Bipolar transistors#

In December 1947, John Bardeen, Walter Brattain and William Shockley discovered the transistor effect and developed the first device. The patent [24] was issued on September 25, 1951. For this invention, the inventors obtained the Nobel Prize in Physics in 1956.

../_images/BJT.svg

Fig. 73 Cross-section of an integrated circuit vertical NPN transistor and a lateral PNP transistor and their schematic symbols.#

Bipolar junction transistors (BJTs) are available as discrete components or as integrated circuit devices. Fig. 73 shows the cross sections of a vertical NPN transistor and a lateral PNP transistor, realized in a simple junction-isolated bipolar integrated circuit process. Both vertical NPN and PNP bipolar transistors are available in so-called complementary bipolar IC processes. In IC technology, the transistors are realized in separate islands, isolated either by depleted PN junctions or by SiO\(_{\text{2}}\). Fig. 73 also shows the symbols that are used for both transistors with and without substrate connection.

Operation#

In the so-called forward active region, the emitter base junction is forward biased and the base collector junction is reverse biased. In an NPN transistor electrons are then injected from the emitter into the base.

Due to this injection the concentration in of electrons in the base at the emitter-base junction increases. This causes a diffusion current of electrons in the base towards the collector. At the base-collector junction the concentration of electrons is zero because the drift field in the reverse biased base-collector junction accelerates them towards the collector terminal. In modern transistors, the recombination of the minorities (electrons) in the base is very low and almost all electrons injected by the emitter will reach the collector terminal. A small reverse injection of holes from the base into the emitter causes a so-called ideal base current. This current can be kept low by keeping the doping level of the base much lower than that of the emitter.

The width of the base-collector depletion layer depends on the collector-to-base voltage. Since the base doping level usually exceeds the collector doping level, the depletion layer extends mostly in the collector region. However, an increase the reverse collector-to-base voltage causes a small decrease of the base width, thereby increasing the collector current. The resulting collector-to-base voltage dependency of the collector current is called the Early effect, named after its discoverer James M. Early

(see [25]).

When the excess minority density in the base exceeds the doping level, we speak of high injection. At high injection levels the majority carrier concentration increases with the excess minority concentration, resulting in a reduction of the emitter efficiency by a factor two.

Gummel-Poon model#

The Gummel-Poon model (see [26]) is a charge-control model of the bipolar transistor that is used in SPICE-like simulation programs. It describes the behavior of bipolar transistors outside the breakdown or strong saturation regions. The operation of parasitic substrate transistors is also not modeled. Since, at high frequencies the substrate current can generally not be neglected, a voltage-dependent junction capacitance is added to the three-terminal transistor model. Four-terminal NPN transistors that are fabricated in standard bipolar processes, have their substrate capacitance connected to the collector. SPICE also has a model for lateral PNP transistors. These transistors have their substrate capacitance connected to the base.

In this section, we will give a short description of this charge-control model and its implementation in SPICE. The Gummel-Poon model for four-terminal NPN transistors is shown in Fig. 74.

../_images/gummel-poon.svg

Fig. 74 SPICE Gummel-Poon model of the four-terminal vertical BJT.#

The charge based Gummel-Poon model is an enhancement of the older Ebers-Moll model that describes the transistor as two merged PN diodes (see \cite[-1cm]{EbersMoll1954}).

Static (DC) operation#

For static operation, the transistor is modeled as a nonlinear resistive network that consist of the intrinsic transistor

with added bulk resistances \(R_{e}\), \(R_{b}\) and \(R_{c}\). The intrinsic transistor is modeled with a number of nonlinear voltage-controlled elements. In the expressions, the Gummel-Poon model parameters are written in SMALL CAPITALS. An overview of all the model parameters is given in section Device parameters.

  1. The ideal base currents consists of injected majorities from the base into the emitter, or from the base into the collector (in reverse operation). They are described by \(I_{be}\) and \(I_{bc}\):

    \[I_{be}=\frac{I_{s}}{\beta_{F}}\left( \exp\left( \frac{V_{b^{\prime} e^{\prime}}}{\text{NF}U_{T}}\right) -1\right) ,\]
    \[I_{bc}=\frac{I_{s}}{\beta_{R}}\left( \exp\left( \frac{V_{b^{\prime} c^{\prime}}}{\text{NR}U_{T}}\right) -1\right) ,\]

    where \(c^{\prime},b^{\prime}\) and \(e^{\prime}\) are the collector, base and emitter of the intrinsic transistor, respectively. The thermal voltage \(U_{T}\) equals \(\frac{kT}{q},\) in which \(k\) represents the Boltzmann constant \(\left( 1.381\times10^{-23}\text{J/K}\right) \) and \(q\) the elementary charge \(\left( 1.602\times10^{-19}\operatorname{C}\right) .\) At room temperature (\(T=293\)K), \(U_{T}\) equals \(25.26\)mV.

