Controlled source

A controlled source (also dependent source , English dependent source ) is an elementary ideal electrical component , which enables the reaction-free coupling of voltages and currents of different branches of an electrical network . Controlled sources are absolutely necessary, for example, in order to represent real non-reversible and / or active two-port (i.e. electronic components such as electron tubes , transistors or operational amplifiers ) as an equivalent circuit diagram . In contrast to independent current and voltage sources , their source current or their source voltage is controlled without power by a control current or a control voltage . Controlled sources are even very rough approximations of real active components. They must not be confused with controllable power supply units .

Linear controlled sources

Two-ports can be classified according to the properties of their chain matrix . This is defined (in matrix form assuming the symmetrical counting arrow system ) by the following chain equations of the two-port theory : ${\displaystyle \mathbf {A} }$

${\displaystyle {\begin{pmatrix}u_{1}\\i_{1}\end{pmatrix}}={\begin{pmatrix}A_{11}&A_{12}\\A_{21}&A_{22}\end{pmatrix}}\cdot {\begin{pmatrix}u_{2}\\-i_{2}\end{pmatrix}}=\mathbf {A} \cdot {\begin{pmatrix}u_{2}\\-i_{2}\end{pmatrix}}}$

The two-port theory understands a linear controlled source as a linear two-port, in whose chain matrix only one element is occupied, i.e. not equal to 0. Therefore there are exactly four differently behaving controlled sources. Your equivalent circuit consists of a control branch (open circuit or short circuit) and a controlled branch (controlled current or voltage source). The four variants result from these combinations. The control branch and the controlled branch are generally galvanically separated . If necessary, a controlled source can be made a three-pole with the help of a "continuous earth line ". Controlled sources do not "consume" any input power. Therefore, any output power can be "set without power". Because of this "infinitely large power gain", controlled sources are active two-port. Due to the vanishing determinant of the chain matrix, they are (e.g. in contrast to the ideal transformer ) free of feedback and therefore irreversible. The single chain parameter describing a linear controlled source may generally be a complex operator in the sense of the complex AC calculation. For many applications, however, a real value is sufficient, whereby the controlled source becomes a resistive two-port, i.e. a two-port without storage behavior. With this restriction and using the symmetrical counting arrow system on the gates, the following typical variants are shown.

Voltage controlled voltage source (USU)

Voltage-controlled voltage source as a two-port

A voltage-controlled voltage source ( English voltage controlled voltage source , VCVS) has a chain matrix of the form

${\displaystyle \mathbf {A} ={\begin{pmatrix}{\frac {1}{\mu }}&0\\0&0\end{pmatrix}}}$

This represents the (positive or negative) voltage translation or voltage gain. This corresponds to the explicit two-port equations ${\displaystyle \mu }$

${\displaystyle i_{1}=0}$

${\displaystyle u_{2}=\mu \cdot u_{1}}$

An example of a voltage-controlled voltage source is a negative feedback operational amplifier, e.g. B. as a so-called electrometer amplifier . With it, the voltage gain is determined by the external resistances.

Voltage controlled current source (USI)

Voltage controlled current source as two-port

A voltage controlled current source ( English voltage controlled current source , VCCS) has a chain matrix of the form

${\displaystyle \mathbf {A} ={\begin{pmatrix}0&-{\frac {1}{S}}\\0&0\end{pmatrix}}}$

It turns the (positive or negative) transfer slope or transconductance . The corresponding explicit Zweitorgleichungen ${\displaystyle S}$

${\displaystyle i_{1}=0}$

${\displaystyle i_{2}=S\cdot u_{1}}$

Examples of voltage-controlled current sources are the ideal electron tube with infinite input and internal resistance as well as the slope and the transconductance amplifier with the "transconductance" . ${\displaystyle S}$ ${\displaystyle S}$

Current controlled voltage source (ISU)

Current-controlled voltage source as a two-port

A current controlled voltage source ( English current controlled voltage source , CCVS) has a chain matrix of the form

${\displaystyle \mathbf {A} ={\begin{pmatrix}0&0\\{\frac {1}{R}}&0\end{pmatrix}}}$

It represents the (positive or negative) transmission resistance (transimpedance). This corresponds to the explicit two-port equations ${\displaystyle R}$

${\displaystyle u_{1}=0}$

${\displaystyle u_{2}=R\cdot i_{1}}$

An example of a current-controlled voltage source from an ideal negative feedback operational amplifier is the transimpedance amplifier .

