# Dependent source

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In the theory of electrical networks, a dependent source is a voltage source or a current source whose value depends on a voltage or current elsewhere in the network. An electrical network is an interconnection of electrical components or a model of such an interconnection, consisting of electrical elements. An electrical circuit is a network consisting of a closed loop, giving a return path for the current. Linear electrical networks, a special type consisting only of sources, linear lumped elements, and linear distributed elements, have the property that signals are linearly superimposable. They are thus more easily analyzed, using powerful frequency domain methods such as Laplace transforms, to determine DC response, AC response, and transient response. A voltage source is a two-terminal device which can maintain a fixed voltage. An ideal voltage source can maintain the fixed voltage independent of the load resistance or the output current. However, a real-world voltage source cannot supply unlimited current. A voltage source is the dual of a current source. Real-world sources of electrical energy, such as batteries, generators, can be modeled for analysis purposes as a combination of an ideal voltage source and additional combinations of impedance elements. A current source is an electronic circuit that delivers or absorbs an electric current which is independent of the voltage across it.

## Contents

Dependent sources are useful, for example, in modelling the behavior of amplifiers. A bipolar junction transistor can be modelled as a dependent current source whose magnitude depends on the magnitude of the current fed into its controlling base terminal. An operational amplifier can be described as a voltage source dependent on the differential input voltage between its input terminals.  Practical circuit elements have properties such as finite power capacity, voltage, current, or frequency limits that mean an ideal source is only an approximate model. Accurate modelling of practical devices requires using several idealized elements in combination. A bipolar junction transistor is a type of transistor that uses both electrons and holes as charge carriers. 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.

## Classification

Dependent sources can be classified as follows:

• Voltage-controlled voltage source: The source delivers the voltage as per the voltage of the dependent element. $V={f_{a}}({v_{x}})$ • Voltage-controlled current source: The source delivers the current as per the voltage of the dependent element. $I={f_{b}}({v_{x}})$ • Current-controlled current source: The source delivers the current as per the current of the dependent element. $I={f_{c}}({i_{x}})$ • Current-controlled voltage source: The source delivers the voltage as per the current of the dependent element. $V={f_{d}}({i_{x}})$  An electric current is the rate of flow of electric charge past a point or region. An electric current is said to exist when there is a net flow of electric charge through a region. In electric circuits this charge is often carried by electrons moving through a wire. It can also be carried by ions in an electrolyte, or by both ions and electrons such as in an ionized gas (plasma).

Dependent sources are not necessarily linear. For example, MOSFET switches can be modeled as a voltage-controlled current source when $V_{\rm {DS}}>V_{\rm {GS}}-V_{T}$ and $V_{\rm {GS}}>V_{T}$ . The metal–oxide–semiconductor field-effect transistor (MOSFET, MOS-FET, or MOS FET), also known as the metal–oxide–silicon transistor (MOS transistor, or MOS), is a type of field-effect transistor that has an insulated gate and is fabricated by the controlled oxidation of a semiconductor, typically silicon. The voltage of the covered gate determines the electrical conductivity of the device; this ability to change conductivity with the amount of applied voltage can be used for amplifying or switching electronic signals. The MOSFET was invented by Egyptian engineer Mohamed M. Atalla and Korean engineer Dawon Kahng at Bell Labs in November 1959. It is the basic building block of modern electronics, and the most widely manufactured device in history, with an estimated total of 13 sextillion (1.3 × 1022) MOSFETs manufactured between 1960 and 2018.

However, the relationship between the current flowing through it and $V_{\rm {DS}}$ is approximately:

$I_{\rm {D}}={\frac {\mu _{n}C_{\rm {ox}}}{2}}{\frac {W}{L}}(V_{\rm {GS}}-V_{\rm {th}})^{2}\left(1+\lambda (V_{\rm {DS}}-V_{\rm {DSsat}})\right).$ In this case, the current is not linear to $V_{\rm {DS}}$ , but rather approximately proportional to the square of $V_{\rm {DS}}-V_{T}$ .