    The forward current gain \(\beta_{F}\) and the reverse current gain \(\beta_{R}\) both depend on temperature. Their values at reference temperature \(T_{0}\) are given by the parameters BF and BR, respectively. Their temperature behavior is modeled with the parameter XTB as

    \[\begin{split}\beta_{F}(T) & =\text{BF}\left( \frac{T}{T_{0}}\right) ^{\text{XTB}},\\ \beta_{R}(T) & =\text{BR}\left( \frac{T}{T_{0}}\right) ^{\text{XTB}}.\end{split}\]
  2. The non-ideal base currents model recombination of minorities in the base. For forward and reverse operation they are given by \(I_{re}\) and \(I_{rc},\) respectively as

    \[I_{re}=\text{ISE}\left( \exp\left( \frac{V_{b^{\prime}e^{\prime}} }{\text{NE}U_{T}}\right) -1\right) ,\]
    \[I_{rc}=\text{ISC}\left( \exp\left( \frac{V_{b^{\prime}c^{\prime}} }{\text{NC}U_{T}}\right) -1\right) .\]
  3. The transport current of minorities from the emitter to the collector is represented by the transport current \(I_{cc}\)

    \[I_{cc}=\frac{Q_{b0}}{Q_{b}}\text{IS}\left( \exp\left( \frac{V_{b^{\prime}e^{\prime}}}{\text{NF}U_{T}}\right) -\exp\left( \frac{V_{b^{\prime}c^{\prime}}}{\text{NR}U_{T}}\right) \right) ,\]

    where \(Q_{b0}\) and \(Q_{b}\) are the total minority charges in the base at zero bias and applied bias \(\left( V_{b^{\prime}e^{\prime}},V_{b^{\prime} c^{\prime}}\right) ,\)\ respectively. Their ratio is defined as

    \[\begin{split}\frac{Q_{b}}{Q_{b0}} & =\frac{1}{2}\left( 1+\frac{V_{c^{\prime}b^{\prime}} }{\text{VAF}}+\frac{V_{e^{\prime}b^{\prime}}}{\text{VAR} }\right) +\\ & \sqrt{\frac{1}{4}\left( 1+\frac{V_{c^{\prime}b^{\prime}}}{\text{VAF}}+\frac{V_{e^{\prime}b^{\prime}}}{\text{VAR} }\right) ^{2}+\frac{I_{s}}{\text{IKF}}\left( \exp\frac{V_{b^{\prime}e^{\prime}}}{\text{NF}U_{T}}-1\right) +\frac{I_{s} }{\text{IKR}}\left( \exp\frac{V_{b^{\prime}c^{\prime}} }{\text{NR}U_{T}}-1\right) }.\nonumber\end{split}\]

The parameters VAF and VAR represent the forward and the reverse Early voltages, respectively. The saturation current \(I_{s}\) is a function of temperature. Its value at the reference temperature \(T_{0} \)\ equals IS. The temperature effects are described by XTI and the bandgap voltage EG:

\[I_{s}\left( T\right) =\text{IS}\left( \frac{T}{T_{0}}\right) ^{\text{XTI}}\exp\left( \frac{\text{EG}}{U_{T}} \frac{\left( T-T_{0}\right) }{T_{0}}\right) .\]

The temperature dependence of the non-ideal base currents \(I_{se}\) and \(I_{sc}\) can differ for the various SPICE versions. We therefore refer to the appropriate reference manual. The bulk resistors \(R_{c}\), \(R_{b}\) and \(R_{e}\) in series with the terminals of the intrinsic transistor are modeled by the parameters RB, RE and RC, respectively. Some SPICE versions support the modeling of temperature effects on these resistors.

The parameters IKF and IKR represent the current at which high injection occurs for forward and reverse operation, respectively.

