadsheet cells holding the initial values to obtain the solution in short order.Note that with the circuit as shown, if VCC changes, the output current will change. Hence, to keep the output current constant despite fluctuations in "V"CC, the circuit should be driven by a constant current source rather than using the resistor "R"1.
Exact solution
The
transcendental equation s above can be solved exactly in terms of theLambert W function .Output impedance
An important property of a current source is its small signal incremental output impedance, which should ideally be infinite. The Widlar circuit introduces local current feedback for transistor . Any increase in the current in "Q"2 increases the voltage drop across "R"2, reducing the "V"BE for "Q"2, thereby countering the increase in current. This feedback means the output impedance of the circuit is increased, because the feedback involving "R"2 forces use of a larger voltage to drive a given current.
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