# Current source

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A current source is an electronic circuit that delivers or absorbs an electric current which is independent of the voltage across it.

## Contents

A current source is the dual of a voltage source. The term current sink is sometimes used for sources fed from a negative voltage supply. Figure 1 shows the schematic symbol for an ideal current source driving a resistive load. There are two types. An independent current source (or sink) delivers a constant current. A dependent current source delivers a current which is proportional to some other voltage or current in the circuit.

## Background

 Voltage source Current source Controlled voltage source Controlled current source Battery of cells Single cell
Figure 2: Source symbols

An ideal current source generates a current that is independent of the voltage changes across it. An ideal current source is a mathematical model, which real devices can approach very closely. If the current through an ideal current source can be specified independently of any other variable in a circuit, it is called an independent current source. Conversely, if the current through an ideal current source is determined by some other voltage or current in a circuit, it is called a dependent or controlled current source. Symbols for these sources are shown in Figure.

The internal resistance of an ideal current source is infinite. An independent current source with zero current is identical to an ideal open circuit. The voltage across an ideal current source is completely determined by the circuit it is connected to. When connected to a short circuit, there is zero voltage and thus zero power delivered. When connected to a load resistance, the voltage across the source approaches infinity as the load resistance approaches infinity (an open circuit).

No physical current source is ideal. For example, no physical current source can operate when applied to an open circuit. There are two characteristics that define a current source in real life. One is its internal resistance and the other is its compliance voltage. The compliance voltage is the maximum voltage that the current source can supply to a load. Over a given load range, it is possible for some types of real current sources to exhibit nearly infinite internal resistance. However, when the current source reaches its compliance voltage, it abruptly stops being a current source.

In circuit analysis, a current source having finite internal resistance is modeled by placing the value of that resistance across an ideal current source (the Norton equivalent circuit). However, this model is only useful when a current source is operating within its compliance voltage.

## Implementations

### Passive current source

The simplest non-ideal current source consists of a voltage source in series with a resistor. The amount of current available from such a source is given by the ratio of the voltage across the voltage source to the resistance of the resistor (Ohm's law; I = V/R). This value of current will only be delivered to a load with zero voltage drop across its terminals (a short circuit, an uncharged capacitor, a charged inductor, a virtual ground circuit, etc.) The current delivered to a load with nonzero voltage (drop) across its terminals (a linear or nonlinear resistor with a finite resistance, a charged capacitor, an uncharged inductor, a voltage source, etc.) will always be different. It is given by the ratio of the voltage drop across the resistor (the difference between the exciting voltage and the voltage across the load) to its resistance. For a nearly ideal current source, the value of the resistor should be very large but this implies that, for a specified current, the voltage source must be very large (in the limit as the resistance and the voltage go to infinity, the current source will become ideal and the current will not depend at all on the voltage across the load). Thus, efficiency is low (due to power loss in the resistor) and it is usually impractical to construct a 'good' current source this way. Nonetheless, it is often the case that such a circuit will provide adequate performance when the specified current and load resistance are small. For example, a 5 V voltage source in series with a 4.7 kΩ resistor will provide an approximately constant current of 1 mA ± 5% to a load resistance in the range of 50 to 450 Ω.

A Van de Graaff generator is an example of such a high voltage current source. It behaves as an almost constant current source because of its very high output voltage coupled with its very high output resistance and so it supplies the same few microamperes at any output voltage up to hundreds of thousands of volts (or even tens of megavolts) for large laboratory versions.

### Active current sources without negative feedback

In these circuits the output current is not monitored and controlled by means of negative feedback.

#### Current-stable nonlinear implementation

They are implemented by active electronic components (transistors) having current-stable nonlinear output characteristic when driven by steady input quantity (current or voltage). These circuits behave as dynamic resistors changing their present resistance to compensate current variations. For example, if the load increases its resistance, the transistor decreases its present output resistance (and vice versa ) to keep up a constant total resistance in the circuit.

Active current sources have many important applications in electronic circuits. They are often used in place of ohmic resistors in analog integrated circuits (e.g., a differential amplifier) to generate a current that depends slightly on the voltage across the load.

The common emitter configuration driven by a constant input current or voltage and common source (common cathode) driven by a constant voltage naturally behave as current sources (or sinks) because the output impedance of these devices is naturally high. The output part of the simple current mirror is an example of such a current source widely used in integrated circuits. The common base, common gate and common grid configurations can serve as constant current sources as well.