Dynamic effects#

Under non-equilibrium conditions, changes in the excess minority base charge and in the charge stored in the base-emitter and base-collector junction capacitances cause dynamic currents. The dynamic parts of the terminal currents \(I_{b},\) \(I_{c}\) and \(I_{e}\) can be obtained as

\[\begin{split}I_{c} & =I_{cc}-I_{bc}-I_{rc}+\frac{dQ_{dc}}{dt}-c_{jc}\frac{dV_{b^{\prime }c^{\prime}}}{dt}+c_{js}\frac{dV_{cs}}{dt},\\ I_{e} & =-I_{cc}-I_{be}-I_{re}+\frac{dQ_{de}}{dt}-c_{je}\frac{dV_{b^{\prime }e^{\prime}}}{dt},\\ I_{b} & =I_{be}+I_{bc}+I_{re}+I_{rc}+\frac{d}{dt}Q_{b}+c_{jc}\frac{dV_{b^{\prime}c^{\prime}}}{dt}+c_{je}\frac{dV_{b^{\prime}e^{\prime}}}{dt},\\ I_{s} & =-c_{js}\frac{dV_{cs}}{dt},\end{split}\]

where \(Q_{de}\) and \(Q_{dc}\) represent the excess minority charges in the base, and \(c_{jc}\) and \(c_{je}\) are the (small-signal) base-collector and base-emitter junction capacitances, respectively. The total excess minority charge depends on the minority transport current \(I_{cc}\) and the minority transit time in the base. These transit times are modeled by \(tf\) and TR for forward and reverse operation, respectively. The associated excess minority charges \(Q_{de}\) and \(Q_{dc}\) can be expressed as

\[\begin{split}Q_{de} & =I_{cc}\tau_{f}\left( \exp\left( \frac{V_{b^{\prime}e^{\prime}} }{U_{T}}\right) \right) ,\\ Q_{dc} & =I_{cc}\text{TR}\left( \exp\left( \frac{V_{b^{\prime }c^{\prime}}}{U_{T}}\right) \right) .\end{split}\]

The forward transit time \(\tau_{f}\) is modeled as a function of both the forward ideal transport current \(I_{tf}\) and the collector to base voltage \(V_{b^{\prime}c^{\prime}}\). At high currents \(\tau_{f}\) increases due to the so-called base push-out or Kirk effect (see [27]):

\[\tau_{f}=\text{TF}\left( 1+\left( \text{XTF}\frac{I_{tf} }{I_{tf}+\text{ITF}}\right) ^{2}\right) \exp\left( \frac{V_{b^{\prime}c^{\prime}}}{1.44\text{VTF}}\right) ,\]

where the ideal forward transport current \(I_{tf}\) is defined as

\[I_{tf}=\text{IS}\exp\left( \frac{V_{b^{\prime}e^{\prime}} }{\text{NF}U_{T}}\right) .\]

The amount of charge stored in the depletion capacitances of PN-junctions depends on the voltage across these junctions. Due to the voltage dependence of the depletion layer width, this charge shows a nonlinear relation with the voltage. In SPICE the \(c_{j}(V)\) relation instead of the \(Q_{j}(V)\) relation is modeled. For reverse biased junctions \(\left( V\leq0\right) \) the value of the depletion capacitance is modeled as:

\[c_{j}(V)=\frac{C_{j}(0)}{\left( 1-\frac{V}{V_{j}}\right) ^{M}}\]

Where \(C_{j}(0)\) is the zero bias depletion capacitance, \(V_{j}\) the built-in junction barrier voltage and \(M\) the junction grading factor, \(M=0.33\) for linear graded junctions and \(M=0.5\) for step junctions. These expressions are also used for weakly forward biased junctions. SPICE uses this expression if the forward junction voltage \(V\) is smaller than FC times the built-in junction voltage. The SPICE parameter FC has a default value of \(0.5\).

The bipolar transistor model has three junction capacitances. Their values for reverse biased junctions are

\[\begin{split}c_{jc} & =\frac{\text{CJC}}{\left( 1-\frac{V_{b^{\prime}c^{\prime }}}{\text{VJC}}\right) ^{\text{MJC}}},\\ c_{je} & =\frac{\text{CJE}}{\left( 1-\frac{V_{b^{\prime}e^{\prime }}}{\text{VJE}}\right) ^{\text{MJE}}},\\ c_{js} & =\frac{\text{CJS}}{\left( 1-\frac{V_{cs}}{\text{VJS}}\right) ^{\text{MJS}}}.\end{split}\]

Model improvements#

One improvement of the enhanced Gummel-Poon model is the modeling of a current-dependent base resistance. This effect, however, is not found from noise measurements. The current-dependency of \(R_{b}\) is characterized by three parameters RB, RBM and IBM that represent the maximum value, the minimum value at high currents and the current where the resistor is halfway its minimum value, respectively. In some SPICE versions, the temperature dependency of the bulk resistors is modeled. Some SPICE versions have model extensions for deep saturation (strongly forward biased collector-base junction).

