Mathematical description of the bipolar transistor

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The physical behavior of the bipolar transistor is essentially based on that of the diode , which means that the corresponding formulas (in a slightly modified form) can also be applied to the bipolar transistor. In addition, there are some other effects such as the current gain to be taken into account.

Formula symbol

The symbols used here are used in the following. For further formula symbols see also " Equivalent circuits of the bipolar transistor ".

Currents

	-Electricity	- Continuous current	-Peak current
Collector-	I C	${\displaystyle I_{CM}}$	${\displaystyle I_{C,\mathrm {max} }}$
Base-	I B	${\displaystyle I_{BM}}$	${\displaystyle I_{B,\mathrm {max} }}$
Emitter	I E	${\displaystyle I_{EM}}$	${\displaystyle I_{E,\mathrm {max} }}$

• Reverse saturation current: ${\displaystyle I_{S}\approx 10^{-16}\dots 10^{-12}\,\mathrm {A} }$
• Collector base reverse current: ${\displaystyle I_{CBO}}$
• Emitter-base reverse current: ${\displaystyle I_{EBO}}$
• Collector-emitter reverse current: resp. ${\displaystyle I_{CEO}}$ ${\displaystyle I_{CES}}$

Tensions

• Collector-emitter voltage: ${\displaystyle U_{\mathrm {CE} }}$
• Base-emitter voltage: ${\displaystyle U_{\mathrm {BE} }}$
• Collector base voltage: ${\displaystyle U_{\mathrm {CB} }}$
• Early voltage: ${\displaystyle U_{A}}$

  ${\displaystyle U_{A,\mathrm {pnp} }\approx 30\dots 75\,\mathrm {V} }$

  ${\displaystyle U_{A,\mathrm {npn} }\approx 30\dots 150\,\mathrm {V} }$

• Emitter-base breakdown voltage : ${\displaystyle U_{(BR)EBO}\approx 5\dots 7\,\mathrm {V} }$
• Collector-base breakdown voltage: ${\displaystyle U_{(BR)CBO}}$

  ${\displaystyle U_{(BR)CBO}\approx 20\dots 80\,\mathrm {V} }$ with low voltage transistors

  ${\displaystyle U_{(BR)CBO}<1{,}3\,\mathrm {kV} }$ with high voltage transistors

Resistances

• Small signal output resistance: ${\displaystyle r_{CE}}$
• Small signal input resistance: ${\displaystyle r_{BE}}$

Services

• Power dissipation: P _V
• Maximum power loss:

  P _tot or P _max (general)

  P _{V, 25 (A)} (ambient air cooled; at 25 ° C)

  P _{V, 25 (C)} (with additional cooling; at 25 ° C)

Other

• Large signal amplification: B.

  ${\displaystyle B\approx 10\dots 100}$ with power transistors

  ${\displaystyle B\approx 100\dots 500}$ with small power transistors

  ${\displaystyle B\approx 500\dots 10^{4}}$with Darlington transistors

• Small signal amplification: ${\displaystyle \beta }$
• Steepness : p
• Backward slope : ${\displaystyle S_{r}}$
• Working point : AP
• Ambient temperature: T _A
• Case temperature: T _C

Current amplification factor

With the bipolar transistor, a distinction is made between the direct current gain factor B (also ) and the differential current gain β (also ). Both can be very different (depending on the structure and doping of the transistor). If no other information on β can be found in the data sheet , the approximation can be used. ${\displaystyle h_{FE}}$ ${\displaystyle h_{fe}}$ ${\displaystyle \beta =B}$

The formula for the DC gain is:

${\displaystyle B=h_{FE}={\frac {I_{\mathrm {C} }}{I_{\mathrm {B} }}}}$

This formula can be used for most calculations, as defined by the Early effect dependence of the DC gain caused B of a low impact. ${\displaystyle U_{CE}}$

Taking into account the early effect one obtains:

${\displaystyle B\left(U_{BE},U_{CE}\right)=B_{0}\left(U_{BE}\right)\,\left(1+{\frac {U_{CE}}{U_{A}}}\right)}$

where represents the ideal current gain without the Early effect. ${\displaystyle B_{0}}$

With alternating current , the differential current gain occurs. This results from:

${\displaystyle \beta ={\frac {\partial I_{\mathrm {C} }}{\partial I_{\mathrm {B} }}}}$ With ${\displaystyle U_{\mathrm {CE} }={\text{konst.}}}$

Insertion gives the relationship between B and ß :

${\displaystyle \beta ={\frac {\partial I_{C}}{\partial \left({\frac {I_{C}}{B\left(I_{C},U_{CE}\right)}}\right)}}={\frac {B}{1-{\frac {I_{C}}{B}}\,{\frac {\partial B}{\partial I_{C}}}}}}$

Is called B as large-signal gain and a small signal gain. ${\displaystyle \beta }$

Large signal equations

The equations of the diode show an exponential dependence of the currents and of the voltage . For normal operation this results in: ${\displaystyle I_{B}}$ ${\displaystyle I_{C}}$ ${\displaystyle U_{BE}}$

${\displaystyle I_{C}=I_{S}\,e^{\frac {U_{BE}}{U_{T}}}\,\left(1+{\frac {U_{CE}}{U_{A}}}\right)}$

${\displaystyle I_{B}={\frac {I_{C}}{B\left(U_{BE},U_{CE}\right)}}={\frac {I_{S}}{B_{0}}}\left(\,e^{\frac {U_{BE}}{U_{T}}}-1\right)}$

Small signal parameters

The partial derivatives at the operating point are referred to as small-signal parameters . These can be determined from the characteristic, but the reading error when using data sheets does not normally result in a usable result. In addition, the corresponding characteristic curves are usually not given.

Steepness

The slope describes the differential change in the collector current and the voltage . ${\displaystyle I_{C}}$ ${\displaystyle U_{BE}}$

${\displaystyle S={\frac {\partial I_{C}}{\partial U_{BE}}}\forall {AP}={\frac {I_{C,AP}}{U_{T}}}}$

Small signal input resistance

The small-signal input resistance describes the differential change in voltage and with the base current . ${\displaystyle r_{BE}}$ ${\displaystyle U_{BE}}$ ${\displaystyle I_{B}}$

${\displaystyle r_{BE}={\frac {\partial U_{BE}}{\partial I_{B}}}\forall {AP}}$

${\displaystyle r_{BE}}$ can also be derived from the slope by converting this formula:

${\displaystyle r_{BE}={\frac {\partial U_{BE}}{\partial I_{C}}}\forall {AP}\cdot {\frac {\partial I_{C}}{\partial I_{B}}}\forall {AP}=\beta {\frac {\partial U_{BE}}{\partial I_{C}}}\forall {AP}={\frac {\beta }{S}}}$

Small signal output resistance

The small-signal output resistance indicates the differential change between the emitter voltage and the collector current . ${\displaystyle r_{CE}}$ ${\displaystyle U_{CE}}$ ${\displaystyle I_{C}}$

${\displaystyle r_{CE}={\frac {\partial U_{CE}}{\partial I_{C}}}\forall {AP}={\frac {U_{A}+U_{CE,AP}}{I_{C,AP}}}{\stackrel {\quad U_{CE,AP}\ll U_{A}\quad }{\approx }}{\frac {U_{A}}{I_{C,AP}}}}$

Backward steepness

The reverse slope describes the differential change between the base current and the collector-emitter voltage . ${\displaystyle S_{r}}$ ${\displaystyle I_{B}}$ ${\displaystyle U_{CE}}$

${\displaystyle S_{r}={\frac {\partial I_{B}}{\partial U_{CE}}}\forall {AP}}$

The backward slope is very low and can therefore mostly be neglected.