A JFET can be made to act as a current source by tying its gate to its source. The current then flowing is the IDSS of the FET. These can be purchased with this connection already made and in this case the devices are called current regulator diodes or constant current diodes or current limiting diodes (CLD). An enhancement-mode N-channel MOSFET (metal-oxide-semiconductor field-effect transistor) can be used in the circuits listed below.

#### Following voltage implementation

An example: bootstrapped current source. [1]

#### Voltage compensation implementation

The simple resistor passive current source is ideal only when the voltage across it is 0; so voltage compensation by applying parallel negative feedback might be considered to improve the source. Operational amplifiers with feedback effectively work to minimise the voltage across their inputs. This results in making the inverting input a virtual ground, with the current running through the feedback, or load, and the passive current source. The input voltage source, the resistor, and the op-amp constitutes an "ideal" current source with value, IOUT = VIN/R. The op-amp voltage-to-current converter in Figure 3, a transimpedance amplifier and an op-amp inverting amplifier are typical implementations of this idea.

#### Current compensation implementation

A typical example are Howland current source [2] and its derivative Deboo integrator. [3] In the last example (Fig. 1), the Howland current source consists of an input voltage source, VIN, a positive resistor, R, a load (the capacitor, C, acting as impedance Z) and a negative impedance converter INIC (R1 = R2 = R3 = R and the op-amp). The input voltage source and the resistor R constitute an imperfect current source passing current, IR through the load (Fig. 3 in the source). The INIC acts as a second current source passing "helping" current, I−R, through the load. As a result, the total current flowing through the load is constant and the circuit impedance seen by the input source is increased. However the Howland current source isn't widely used because it requires the four resistors to be perfectly matched, and its impedance drops at high frequencies. [4]

### Current sources with negative feedback

They are implemented as a voltage follower with series negative feedback driven by a constant input voltage source (i.e., a negative feedback voltage stabilizer). The voltage follower is loaded by a constant (current sensing) resistor acting as a simple current-to-voltage converter connected in the feedback loop. The external load of this current source is connected somewhere in the path of the current supplying the current sensing resistor but out of the feedback loop.

The voltage follower adjusts its output current IOUT flowing through the load so that to make the voltage drop VR = IOUTR across the current sensing resistor R equal to the constant input voltage VIN. Thus the voltage stabilizer keeps up a constant voltage drop across a constant resistor; so, a constant current IOUT = VR/R = VIN/R flows through the resistor and respectively through the load.

If the input voltage varies, this arrangement will act as a voltage-to-current converter (voltage-controlled current source, VCCS); it can be thought as a reversed (by means of negative feedback) current-to-voltage converter. The resistance R determines the transfer ratio (transconductance).

Current sources implemented as circuits with series negative feedback have the disadvantage that the voltage drop across the current sensing resistor decreases the maximal voltage across the load (the compliance voltage).

#### Simple transistor current sources

##### Constant current diode

The simplest constant-current source or sink is formed from one component: a JFET with its gate attached to its source. Once the drain-source voltage reaches a certain minimum value, the JFET enters saturation where current is approximately constant. This configuration is known as a constant-current diode, as it behaves much like a dual to the constant voltage diode (Zener diode) used in simple voltage sources.

Due to the large variability in saturation current of JFETs, it is common to also include a source resistor (shown in the adjacent image) which allows the current to be tuned down to a desired value.

##### Zener diode current source

In this bipolar junction transistor (BJT) implementation (Figure 4) of the general idea above, a Zener voltage stabilizer (R1 and DZ1) drives an emitter follower (Q1) loaded by a constant emitter resistor (R2) sensing the load current. The external (floating) load of this current source is connected to the collector so that almost the same current flows through it and the emitter resistor (they can be thought of as connected in series). The transistor, Q1, adjusts the output (collector) current so as to keep the voltage drop across the constant emitter resistor, R2, almost equal to the relatively constant voltage drop across the Zener diode, DZ1. As a result, the output current is almost constant even if the load resistance and/or voltage vary. The operation of the circuit is considered in details below.

A Zener diode, when reverse biased (as shown in the circuit) has a constant voltage drop across it irrespective of the current flowing through it. Thus, as long as the Zener current (IZ) is above a certain level (called holding current), the voltage across the Zener diode (VZ) will be constant. Resistor, R1, supplies the Zener current and the base current (IB) of NPN transistor (Q1). The constant Zener voltage is applied across the base of Q1 and emitter resistor, R2.