Small-signal dynamic model#

../_images/hy-pi.svg

Fig. 75 Small-signal equivalent circuit of the bipolar transistor in the active forward region. In the Gummel-Poon model, the collector base resistance \(r_{\mu}\) is \(\infty\).#

For small excursions from an operating point \(Q\), at which the DC operating currents and voltages are given by \(I_{B},\ I_{C},\ V_{BE}\) and \(V_{CE}\), we can use a small-signal equivalent circuit. The values of the small-signal model parameters are obtained from differentiation of the \(v-i\) and \(q-v\) relations in the operating point \(Q\). The small-signal equivalent model of the BJT is shown in Fig. 75. The Gummel-Poon model does not include \(r_{\mu}\). This small-signal model is usually referred to as the hybrid-\(\pi\) small-signal equivalent circuit, its parameters are

\(r_{b}=\left. R_{b}\right\vert _{Q}\)

\(r_{\pi}=\left. \frac{\partial V_{b\prime e\prime}}{\partial I_{b}}\right\vert _{Q}\)

\(c_{bc}=\left. \left( \frac{\partial Q_{dc}}{\partial V_{b\prime c\prime}}+c_{jc} \text{XCJC}\right) \right\vert _{Q}\)

\(R_{e}=\text{R}\)E

\(g_{m}=\left. \frac{\partial I_{c} }{\partial V_{b\prime e\prime}}\right\vert _{Q}\)

\(c_{bx}=\left. \left( 1-\text{XCJC}\right) c_{jc}\right\vert _{Q}\)

\(R_{c}=\) RC

\(r_{o}=\left. \frac{\partial V_{c\prime e\prime} }{\partial I_{c}}\right\vert _{Q}\)

\(c_{\pi}=\left. \left( \frac{\partial Q_{de}}{\partial V_{b\prime e\prime}}+c_{je}\right) \right\vert _{Q}\)

\(r_{\mu}=\left. \frac{\partial V_{c\prime e\prime}}{\partial I_{b} }\right\vert _{Q}\)

\(c_{js}=\left. c_{js}\right\vert _{Q}\)

Stationary noise model#

../_images/hy-pi-noise.svg

Fig. 76 Small signal hybrid \(\pi\) equivalent circuit with noise sources.#

In SPICE, frequency-domain noise analysis can be performed with a small-signal noise analysis. For this purpose independent and stationary noise sources are added to the small-signal hybrid-\(\pi\) equivalent circuit from Fig. 75. The resulting noise model is shown in Fig. 76. The noise sources \(v_{b}\), \(v_{e}\) and \(v_{c}\) represent the thermal noise of the bulk resistors. These sources have a Gaussian amplitude distribution function and a uniform spectrum. Their spectral densities are

\[\begin{split}S_{vb} & =4kTr_{b}\quad\text{[V}^{2}/\text{Hz],}\\ S_{ve} & =4kTR_{e}\quad\text{[V}^{2}/\text{Hz],}\\ S_{vc} & =4kTR_{c}\quad\text{[V}^{2}/\text{Hz].}\end{split}\]

The current sources \(i_{b}\) and \(i_{c}\) represent the shot noise of the base current and the collector current in the operating point, respectively. These sources have a Gaussian amplitude distribution function and a uniform spectrum. Their spectral densities are given by

(24)#\[\begin{split}S_{ib} & =2qI_{B}\quad\text{[A}^{2}/\text{Hz]}\label{eq-Sib}\\ S_{ic} & =2qI_{C}\quad\text{[A}^{2}/\text{Hz]}\end{split}\]

The noise source \(i_{bf}\) represents the flicker noise associated with the base current. This source has a Gaussian amplitude distribution function and a power spectrum that is inversely proportional with frequency. The spectral density \(S_{ibf}\) of this source is modeled as

(25)#\[S_{ibf}=\frac{\text{KF}I_{B}^{\text{AF}}}{f}\quad \text{[A}^{2}/\text{Hz],} \label{eq-Sibf}\]

where KF and AF are the model parameters for this \(1/f\) noise term.

The noise current associated with the base current is often described with the aid of the noise corner frequency \(f_{\ell}\) of the \(1/f\) noise:

\[S_{ibt}=2qI_{B}\left( 1+\frac{f_{\ell}}{f}\right) \quad\text{[A} ^{2}/\text{Hz]},\]

where the corner frequency \(f_{\ell}\) of the \(\frac{1}{f}\) noise is obtained from ((24)) and ((25)) as

\[f_{\ell}=\frac{\text{KF}}{2q}I_{B}^{\left( \text{AF }-1\right) }\quad\text{[Hz].}\]

Low-noise BJTs can have \(f_{\ell}\) below 100Hz. This cut-off frequency strongly depends on the applied technology. In modern RF IC processes it may exceed 10kHz.