${\displaystyle S_{r}\approx 0}$

Small signal equations

The small-signal equations are obtained from the small-signal parameters :

${\displaystyle i_{B}={\frac {1}{r_{BE}}}\,u_{BE}+S_{r}\,u_{CE}}$

${\displaystyle i_{C}=S\,u_{BE}+{\frac {1}{r_{CE}}}\,u_{CE}}$

cooling

The calculation of the cooling of a transistor comes from the theory of heat . The heat generated in the barrier layer ( junction ) has over the substrate to the housing ( case ) with the temperature , then via the heat sink ( heat sink ) with the temperature (and then to the surrounding ambient ) with the temperature can be derived. The resulting heat flow ( Φ ) corresponds to the power converted in the transistor . ${\displaystyle T_{J}}$ ${\displaystyle T_{C}}$ ${\displaystyle T_{H}}$ ${\displaystyle T_{A}}$ ${\displaystyle P_{V}}$

${\displaystyle P_{V}=U_{CE}\,I_{C}=\Phi ={\frac {T_{J}-T_{A}}{R_{th,JA}}}}$

The amount of heat contained in the individual bodies (substrate, housing, heat sink, environment) results from: ${\displaystyle Q_{\mathrm {th} }}$

${\displaystyle Q_{\mathrm {th} }=C_{\mathrm {th} }\,T_{\mathrm {th} }}$

Whereby represents the heat capacity of the respective body in which the heat is stored. If too much power is converted in the transistor, the heat cannot flow away quickly enough and the temperature of the individual layers increases. In addition, the ambient temperature must not be too high so that the heat can flow away. In pulsed operation, the maximum power is exceeded for a short time, but since the layers have the opportunity to cool down, the maximum permissible temperature is not exceeded. ${\displaystyle C_{\mathrm {th} }}$

${\displaystyle P_{V,\mathrm {max} ,\mathrm {puls} }\left(t_{P},D\right)={\frac {T_{J,\mathrm {max} }-T_{A,\mathrm {max} }}{R_{\mathrm {th} ,JA,\mathrm {puls} }\left(t_{p},D\right)}}}$

Here is the pulse duration, the repetition frequency and D is the duty cycle . ${\displaystyle t_{p}}$ ${\displaystyle f_{w}}$

${\displaystyle \lim _{t_{p}\to 0}{\frac {P_{V,\mathrm {max} ,\mathrm {puls} }}{P_{V,\mathrm {max} ,\mathrm {stat} }}}={\frac {1}{D}}={\frac {1}{t_{p}\,f_{w}}}}$

Limit data

A transistor has various characteristics that must not be exceeded during operation. This includes limit voltages, limit currents and the maximum permissible power loss. If these values ​​are exceeded, a breakthrough occurs in which the semiconductor material in the transistor melts and thus becomes permanently conductive or evaporates. When the semiconductor material evaporates, the resulting gas pressure can blow open the transistor housing. The values ​​of pnp and npn transistors differ in sign , but not in amounts.

The designation of the breakdown voltages and currents are made up of the respective symbols (voltage = U; current = I), the designation BR for breakthrough ( break down ), the indication of connections to which the value relates (C; B; E ), and an addition, which stands for the type of load on the transistor.

additive	meaning	output
S.	shorted	shorted
O	open ; open	unencumbered
R.	resistor	burdened

Workspace

Limitation of the transistor working range

Output characteristic field of an npn transistor

Operating limits of a pnp Darlington power transistor type BDV66C

A bipolar transistor has a working area (SOA, Safe Operation Area), which is essentially limited by the following sizes:

• maximum permissible collector current ${\displaystyle I_{C,\mathrm {max} }}$
• maximum collector-emitter voltage (no-load operation), as referred to ${\displaystyle U_{CE,O}}$ ${\displaystyle U_{CE,\mathrm {max} }}$
• maximum junction temperature ${\displaystyle \vartheta _{j,\mathrm {max} }}$

Since the junction temperature cannot be measured directly, the maximum power loss at a given ambient or housing temperature is given in data sheets . ${\displaystyle P_{\mathrm {tot,max} }}$