Voltage across R2 (VR2) is given by VZVBE, where VBE is the base-emitter drop of Q1. The emitter current of Q1 which is also the current through R2 is given by

${\displaystyle I_{\text{R2}}(=I_{\text{E}}=I_{\text{C}})={\frac {V_{\text{R2}}}{R_{\text{R2}}}}={\frac {V_{\text{Z}}-V_{\text{BE}}}{R_{\text{R2}}}}.}$

Since VZ is constant and VBE is also (approximately) constant for a given temperature, it follows that VR2 is constant and hence IE is also constant. Due to transistor action, emitter current, IE, is very nearly equal to the collector current, IC, of the transistor (which in turn, is the current through the load). Thus, the load current is constant (neglecting the output resistance of the transistor due to the Early effect) and the circuit operates as a constant current source. As long as the temperature remains constant (or doesn't vary much), the load current will be independent of the supply voltage, R1 and the transistor's gain. R2 allows the load current to be set at any desirable value and is calculated by

${\displaystyle R_{\text{R2}}={\frac {V_{\text{Z}}-V_{\text{BE}}}{I_{\text{R2}}}}}$

where VBE is typically 0.65 V for a silicon device. [5]

(IR2 is also the emitter current and is assumed to be the same as the collector or required load current, provided hFE is sufficiently large). Resistance, RR1, at resistor, R1, is calculated as

${\displaystyle R_{\text{R1}}={\frac {V_{\text{S}}-V_{\text{Z}}}{I_{\text{Z}}+K\cdot I_{\text{B}}}}}$

where K = 1.2 to 2 (so that RR1 is low enough to ensure adequate IB),

${\displaystyle I_{\text{B}}={\frac {I_{\text{C}}}{h_{FE,{\text{min}}}}}}$

and hFE,min is the lowest acceptable current gain for the particular transistor type being used.

##### LED current source

The Zener diode can be replaced by any other diode; e.g., a light-emitting diode LED1 as shown in Figure 5. The LED voltage drop (VD) is now used to derive the constant voltage and also has the additional advantage of tracking (compensating) VBE changes due to temperature. RR2 is calculated as

${\displaystyle R_{\text{R2}}={\frac {V_{\text{D}}-V_{\text{BE}}}{I_{\text{R2}}}}}$

and R1 as

${\displaystyle R_{\text{R1}}={\frac {V_{\text{S}}-V_{\text{D}}}{I_{\text{D}}+K\cdot I_{\text{B}}}}}$, where ID is the LED current
##### Transistor current source with diode compensation

Temperature changes will change the output current delivered by the circuit of Figure 4 because VBE is sensitive to temperature. Temperature dependence can be compensated using the circuit of Figure 6 that includes a standard diode, D, (of the same semiconductor material as the transistor) in series with the Zener diode as shown in the image on the left. The diode drop (VD) tracks the VBE changes due to temperature and thus significantly counteracts temperature dependence of the CCS.

Resistance R2 is now calculated as

${\displaystyle R_{2}={\frac {V_{\text{Z}}+V_{\text{D}}-V_{BE}}{I_{\text{R2}}}}}$

Since VD = VBE = 0.65 V, [6]

${\displaystyle R_{2}={\frac {V_{\text{Z}}}{I_{\text{R2}}}}}$

(In practice, VD is never exactly equal to VBE and hence it only suppresses the change in VBE rather than nulling it out.)

R1 is calculated as

${\displaystyle R_{1}={\frac {V_{\text{S}}-V_{\text{Z}}-V_{\text{D}}}{I_{\text{Z}}+K\cdot I_{\text{B}}}}}$

(the compensating diode's forward voltage drop, VD, appears in the equation and is typically 0.65 V for silicon devices. [6] )

##### Current mirror with emitter degeneration

Series negative feedback is also used in the two-transistor current mirror with emitter degeneration. Negative feedback is a basic feature in some current mirrors using multiple transistors, such as the Widlar current source and the Wilson current source.

##### Constant current source with thermal compensation

One limitation with the circuits in Figures 5 and 6 is that the thermal compensation is imperfect. In bipolar transistors, as the junction temperature increases the Vbe drop (voltage drop from base to emitter) decreases. In the two previous circuits, a decrease in Vbe will cause an increase in voltage across the emitter resistor, which in turn will cause an increase in collector current drawn through the load. The end result is that the amount of 'constant' current supplied is at least somewhat dependent on temperature. This effect is mitigated to a large extent, but not completely, by corresponding voltage drops for the diode, D1, in Figure 6, and the LED, LED1, in Figure 5. If the power dissipation in the active device of the CCS is not small and/or insufficient emitter degeneration is used, this can become a non-trivial issue.