Device parameters#

In this section, we will give an overview of the Gummel-Poon model parameters. Both NPN and PNP devices have positive valued parameters. SPICE has different BJT models for vertical and for lateral transistors. The supported BJT types are listed in Table 7.

Table 7 Types of BJTs supported in {SPICE}.#

model

description

NPN

vertical NPN transistor with collector-substrate capacitance

PNP

vertical PNP transistor with collector-substrate capacitance

LPNP

lateral PNP transistor with base-substrate capacitance

The SPICE parameters of the Gummel-Poon model are listed in Table 8. There are two arguments AREA and TEMP that represent the device emitter area scaling factor and the device temperature, respectively.

Simulated device characteristics#

Fig. 77 shows a test bench for simulation of the device characteristics. The netlist file is shown below:

 1DCchars
 2* FILE: myNPN_DCchars.cir
 3* LTspice circuit file
 4VBE 1 0 0
 5VCE 2 0 0
 6Q1 2 1 0 myNPN
 7.model myNPN NPN
 8+ IS=0.5f BF=100 NF=1 IKF=100m ISE=10f NE=2 RB=10 VAF=20
 9+ TF=1n CJE=5p CJC=1p VJE=0.6 VJC=0.8 XTF=1 VTF=2 ITF=20m
10.dc VCE 0 5 10m VBE 0.6 0.65 10m
11* .dc VBE 0 0.65 10m VCE 1 5 1
12.end

The simulation results are shown in Fig. 79. These figures show that the BJT behaves as a voltage-controlled current source. In the Gummel-Poon model the input current \(I_{B}\) only depends on the input voltage. The output current is controlled by the input voltage. Over a wide range of output voltages it shows a weak dependence of the output voltage. For the forward active region, this is characterized with the forward Early voltage VAF. The value of this parameter can be found from the intersection point of the extrapolated \(I_{C}\left( V_{CE}\right) \) characteristic and the \(x\)-axis, as shown in Fig. 78.

Table 8 BJT Gummel-Poon model parameters. An asterix (*) in the {AREA} column indicates scaling of the corresponding parameter with AREA.#

name

description

unit

default

AREA

IS

transport saturation current

A

10\(^{-16}\)

NF

forward emission coefficient

1

VAF

forward early voltage

V

\(\infty\)

IKF

forward high-injection knee current

A

\(\infty\)

BF

ideal forward DC current gain factor

100

ISE

base-emitter leakage saturation current

A

0

NE

base-emitter leakage emission coefficient

1.2

EG

bandgap voltage

V

1.11

XTI

temperature coefficient (IS)

3

NR

reverse emission coefficient

1

VAR

reverse early voltage

V

\(\infty\)

IKR

reverse high-injection knee current

A

\(\infty\)

BR

ideal reverse DC current gain factor

1

ISC

base-collector leakage saturation current

A

0

NC

base-collector leakage emission coefficient

2

XTB

temperature coefficient (BF; BR; ISE; ISC)

0

RB

zero-bias base resistance

\(\Omega\)

0

IRB

current at which \(R_{b}\) falls halfway to RBM

A

\(\infty\)

RBM

minimum value of \(R_{b}\) at high current

\(\Omega\)

RB

RE

emitter bulk resistance

\(\Omega\)

0

RC

collector bulk resistance

\(\Omega\)

0

TNOM

temperature for parameter measurement

\(^{\text{o}}\)C

27

TF

ideal forward transit time

s

0

XTF

coefficient for bias dependence of TF

0

VTF

voltage describing \(V_{BC}\) dependence of TF

V

\(\infty\)

ITF

high-current parameter for effect on TF

A

0

PTF

excess phase at \(f=1/(2\pi\)TF\()\ [\)Hz\(]\)

deg

0

TR

ideal reverse transit time

s

0

CJE

base-emitter zero-bias depletion capacitance

F

0

VJE

base-emitter built-in potential

V

0.75

MJE

base-emitter junction grading coefficient

0.33

CJC

base-collector zero-bias depletion capacitance

F

0

VJC

base-collector built-in potential

V

0.75

MJC

base-collector junction grading coefficient

0.33

XCJC

fraction of \(C_{bc}\) connected to internal base

1

CJS

collector-substrate zero-bias depletion capacitance

F

0

VJS

collector-substrate built-in potential

V

0.75

MJS

collector-substrate junction grading coefficient

0

FC

forward bias depletion capacitance coefficient

0.5

KF

flicker noise coefficient

0

AF

flicker noise exponent

1

../_images/myNPN_DCchars.svg

Fig. 79 DC input and output characteristic of the transistor “myNPN”, obtained from simulation with the circuit from Fig. 77.#