In the case of power transistors in particular, there is another limit, the breakthrough of the second type ( second breakdown or secondary breakdown ). In the case of power transistors, the semiconductor material necessarily has a larger volume than z. B. with small signal transistors. Inhomogeneities therefore occur increasingly within the semiconductor material, which means that in some volume elements a higher power loss is converted into heat than in other volume elements. If the power loss is sufficiently high, but it is still below the maximum value , the temperature in some volume elements increases to such an extent that the semiconductor material in the volume elements concerned becomes unstable. If the collector-emitter voltage is sufficiently high, a local breakdown occurs in the affected volume elements, which destroys them. The failure of individual volume elements increases the power loss and thus the temperature in all other volume elements. There are further local breakthroughs with destruction of the affected volume elements. The effect continues like a cascade and ultimately leads to the destruction of the semiconductor material. ${\displaystyle P_{\mathrm {tot,max} }}$

Tensions

Base-emitter breakdown voltage

In the base-emitter diode of the pnp transistor, the base-emitter breakdown voltage for most transistors is in the range between 5 and 7 volts . Since npn transistors are usually not operated with a negative BE voltage, this information is usually not relevant. This voltage is the lowest limit voltage of a transistor. ${\displaystyle U_{(BR)EBO}}$

Collector base breakdown voltage

The collector-base breakdown voltage indicates when the collector diode breaks down in reverse operation. Since the collector diode must be blocked in normal operation, this voltage must not be exceeded in normal operation. This voltage is the greatest limit voltage of a transistor. ${\displaystyle U_{(BR)CBO}}$

Collector-emitter voltage

In practice, the maximum collector-emitter voltage is particularly important. Above a certain collector-emitter voltage, a breakdown occurs, through which the collector current increases very sharply and thus causes the destruction of the transistor. ${\displaystyle U_{CE}}$

In general:

${\displaystyle U_{(BR)CEO}<U_{(BR)CER}<U_{(BR)CES}<U_{(BR)CBO}}$

for npn transistors (U _BR > 0 V) ​​and vice versa

${\displaystyle U_{(BR)CEO}>U_{(BR)CER}>U_{(BR)CES}>U_{(BR)CBO}}$

for pnp transistors (U _BR <0 V).

Currents

In the limiting currents , a distinction between the maximum continuous currents ( continuous currents ) and peak currents ( peak currents ). The maximum continuous currents are called , and . The peak currents , and called and apply for certain, specified in the data sheet, pulse durations and pulse repetition rates. The peak currents are usually 1.2 to 2 times as large as the continuous currents. ${\displaystyle I_{B,\mathrm {max} }}$ ${\displaystyle I_{C,\mathrm {max} }}$ ${\displaystyle I_{E,\mathrm {max} }}$ ${\displaystyle I_{CM}}$ ${\displaystyle I_{BM}}$ ${\displaystyle I_{EM}}$

The reverse currents ( cut-off currents ) are with and as well as or referred. These currents occur at the emitter or collector diode when somewhat less than the respective diffusion voltages are applied to it (i.e. the diode is currently not turning on). The following applies here: ${\displaystyle I_{EBO}}$ ${\displaystyle I_{CBO}}$ ${\displaystyle I_{CEO}}$ ${\displaystyle I_{CES}}$

${\displaystyle I_{CES}<I_{CEO}}$

power

The power loss of the transistor results from:

${\displaystyle P_{V}=U_{CE}\,I_{C}+U_{BE}\,I_{B}\approx U_{CE}\,I_{C}}$

The maximum power loss or is one of the most important characteristics of a transistor. The temperature in the barrier layer increases by the value at which the heat can be released to the environment via the housing and the heat sink . This temperature must not exceed the material-dependent limit value. The following applies to silicon : ${\displaystyle P_{\mathrm {tot} }}$ ${\displaystyle P_{\mathrm {max} }}$

${\displaystyle T_{\mathrm {max,SI} }=175\ \mathrm {{}^{\circ }C} }$

In practice, to be on the safe side, a limit value of 150 ° C is used to prevent the silicon from melting prematurely.