Imagine in Figure 5, at power up, that the LED has 1 V across it driving the base of the transistor. At room temperature there is about 0.6 V drop across the Vbe junction and hence 0.4 V across the emitter resistor, giving an approximate collector (load) current of 0.4/Re amps. Now imagine that the power dissipation in the transistor causes it to heat up. This causes the Vbe drop (which was 0.6 V at room temperature) to drop to, say, 0.2 V. Now the voltage across the emitter resistor is 0.8 V, twice what it was before the warmup. This means that the collector (load) current is now twice the design value! This is an extreme example of course, but serves to illustrate the issue.

The circuit to the left overcomes the thermal problem (see also, current limiting). To see how the circuit works, assume the voltage has just been applied at V+. Current runs through R1 to the base of Q1, turning it on and causing current to begin to flow through the load into the collector of Q1. This same load current then flows out of Q1's emitter and consequently through Rsense to ground. When this current through Rsense to ground is sufficient to cause a voltage drop that is equal to the Vbe drop of Q2, Q2 begins to turn on. As Q2 turns on it pulls more current through its collector resistor, R1, which diverts some of the injected current in the base of Q1, causing Q1 to conduct less current through the load. This creates a negative feedback loop within the circuit, which keeps the voltage at Q1's emitter almost exactly equal to the Vbe drop of Q2. Since Q2 is dissipating very little power compared to Q1 (since all the load current goes through Q1, not Q2), Q2 will not heat up any significant amount and the reference (current setting) voltage across Rsense will remain steady at ≈0.6 V, or one diode drop above ground, regardless of the thermal changes in the Vbe drop of Q1. The circuit is still sensitive to changes in the ambient temperature in which the device operates as the BE voltage drop in Q2 varies slightly with temperature.

#### Op-amp current sources

The simple transistor current source from Figure 4 can be improved by inserting the base-emitter junction of the transistor in the feedback loop of an op-amp (Figure 7). Now the op-amp increases its output voltage to compensate for the VBE drop. The circuit is actually a buffered non-inverting amplifier driven by a constant input voltage. It keeps up this constant voltage across the constant sense resistor. As a result, the current flowing through the load is constant as well; it is exactly the Zener voltage divided by the sense resistor. The load can be connected either in the emitter (Figure 7) or in the collector (Figure 4) but in both the cases it is floating as in all the circuits above. The transistor is not needed if the required current doesn't exceed the sourcing ability of the op-amp. The article on current mirror discusses another example of these so-called gain-boosted current mirrors.

#### Voltage regulator current sources

The general negative feedback arrangement can be implemented by an IC voltage regulator (LM317 voltage regulator on Figure 8). As with the bare emitter follower and the precise op-amp follower above, it keeps up a constant voltage drop (1.25 V) across a constant resistor (1.25 Ω); so, a constant current (1 A) flows through the resistor and the load. The LED is on when the voltage across the load exceeds 1.8 V (the indicator circuit introduces some error). The grounded load is an important advantage of this solution.

#### Curpistor tubes

Nitrogen-filled glass tubes with two electrodes and a calibrated Becquerel (fissions per second) amount of 226Ra offer a constant number of charge carriers per second for conduction, which determines the maximum current the tube can pass over a voltage range from 25 to 500 V. [7]

## Current and voltage source comparison

Most sources of electrical energy (mains electricity, a battery, etc.) are best modeled as voltage sources. Such sources provide constant voltage, which means that as long as the current drawn from the source is within the source's capabilities, its output voltage stays constant. An ideal voltage source provides no energy when it is loaded by an open circuit (i.e., an infinite impedance), but approaches infinite power and current when the load resistance approaches zero (a short circuit). Such a theoretical device would have a zero ohm output impedance in series with the source. A real-world voltage source has a very low, but non-zero output impedance: often much less than 1 ohm.

Conversely, a current source provides a constant current, as long as the load connected to the source terminals has sufficiently low impedance. An ideal current source would provide no energy to a short circuit and approach infinite energy and voltage as the load resistance approaches infinity (an open circuit). An ideal current source has an infinite output impedance in parallel with the source. A real-world current source has a very high, but finite output impedance. In the case of transistor current sources, impedances of a few megohms (at DC) are typical.