As for the case of linear dependent sources, the proportionality constant between dependent and independent variables is dimensionless if they are both currents (or both voltages). A voltage controlled by a current has a proportionality factor expressed in units of resistance (ohms), and this constant is sometimes called "transresistance". A current controlled by a voltage has the units of conductance (siemens), and is called "transconductance". Transconductance is a commonly used specification for measuring the performance of field effect transistors and vacuum tubes. The ohm is the SI derived unit of electrical resistance, named after German physicist Georg Simon Ohm. Although several empirically derived standard units for expressing electrical resistance were developed in connection with early telegraphy practice, the British Association for the Advancement of Science proposed a unit derived from existing units of mass, length and time and of a convenient size for practical work as early as 1861. The definition of the ohm was revised several times. Today, the definition of the ohm is expressed from the quantum Hall effect.

The siemens is the derived unit of electric conductance, electric susceptance, and electric admittance in the International System of Units (SI). Conductance, susceptance, and admittance are the reciprocals of resistance, reactance, and impedance respectively; hence one siemens is redundantly equal to the reciprocal of one ohm, and is also referred to as the mho. The 14th General Conference on Weights and Measures approved the addition of the siemens as a derived unit in 1971. In electronics, a vacuum tube, an electron tube, or valve or, colloquially, a tube, is a device that controls electric current flow in a high vacuum between electrodes to which an electric potential difference has been applied. In electrical engineering, ground or earth is the reference point in an electrical circuit from which voltages are measured, a common return path for electric current, or a direct physical connection to the earth.

Mathematical methods are integral to the study of electronics. Open-circuit voltage is the difference of electrical potential between two terminals of a device when disconnected from any circuit. There is no external load connected. No external electric current flows between the terminals. Alternatively, the open-circuit voltage may be thought of as the voltage that must be applied to a solar cell or a battery to stop the current. It is sometimes given the symbol Voc. In network analysis this voltage is also known as the Thévenin voltage.

## Related Research Articles The junction gate field-effect transistor is one of the simplest types of field-effect transistor. JFETs are three-terminal semiconductor devices that can be used as electronically-controlled switches, amplifiers, or voltage-controlled resistors. The electrical resistance of an object is a measure of its opposition to the flow of electric current. The inverse quantity is electrical conductance, and is the ease with which an electric current passes. Electrical resistance shares some conceptual parallels with the notion of mechanical friction. The SI unit of electrical resistance is the ohm (Ω), while electrical conductance is measured in siemens (S).

Electrical elements are conceptual abstractions representing idealized electrical components, such as resistors, capacitors, and inductors, used in the analysis of electrical networks. All electrical networks can be analyzed as multiple electrical elements interconnected by wires. Where the elements roughly correspond to real components the representation can be in the form of a schematic diagram or circuit diagram. This is called a lumped-element circuit model. In other cases infinitesimal elements are used to model the network, in a distributed-element model. 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 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. In electronics, slew rate is defined as the change of voltage or current, or any other electrical quantity, per unit of time. Expressed in SI units, the unit of measurement is volts/second or amperes/second or the unit being discussed,.