The data sheet specifies the maximum power loss for two cases:

1. ${\displaystyle P_{V,25(A)}}$

   Ambient -air-cooled ( free-air-cooled ) operation with upright mounting on a circuit board at an ambient temperature of . In the case of low-power transistors without attachment for an additional heat sink, only this value is specified in the data sheet, as this applies in this case . ${\displaystyle T_{A}=25\ \mathrm {{}^{\circ }C} }$ ${\displaystyle P_{\mathrm {tot} }=P_{V,25(A)}}$

2. ${\displaystyle P_{V,25(C)}}$

   Operation at a case temperature of . The necessary cooling measures are usually not specified here. In the case of power transistors that may only be operated with a heat sink, only this value is specified in the data sheet . ${\displaystyle T_{C}=25\ \mathrm {{}^{\circ }C} }$ ${\displaystyle P_{\mathrm {tot} }=P_{V,25(C)}}$

Since the maximum permissible power decreases with increasing temperature, the so-called power derating curve is often given in the data sheet, which depends on or . ${\displaystyle P_{\mathrm {tot} }}$ ${\displaystyle P_{\mathrm {tot} }}$ ${\displaystyle T_{A}}$ ${\displaystyle T_{C}}$

Temperature dependence

The characteristics of a transistor are strongly dependent on the temperature of the transistor. The dependence of the relationship between collector current and base-emitter voltage on the temperature T is particularly important: ${\displaystyle I_{C}}$ ${\displaystyle U_{BE}}$

${\displaystyle I_{C}\left(U_{BE},T\right)=I_{S}\left(T\right)\,e^{\frac {U_{BE}}{U_{T}\left(T\right)}}\,\left(1+{\frac {U_{CE}}{U_{A}}}\right)}$

The reason for this is the temperature dependence of the reverse current and the temperature voltage : ${\displaystyle I_{S}}$ ${\displaystyle U_{T}}$

${\displaystyle U_{T}(T)={\frac {k\,T}{q}}=86{,}142\cdot 10^{-6}\,{\frac {\rm {V}}{\rm {K}}}\,T}$

${\displaystyle I_{S}(T)=I_{S}\left(T_{0}\right)\,e^{\left({\frac {T}{T_{0}}}-1\right)\,{\frac {U_{G}\left(T\right)}{U_{T}\left(T\right)}}}\,{\left({\frac {T}{T_{0}}}\right)}^{x_{T,I}}}$ With ${\displaystyle x_{T,I}\approx 3}$

Here k is the Boltzmann constant , q is the elementary charge and the band gap voltage of silicon. Since the temperature dependence of U _{G is} only very slight, it is not taken into account in practice. ${\displaystyle U_{G}={\tfrac {1{,}12\,\mathrm {eV} }{q}}=0{,}7\,\mathrm {V} }$

Differentiation gives:

${\displaystyle {\frac {1}{I_{S}}}\,{\frac {\delta I_{S}}{\delta T}}={\frac {1}{T}}\,\left(3+{\frac {U_{G}}{U_{T}}}\right){\begin{matrix}{}_{T=300\,\mathrm {K} }\\\approx \\\,\end{matrix}}0{,}15\,\mathrm {K} ^{-1}}$

${\displaystyle {\frac {1}{I_{C}}}\,{\frac {\delta I_{C}}{\delta T}}\forall \left(U_{BE}={const.}\right)={\frac {1}{T}}\,\left(3+{\frac {U_{G}-U_{BE}}{U_{T}}}\right){\begin{matrix}{}_{T=300\,\mathrm {K} }\\{}_{U_{BE}=0{,}7\,\mathrm {V} }\\\approx \\\,\\\,\end{matrix}}0{,}065\,\mathrm {K} ^{-1}}$