An ideal current source cannot be connected to an ideal open circuit because this would create the paradox of running a constant, non-zero current (from the current source) through an element with a defined zero current (the open circuit). Also, a current source should not be connected to another current source if their currents differ but this arrangement is frequently used (e.g., in amplifying stages with dynamic load, CMOS circuits, etc.)

Similarly, an ideal voltage source cannot be connected to an ideal short circuit (R = 0), since this would result a similar paradox of finite non-zero voltage across an element with defined zero voltage (the short circuit). Also, a voltage source should not be connected to another voltage source if their voltages differ but again this arrangement is frequently used (e.g., in common base and differential amplifying stages).

Contrary, current and voltage sources can be connected to each other without any problems, and this technique is widely used in circuitry (e.g., in cascode circuits, differential amplifier stages with common emitter current source, etc.)

Because no ideal sources of either variety exist (all real-world examples have finite and non-zero source impedance), any current source can be considered as a voltage source with the same source impedance and vice versa. These concepts are dealt with by Norton's and Thévenin's theorems.

Charging of capacitor by constant current source and by voltage source is different. Linearity is maintained for constant current source charging of capacitor with time, whereas voltage source charging of capacitor is exponential with time. This particular property of constant current source helps for proper signal conditioning with nearly zero reflection from load.

## Related Research Articles

An amplifier, electronic amplifier or (informally) amp is an electronic device that can increase the power of a signal. It is a two-port electronic circuit that uses electric power from a power supply to increase the amplitude of a signal applied to its input terminals, producing a proportionally greater amplitude signal at its output. The amount of amplification provided by an amplifier is measured by its gain: the ratio of output voltage, current, or power to input. An amplifier is a circuit that has a power gain greater than one.

A multivibrator is an electronic circuit used to implement a variety of simple two-state devices such as relaxation oscillators, timers and flip-flops. It consists of two amplifying devices cross-coupled by resistors or capacitors. The first multivibrator circuit, the astable multivibrator oscillator, was invented by Henri Abraham and Eugene Bloch during World War I. They called their circuit a "multivibrator" because its output waveform was rich in harmonics.

An operational amplifier is a DC-coupled high-gain electronic voltage amplifier with a differential input and, usually, a single-ended output. In this configuration, an op-amp produces an output potential that is typically hundreds of thousands of times larger than the potential difference between its input terminals. Operational amplifiers had their origins in analog computers, where they were used to perform mathematical operations in many linear, non-linear, and frequency-dependent circuits.

A Negative-feedback amplifier is an electronic amplifier that subtracts a fraction of its output from its input, so that negative feedback opposes the original signal. The applied negative feedback can improve its performance and reduces sensitivity to parameter variations due to manufacturing or environment. Because of these advantages, many amplifiers and control systems use negative feedback.

In electronics, a linear regulator is a system used to maintain a steady voltage. The resistance of the regulator varies in accordance with the load resulting in a constant voltage output. The regulating device is made to act like a variable resistor, continuously adjusting a voltage divider network to maintain a constant output voltage and continually dissipating the difference between the input and regulated voltages as waste heat. By contrast, a switching regulator uses an active device that switches on and off to maintain an average value of output. Because the regulated voltage of a linear regulator must always be lower than input voltage, efficiency is limited and the input voltage must be high enough to always allow the active device to drop some voltage.

In electronics, a common-base amplifier is one of three basic single-stage bipolar junction transistor (BJT) amplifier topologies, typically used as a current buffer or voltage amplifier.

A differential amplifier is a type of electronic amplifier that amplifies the difference between two input voltages but suppresses any voltage common to the two inputs. It is an analog circuit with two inputs and and one output in which the output is ideally proportional to the difference between the two voltages

In electronics, a Schmitt trigger is a comparator circuit with hysteresis implemented by applying positive feedback to the noninverting input of a comparator or differential amplifier. It is an active circuit which converts an analog input signal to a digital output signal. The circuit is named a "trigger" because the output retains its value until the input changes sufficiently to trigger a change. In the non-inverting configuration, when the input is higher than a chosen threshold, the output is high. When the input is below a different (lower) chosen threshold the output is low, and when the input is between the two levels the output retains its value. This dual threshold action is called hysteresis and implies that the Schmitt trigger possesses memory and can act as a bistable multivibrator. There is a close relation between the two kinds of circuits: a Schmitt trigger can be converted into a latch and a latch can be converted into a Schmitt trigger.