Gummel plot#

The \(I_{B}\left( V_{BE}\right) \) characteristic and the \(I_{C}\left( V_{BE}\right) \) characteristic plotted together on a semi-logarithmic scale is called the Gummel plot.

It shows the exponential relation between the base to emitter voltage and the base current components. Fig. 80 shows the simulation setup for this plot. The SPICE netlist file is shown below:

 1* FILE: gummelPlot.cir
 2* LTspice circuit file
 3VBE b 0 0
 4VCB c b 0
 5Q1 c b 0 myNPN
 6.model myNPN NPN
 7+ IS=0.5f BF=100 NF=1 IKF=100m ISE=10f NE=2 RB=10
 8.options gmin=1f
 9.dc VBE 0 1 10m
10.end
11

Fig. 81 shows the resulting Gummel plot. It shows the exponential relation between the base-emitter voltage and the collector current over many decades. The parameter BF models the ratio between the ideal collector current and the ideal base current. At very low current levels, the non-ideal base current models the recombination of minorities in the base. At high collector currents, high injection reduces the slope of the \(I_{C}\left( V_{BE}\right) \) curve by a factor two, while the voltage drop across the bulk resistors causes a deviation of the ideal exponential relation.

../_images/myNPN_GummelPlot.svg

Fig. 81 Gummel plot of the transistor “myNPN”, obtained by simulation with the circuit from figure Fig. 80.#

Definition of \(\beta_{DC}\)#

The DC current gain \(\beta_{DC}\) of a bipolar transistor is defined as the ratio of the DC collector current \(I_{C}\) and the DC base current \(I_{B}\):

\[\beta_{DC}=\frac{I_{C}}{I_{B}}.\]

The DC current gain is a function of the collector current. This can be seen from the Gummel plot. Due to the Early effect, it also increases with the collector-emitter voltage.

Definition of \(\beta_{AC}\)#

The small-signal current gain is somewhat misleadingly denoted by \(\beta_{AC} \). This small-signal quantity is defined for infinitesimally small excursions from the operating point:

\[\beta_{AC}=\left. \frac{\partial I_{c}}{\partial I_{b}}\right\vert _{Q},\]

alternatively we may write

\[\beta_{AC}=\left. \frac{\partial I_{c}}{\partial V_{b\prime e\prime} }\right\vert _{Q}\cdot\left. \frac{\partial V_{b\prime e\prime}}{\partial I_{b}}\right\vert _{Q}=g_{m}r_{\pi}.\]

Fig. 82 shows the DC current gain and the static (zero-frequency) forward small-signal current gain \(\beta_{AC}\) versus the collector current of myNPN for \(V_{CB}=0\)V.

../_images/myNPN_BetaDC_BetaAC.svg

Fig. 82 The forward DC current gain \(\beta_{DC} \) and the forward static small-signal current gain \(\beta_{AC}\) of “myNPN” as a function of the collector current.#

Dynamic behavior of \(\beta_{AC}\)#

At high frequencies the small-signal current gain \(\beta_{AC}\) drops below its static value. This is caused by charge storage in the base due to the finite transit time for minorities and in the depletion regions. The cut-off frequency of a bipolar transistor is defined as the frequency at which the magnitude of \(\beta_{AC}\) equals unity. The cut-off frequency \(\omega_{T}\) is a figure of merit for the speed limitation of the bipolar transistor.

../_images/BJT-char-setup-FT.svg

Fig. 83 Simulation test bench for determination of the cut-off frequency \(f_{T}\) as a function of the DC collector current.#

Fig. 83 shows the simulation test bench for the determination of the cut-off frequency using SPICE. The voltage amplifier \(E_{1}\) provides the driving conditions for the bipolar transistor such that its operating collector current and collector-emitter voltage equal \(I_{C}\) and \(V_{CE}\), respectively. The large capacitor \(C_{1}\) prevents feedback at signal frequencies of interest, while it concurrently provides the condition for measuring the short circuit collector current. Below is the LTSPICE\ listing of the netlist file of the circuit from Fig. 83:

 1myNPN_FT
 2* FILE: myNPN_FT.cir
 3* LTspice circuit file for plotting fT(Ic)
 4V1 4 1 3
 5V2 2 3 DC 0 AC 1
 6I1 0 4 {Ic}
 7C1 1 0 1
 8E1 2 0 1 0 1
 9Q1 4 3 0 myNPN
10.model myNPN NPN
11+ IS=0.5f BF=100 NF=1 IKF=100m ISE=10f NE=2 RB=10
12+ TF=1n CJE=5p CJC=1p VJE=0.6 VJC=0.8 XTF=1 VTF=2 ITF=20m
13.ac dec 50 1 1G
14* LTspice syntax for plotting fT(Ic)
15.step dec param Ic 100n 10m 10
16.meas AC fT FIND Frequency WHEN dB(I(V1)/I(V2))=0 CROSS=1
17* Run this netlist and press CTRL + L in the trace window
18* This will bring up the output file (Spice error log) window
19* In this window right-click "Plot .step'ed .meas data"
20.end
../_images/myNPN_FT.svg

Fig. 84 Plot of FT versus frequency of “myNPN”, obtained with the simulation test bench from Fig. 83.#

The simulation results for myNPN are shown in Fig. 84. At low currents the cut-off frequency is dominated by the depletion capacitance, while at high frequencies it is limited by the forward transit time TF. The maximum value \(f_{T\max}\) of the cut-off frequency can be estimated from TF:

\[f_{T\max}\approx\frac{1}{2\pi\text{TF}}.\]

At high collector currents, the base region tends to extend towards the collector. This base push-out or Kirk-effect (see \cite[-0.5cm]{Kirk1962}) causes a reduction of the cut-off frequency at high collector currents.

Operating point information#

The current-drive capability and voltage-drive capability of a BJT depend on the DC collector current \(I_{C}\) and the collector-emitter voltage \(V_{CE}\), respectively. Many other performance aspects, such as, the noise performance and the cut-off frequency also show a direct relation with the collector current. As a consequence, we usually want to fix the operating point through fixing \(I_{C}\) and \(V_{CE}\).

A method for fixing the operating point of nonlinear resistive multi-terminal devices has been discussed in Chapter Amplification Mechanism. According to the presented method, fixing the operating point of a BJT by means of \(I_{C}\) and \(V_{CE},\) requires the addition of a voltage source \(V_{CE}\) between the collector and the output and a current source \(I_{C}\) that flows from the emitter to the collector.

In order to obtain zero output voltage and zero output current for all DC input and output terminations, a voltage source \(V_{BE}\) has to be placed in series with the base and a current source \(I_{B}\) has to be connected in parallel with the base-emitter junction. The values of these input sources depend on the required values of \(I_{C}\) and \(V_{CE}\), on the DC characteristics of the device and on the operating temperature. They can be determined with the aid of the circuit from Fig. 85 . The nullator at the output port (collector-emitter) sets the condition for zero output voltage and zero output current, while the norator at the input port delivers the correct driving quantities to satisfy these conditions. The nullor is not available in SPICE but it can be implemented with two unity-gain voltage-controlled voltage sources as illustrated in this figure.

SPICE returns small-signal parameters in a certain operating point as the result of an operating point analysis (.op statement). The parameters returned depend on the device model and the SPICE version. Tabel-BipOPversions gives an overview of the small-signal parameters of the Gummel-Poon model returned by different SPICE versions.

Below the listing of a circuit for determination of the small-signal parameters of myNPN for \(I_{C}=1\)mA and \(V_{CE}=3\)V.

 1* FILE: myNPN_OP.cir
 2* SPICE circuit file
 3*
 4* Transistor with VCE and IC definition
 5Q1 1 2 0 myNPN
 6VCE 1 3 {V_CE}
 7IC 0 3 {I_C}
 8*
 9* nullor
10E1 4 0 3 0 1
11E2 2 0 2 4 1
12*
13.model myNPN NPN
14+ IS=0.5f BF=100 NF=1 IKF=100m ISE=10f NE=2 RB=10
15+ TF=1n CJE=5p CJC=1p VJE=0.6 VJC=0.8 XTF=1 VTF=2 ITF=20m
16*
17.param V_CE=3 I_C=1m
18.op
19.end

If simulators do not provide the operating point information for the hybrid-\(\pi\) equivalent circuit from Fig. 75, the designer can:

  1. Estimate it from the characteristics given in data sheets or design manuals

  2. Obtain it from AC (small-signal) analysis in the desired operating point.

Table 9 Small-signal parameters of the Gummel-Poon model, returned from an operating point analysis with various simulators.#

name

\textbf{description }(see Fig. 75)