This means that with a temperature increase of only 1.065 times it increases. In addition, the collector current doubles as soon as the temperature has risen by approx . The operating point cannot therefore be set via , since it must be kept as constant as possible when the temperature changes. ${\displaystyle I_{C}}$ ${\displaystyle 1\,\mathrm {K} }$ ${\displaystyle I_{C}}$ ${\displaystyle 15\,\mathrm {K} }$ ${\displaystyle U_{BE,A}}$ ${\displaystyle I_{C,A}}$

In the event that the temperature is only slightly dependent, one can approximately determine the temperature dependence of : ${\displaystyle I_{C,A}}$ ${\displaystyle U_{BE}}$

${\displaystyle {\frac {\delta U_{BE}}{\delta T}}\forall \left(I_{C}={const.}\right)={\frac {U_{BE}-U_{G}-3\,U_{T}}{T}}{\begin{matrix}{}_{T=300\,\mathrm {K} }\\{}_{U_{BE}=0{,}7\,\mathrm {V} }\\\approx \\\,\\\,\end{matrix}}-1{,}7\cdot 10^{-3}\,{\frac {V}{K}}}$

The current gain is also temperature dependent. The following applies here:

${\displaystyle B(T)=B\left(T_{0}\right)\,e^{\left({\frac {T}{T_{0}}}-1\right)\,{\frac {\Delta U_{\mathrm {dot} }}{U_{T}\left(T\right)}}}\approx B\left(T_{0}\right)\,{\left({\frac {T}{T_{0}}}\right)}^{x_{T,B}}}$ With ${\displaystyle x_{T,B}\approx 1{,}5}$

Here is a constant that depends on the material. The following applies to silicon . In practice, T = 300 K results in : ${\displaystyle U_{\mathrm {dot} }}$ ${\displaystyle U_{\mathrm {dot} }\approx 44\,\mathrm {mV} }$

${\displaystyle {\frac {1}{B}}\,{\frac {\delta B}{\delta T}}={\frac {\Delta U_{\mathrm {dot} }}{U_{T}\,T}}\approx 5{,}6\cdot 10^{-3}\,\mathrm {K} ^{-1}}$

And for the approximation:

${\displaystyle {\frac {1}{B}}\,{\frac {\delta B}{\delta T}}={\frac {x_{T,B}}{T}}\approx 5\cdot 10^{-3}\,\mathrm {K} ^{-1}}$

Four-pole representation

According to the four-pole theory , every electronic component can be treated as a four-pole. In the case of the transistor, the small-signal equations are represented in matrix form:

${\displaystyle {\begin{pmatrix}i_{B}\\I_{C}\end{pmatrix}}={\begin{pmatrix}{\frac {1}{r_{BE}}}&S_{r}\\S&{\frac {1}{r_{CE}}}\end{pmatrix}}\,{\begin{pmatrix}u_{BE}\\u_{CE}\end{pmatrix}}}$

or in the master value display with the Y matrix Y _e :

${\displaystyle {\begin{pmatrix}i_{B}\\I_{C}\end{pmatrix}}=\mathbf {Y} _{e}\,{\begin{pmatrix}u_{BE}\\u_{CE}\end{pmatrix}}={\begin{pmatrix}y_{11,e}&y_{12,e}\\y_{21,e}&y_{22,e}\end{pmatrix}}\,{\begin{pmatrix}u_{BE}\\u_{CE}\end{pmatrix}}}$

Alternatively, you can also use the hybrid representation with the H-matrix H _e :

${\displaystyle {\begin{pmatrix}u_{BE}\\I_{C}\end{pmatrix}}=\mathbf {H} _{e}\,{\begin{pmatrix}i_{B}\\u_{CE}\end{pmatrix}}={\begin{pmatrix}h_{11,e}&h_{12,e}\\h_{21,e}&h_{22,e}\end{pmatrix}}\,{\begin{pmatrix}i_{B}\\u_{CE}\end{pmatrix}}}$

The index e here means that the transistor is operated in an emitter circuit .