Transconductance, also infrequently called mutual conductance, is the electrical characteristic relating the current through the output of a device to the voltage across the input of a device. Conductance is the reciprocal of resistance. The asymptotic gain model is a representation of the gain of negative feedback amplifiers given by the asymptotic gain relation: In electronics, a common-source amplifier is one of three basic single-stage field-effect transistor (FET) amplifier topologies, typically used as a voltage or transconductance amplifier. The easiest way to tell if a FET is common source, common drain, or common gate is to examine where the signal enters and leaves. The remaining terminal is what is known as "common". In this example, the signal enters the gate, and exits the drain. The only terminal remaining is the source. This is a common-source FET circuit. The analogous bipolar junction transistor circuit may be viewed as a transconductance amplifier or as a voltage amplifier.. As a transconductance amplifier, the input voltage is seen as modulating the current going to the load. As a voltage amplifier, input voltage modulates the current flowing through the FET, changing the voltage across the output resistance according to Ohm's law. However, the FET device's output resistance typically is not high enough for a reasonable transconductance amplifier, nor low enough for a decent voltage amplifier. Another major drawback is the amplifier's limited high-frequency response. Therefore, in practice the output often is routed through either a voltage follower, or a current follower, to obtain more favorable output and frequency characteristics. The CS–CG combination is called a cascode amplifier. In electronics, a chopper circuit is used to refer to numerous types of electronic switching devices and circuits used in power control and signal applications. A chopper is a device that converts fixed DC input to a variable DC output voltage directly. Essentially, a chopper is an electronic switch that is used to interrupt one signal under the control of another. One of several short-channel effects in MOSFET scaling, channel length modulation (CLM) is a shortening of the length of the inverted channel region with increase in drain bias for large drain biases. The result of CLM is an increase in current with drain bias and a reduction of output resistance. Channel length modulation occurs in all field effect transistors, not just MOSFETs. In electronics, a current divider is a simple linear circuit that produces an output current (IX) that is a fraction of its input current (IT). Current division refers to the splitting of current between the branches of the divider. The currents in the various branches of such a circuit will always divide in such a way as to minimize the total energy expended.

A charge amplifier is an electronic current integrator that produces a voltage output proportional to the integrated value of the input current, or the total charge injected. The operational transconductance amplifier (OTA) is an amplifier whose differential input voltage produces an output current. Thus, it is a voltage controlled current source (VCCS). There is usually an additional input for a current to control the amplifier's transconductance. The OTA is similar to a standard operational amplifier in that it has a high impedance differential input stage and that it may be used with negative feedback. Current differencing transconductance amplifier (CDTA) is a new active circuit element.

The hybrid-pi model is a popular circuit model used for analyzing the small signal behavior of bipolar junction and field effect transistors. Sometimes it is also called Giacoletto model because it was introduced by L.J. Giacoletto in 1969. The model can be quite accurate for low-frequency circuits and can easily be adapted for higher frequency circuits with the addition of appropriate inter-electrode capacitances and other parasitic elements.

The return ratio of a dependent source in a linear electrical circuit is the negative of the ratio of the current (voltage) returned to the site of the dependent source to the current (voltage) of a replacement independent source. The terms loop gain and return ratio are often used interchangeably; however, they are necessarily equivalent only in the case of a single feedback loop system with unilateral blocks.

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. In electronics, a transimpedance amplifier, (TIA) is a current to voltage converter, almost exclusively implemented with one or more operational amplifiers. The TIA can be used to amplify the current output of Geiger–Müller tubes, photo multiplier tubes, accelerometers, photo detectors and other types of sensors to a usable voltage. Current to voltage converters are used with sensors that have a current response that is more linear than the voltage response. This is the case with photodiodes where it is not uncommon for the current response to have better than 1% nonlinearity over a wide range of light input. The transimpedance amplifier presents a low impedance to the photodiode and isolates it from the output voltage of the operational amplifier. In its simplest form a transimpedance amplifier has just a large valued feedback resistor, Rf. The gain of the amplifier is set by this resistor and because the amplifier is in an inverting configuration, has a value of -Rf. There are several different configurations of transimpedance amplifiers, each suited to a particular application. The one factor they all have in common is the requirement to convert the low-level current of a sensor to a voltage. The gain, bandwidth, as well as current and voltage offsets change with different types of sensors, requiring different configurations of transimpedance amplifiers.

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2. PR Gray; PJ Hurst; SH Lewis; RG Meyer. §1.5.2 p. 45. ISBN   0-471-32168-0.
3. A. S. Sedra; K.C. Smith (2004). Microelectronic circuits (Fifth ed.). New York: Oxford. p. 552. ISBN   0-19-514251-9.