Current Controlled Power Source (ISI)

Current-controlled power source as a two-port

A current-controlled current source ( English current controlled current source , CCCS) has a chain matrix of the form

${\displaystyle \mathbf {A} ={\begin{pmatrix}0&0\\0&-{\frac {1}{\beta }}\end{pmatrix}}}$

This represents the (positive or negative) current translation or current gain. This corresponds to the explicit two-gate equations ${\displaystyle \beta }$

${\displaystyle u_{1}=0}$

${\displaystyle i_{2}=\beta \cdot i_{1}}$

Examples of current-controlled current sources are the ideal bipolar transistor in emitter circuit with vanishing input and vanishing internal resistance as well as current amplification and the so-called current mirror circuit . ${\displaystyle \beta }$

Nonlinear Controlled Sources

In general, a non-linear dependence of the controlled variable on the controlled variable of a controlled source is permitted. The essential linear two-port equation is replaced by a non-linear function in this case . Despite this generalization, the four basic variants mentioned above remain. For example, a non-linear voltage-controlled current source can be described by the following system of equations :

${\displaystyle i_{1}=0}$

${\displaystyle i_{2}=f(u_{1})}$

The non-linear function can be given, for example, in the form of a (non-linear) characteristic curve . Non-linear controlled sources are used in equivalent circuit diagrams of components that have a substantial non-linearity that must not be neglected or is functionally “wanted”. A non-linear controlled source can itself be replaced by a combination of a linear controlled source and a resistive non - linear two - terminal network. Continuous non-linear sources can be linearized as usual in small-signal operation. ${\displaystyle f}$

Circuit symbols

Hybrid equivalent circuit diagram of a resistive two-port

In contrast to independent sources, inverted squares are used as circuit symbols for controlled sources. With a voltage source a plus sign shows the "positive pole", with a current source an arrow shows the direction of the current. The control is expressed through the labeling, in that the source current or the source voltage (linear or non-linear) depends on a control current or a control voltage of the network . The picture on the right shows the hybrid equivalent circuit diagram of a bipolar transistor for low-frequency signals with a low amplitude . The equivalent circuit of this resistive two-port is based on the hybrid form of the associated two-port equations. This system of equations is called hybrid , because current and voltage appear "mixed" on both sides with regard to input and output:

${\displaystyle u_{1}=h_{11}\cdot i_{1}+h_{12}\cdot u_{2}}$

${\displaystyle i_{2}=h_{21}\cdot i_{1}+h_{22}\cdot u_{2}}$

This “mixture” can also be seen in the equivalent circuit diagram, in which a resistor is connected in series with a voltage-controlled voltage source in the left branch, but a conductance with a current-controlled current source is connected in parallel in the right branch . The two-port equations can be "read off" directly.

Calculation of linear networks with controlled sources

There are several so-called "simplified calculation methods" for calculating linear networks . If these networks have several (independent) current and / or voltage sources, the calculation is usually carried out individually for each source in accordance with the superposition theorem, with the other sources being replaced by short circuits or open circuits . At the end, all individual results are "superimposed" (added) to the overall result. However, this procedure must not be used for the controlled sources in the network. These must always be explicitly included in the invoice.

Energy balance of controlled sources

Just as independent sources are active two-terminal devices, controlled sources are active two-port devices. They can therefore (appropriately wired) emit electrical energy permanently and must therefore have an "internal energy source". This explanation is sufficient for use as an ideal fictitious component within equivalent circuits. However, if (for whatever purpose) controlled sources are simulated by real components (e.g. operational amplifiers), then - as with any power amplifier - this energy is drawn from the supply voltage of the corresponding functional unit .

Real controlled sources

Real voltage-controlled voltage source as a two-port

In the literature on electrical networks, in addition to the ideal controlled source (discussed in this article), the term real controlled source can also be found. By definition, this has a finite, non-zero input resistance or conductance and a finite, non-zero output internal resistance or conductance. This is expressed in their equivalent circuit diagram. Such a real controlled source represents a general reaction-free two-port. If the real controlled source can be described as a linear two-port, then the chain matrix is ​​“fully occupied” and its determinant is zero. Therefore, all four of the above-mentioned variants can be converted into one another and reduced to a single type of real controlled source. The inverse hybrid equations can be read directly from the example shown in the picture:

${\displaystyle u_{2}=\mu \cdot u_{1}+R_{i}\cdot i_{2}}$

${\displaystyle i_{1}={\frac {u_{1}}{R_{e}}}}$

The chain equations follow through transformation

${\displaystyle u_{1}={\frac {1}{\mu }}\cdot u_{2}-{\frac {R_{i}}{\mu }}\cdot i_{2}}$

${\displaystyle i_{1}={\frac {1}{\mu \cdot R_{e}}}\cdot u_{2}-{\frac {R_{i}}{\mu \cdot R_{e}}}\cdot i_{2}}$

The full population of the chain matrix and its vanishing determinant can easily be verified from this.

See also

• Nullor
• Impedance converter
• Impedance inverter

literature

• Klaus Lunze : Theory of AC circuits . 8th edition. Verlag Technik GmbH, Berlin 1991, ISBN 3-341-00984-1 .
• Reinhold Paul: Electrical engineering basic textbook Volume 2: Networks . 3. Edition. Springer, 1996, ISBN 978-3-540-55866-8 .

Individual evidence

1. ↑ DIN EN 60375: 2004: Agreements for electrical circuits and magnetic circuits (IEC 60375: 2003)