The following applies to the conversion:

${\displaystyle r_{BE}=h_{11,e}={\frac {1}{y_{11,e}}}}$

${\displaystyle \beta =h_{21,e}={\frac {y_{21,e}}{y_{11,e}}}}$

${\displaystyle S={\frac {h_{21,e}}{h_{11,e}}}=y_{21,e}}$

${\displaystyle S_{r}=-{\frac {h_{12,e}}{h_{11,e}}}=y_{12,e}}$

${\displaystyle r_{CE}={\frac {h_{11,e}}{h_{11,e}\,h_{22,e}-h_{12,e}\,h_{21,e}}}={\frac {1}{y_{22,e}}}}$

Equivalent circuit diagram with h parameters. The current source behaves like a current sink. In order for it to flow, the transistor must be operated in a suitable circuit , which is fed by an energy source. ${\displaystyle i_{2}}$

Four-quadrant characteristic field with identification of the h-parameters

Hybrid equivalent circuit

The h-parameters can be determined as follows:

• Short-circuit input impedance at , or . ${\displaystyle u_{2}=0}$ ${\displaystyle r_{BE}}$

${\displaystyle h_{11,e}={\frac {u_{1}}{i_{1}}}={\operatorname {d} \!U_{1} \over \operatorname {d} \!I_{1}}{\bigg |}_{AP}}$

• No-load voltage feedback at , ${\displaystyle i_{1}=0}$

${\displaystyle h_{12,e}={\frac {u_{1}}{u_{2}}}={\operatorname {d} \!U_{1} \over \operatorname {d} \!U_{2}}{\bigg |}_{AP}}$

• Short-circuit current gain at , is specified in data sheets rather than (forward emitter). ${\displaystyle u_{2}=0}$ ${\displaystyle h_{FE}}$

${\displaystyle h_{21,e}={\frac {i_{2}}{i_{1}}}={\operatorname {d} \!I_{2} \over \operatorname {d} \!I_{1}}{\bigg |}_{AP}}$

• Idle output conductance at . ${\displaystyle i_{1}=0}$

${\displaystyle h_{22,e}={\frac {i_{2}}{u_{2}}}={\operatorname {d} \!I_{2} \over \operatorname {d} \!U_{2}}{\bigg |}_{AP}}$ The values ​​can be read directly from the diagrams in the four-quadrant characteristic field.

Interpretation using Kirchhoff's equations

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

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

Application emitter circuit without negative feedback

In this case the emitter is fully connected to GND, the collector resistance can also be drawn to GND in the small-signal equivalent circuit diagram and must be connected in parallel with any load resistance that may be present. This sum resistance is referred to as . The voltage is applied to this resistor , so the following applies . ${\displaystyle R_{L}}$ ${\displaystyle u_{2}}$ ${\displaystyle u_{2}=-i_{2}\cdot R_{L}}$

The operational current gain can be calculated by transforming the above equations:

${\displaystyle v_{I}={\frac {i_{2}}{i_{1}}}={\frac {h_{21}}{1+h_{22}\cdot R_{L}}}}$

The operational voltage gain can also be calculated from the equations above:

${\displaystyle v_{U}={\frac {u_{2}}{u_{1}}}=-{\frac {h_{21}\cdot R_{L}}{(h_{11}\cdot h_{22}-h_{12}\cdot h_{21})\cdot R_{L}+h_{11}}}}$

In certain literature the simplification is made, thus: ${\displaystyle \Delta h=h_{11}\cdot h_{22}-h_{12}\cdot h_{21}}$

${\displaystyle v_{U}=-{\frac {h_{21}\cdot R_{L}}{\Delta h\cdot R_{L}+h_{11}}}}$

Individual evidence

1. ↑ Ulrich Tietze, Christoph Schenk: Semiconductor circuit technology . 12th edition. Springer, 2002, ISBN 978-3-540-42849-7 , pp. 55-56 .