A buffer amplifier is one that provides electrical impedance transformation from one circuit to another, with the aim of preventing the signal source from being affected by whatever currents that the load may be produced with. The signal is 'buffered from' load currents. Two main types of buffer exist: the voltage buffer and the current buffer.

In electronics, a common-emitter amplifier is one of three basic single-stage bipolar-junction-transistor (BJT) amplifier topologies, typically used as the voltage amplifier.

Brokaw bandgap reference is a voltage reference circuit widely used in integrated circuits, with an output voltage around 1.25 V with low temperature dependence. This particular circuit is one type of a bandgap voltage reference, named after Paul Brokaw, the author of its first publication.

In electronics, a common collector amplifier is one of three basic single-stage bipolar junction transistor (BJT) amplifier topologies, typically used as a voltage buffer.

A current mirror is a circuit designed to copy a current through one active device by controlling the current in another active device of a circuit, keeping the output current constant regardless of loading. The current being "copied" can be, and sometimes is, a varying signal current. Conceptually, an ideal current mirror is simply an ideal inverting current amplifier that reverses the current direction as well. Or it can consist of a current-controlled current source (CCCS). The current mirror is used to provide bias currents and active loads to circuits. It can also be used to model a more realistic current source.

A Widlar current source is a modification of the basic two-transistor current mirror that incorporates an emitter degeneration resistor for only the output transistor, enabling the current source to generate low currents using only moderate resistor values.

A Wilson current mirror is a three-terminal circuit that accepts an input current at the input terminal and provides a "mirrored" current source or sink output at the output terminal. The mirrored current is a precise copy of the input current. It may be used as a Wilson current source by applying a constant bias current to the input branch as in Fig. 2. The circuit is named after George R. Wilson, an integrated circuit design engineer who worked for Tektronix. Wilson devised this configuration in 1967 when he and Barrie Gilbert challenged each other to find an improved current mirror overnight that would use only three transistors. Wilson won the challenge.

An active load or dynamic load is a component or a circuit that functions as a current-stable nonlinear resistor.

In graphical analysis of nonlinear electronic circuits, a load line is a line drawn on the characteristic curve, a graph of the current vs. the voltage in a nonlinear device like a diode or transistor. It represents the constraint put on the voltage and current in the nonlinear device by the external circuit. The load line, usually a straight line, represents the response of the linear part of the circuit, connected to the nonlinear device in question. The points where the characteristic curve and the load line intersect are the possible operating point(s) of the circuit; at these points the current and voltage parameters of both parts of the circuit match.

Bipolar transistors must be properly biased to operate correctly. In circuits made with individual devices, biasing networks consisting of resistors are commonly employed. Much more elaborate biasing arrangements are used in integrated circuits, for example, bandgap voltage references and current mirrors. The voltage divider configuration achieves the correct voltages by the use of resistors in certain patterns. By selecting the proper resistor values, stable current levels can be achieved that vary only little over temperature and with transistor properties such as β.

Baker clamp is a generic name for a class of electronic circuits that reduce the storage time of a switching bipolar junction transistor (BJT) by applying a nonlinear negative feedback through various kinds of diodes. The reason for slow turn-off times of saturated BJTs is the stored charge in the base. It must be removed before the transistor will turn off since the storage time is a limiting factor of using bipolar transistors and IGBTs in fast switching applications. The diode-based Baker clamps prevent the transistor from saturating and thereby accumulating a lot of stored charge.

The Miller theorem refers to the process of creating equivalent circuits. It asserts that a floating impedance element, supplied by two voltage sources connected in series, may be split into two grounded elements with corresponding impedances. There is also a dual Miller theorem with regards to impedance supplied by two current sources connected in parallel. The two versions are based on the two Kirchhoff's circuit laws.

## References

1. Widlar bilateral current source Archived 2011-06-07 at the Wayback Machine
2. "AN-1515 A Comprehensive Study of the Howland Current Pump" (PDF) (PDF). Texas Instruments, Inc. 2013.
3. Horowitz, Paul; Winfield Hill (1989). . UK: Cambridge University Press. pp.  182. ISBN   0521370957.
4. The value for VBE varies logarithmically with current level: for more detail see diode modelling.
5. See above note on logarithmic current dependence.
6. "Tung-Sol: Curpistor, minute current regulator data sheet" (PDF). Retrieved 26 May 2013.
• "Current Sources & Voltage References" Linden T. Harrison; Publ. Elsevier-Newnes 2005; 608-pages; ISBN   0-7506-7752-X