LTspice

SIMetrix

Pspice

ngspice

\(\beta_{DC}\)

DC current gain

BetaDC

BetaDC

BETADC

\(\beta_{AC}\)

Zero frequency small-signal current gain

BetaAC

BetaAC

BETAAC

\(c_{bc}\)

Small-signal internal base-collector capacitance

Cbc

Cjc

CBC

cmu

\(c_{\pi}\)

Small-signal base-emitter capacitance

Cbe

Cje

CBE

cpi

\(c_{cs}\)

Small-signal collector-substrate capacitance

Cjs

Cjs

CJS

csub

\(c_{bx}\)

Small-signal external base-collector capacitance

Cbx

Cxjc

CBX

cbx

\(g_{m}\)

Forward transconductance

Gm

Gm

GM

gm

\(I_{B}\)

DC base current

Ib

IB

IB

ib

\(I_{C}\)

DC collector current

Ic

IC

IC

ic

\(I_{E}\)

DC collector current

IE

ie

\(I_{S}\)

DC substrate current

IS

\(P\)

DC power dissipation

Power

\(r_{b}\)

Small-signal base resistance

Rx

Rbase

\(g_{b}=\frac{1}{r_{b}}\)

Small-signal base conductance

GX

gx

\(r_{o}\)

Small-signal output resistance

Ro

Ro

RO

\(g_{o}=\frac{1}{r_{o}}\)

Small-signal output conductance

go

\(r_{\pi}\)

Small-signal internal base-emitter resistance

Rpi

Rpi

RPI

\(g_{\pi}=\frac{1}{r_{\pi}}\)

Small-signal internal base-emitter conductance

gpi

\(r_{\mu}\)

Reverse resistance

\(g_{\mu}=\frac{1}{r_{\mu}}\)

Reverse conductance

gmu

\(F_{T}\)

Cut-off frequency

Ft

FT

\(V_{BE}\)

DC base-emitter voltage

Vbe

Vbe

VBE

vbe

\(V_{CB}\)

DC collector-base voltage

Vbc

Vcb

VBC

vbc

\(V_{CE}\)

DC collector-emitter voltage

Vce

Vce

VCE

Other models#

For modern SiGe RF IC processes the Vertical Bipolar Inter-Company (VBIC) model of the BJT is often used. The model is a public domain replacement for the Gummel-Poon model. It has improved modeling of the saturation region and includes modeling of breakdown and thermal effects. For more information, the reader is referred to literature [28].

Simplified models for hand calculations#

Complete SPICE models are suitable for numerical simulations but they are too complex to provide design information from analytical expressions. For this purpose we need simplified models that are suited for hand calculations. In this section, we will derive these models from the Gummel-Poon model. We will introduce a large-signal static model, a small-signal dynamic model and a noise model that can be used for analytical determination of the operating point, the dynamic small-signal transfer and the noise behavior of the bipolar transistor, respectively.

DC behavior#

The static large-signal behavior

of a bipolar transistor in the active forward operating region can roughly be predicted from the model shown in Fig. 86. In this model, the bulk resistances, the high-injection effect, the non-ideal base current and reverse operation have not been modeled. The model describes port currents of a nonlinear resistive two-port as a function of the port voltages:

\[\begin{split}I_{C} & =\text{IS}\left( \exp\left( \frac{V_{BE}}{U_{T}}\right) \right) \left( 1+\frac{V_{CE}}{\text{VAF}}\right) ,\\ I_{B} & =\frac{I_{C}}{\text{BF}}.\end{split}\]

The active forward operating range is limited by

\[\begin{split}V_{BE} & >0,\\ V_{CE} & >V_{CE,SAT}.\end{split}\]

Where \(V_{CE,SAT}\) is the collector-emitter voltage where the base-collector junction is forward biased at a current that cannot longer be neglected with respect to the current in the forward biased base-emitter junction.

A simple model for hand calculations that includes operation in the forward saturation region can be formulated as

\[\begin{split}I_{C} & =\text{IS}\left( \left( \exp\frac{V_{BE}}{U_{T}}\right) -\left( \exp\frac{V_{BC}}{U_{T}}\right) \right) \left( 1+\frac{V_{CE} }{\text{VAF}}\right) ,\\ I_{B} & =\frac{\text{IS}}{\text{BETAF}}\left( \exp \frac{V_{BE}}{U_{T}}-1\right) -\frac{\text{IS}}{\text{BETAR }}\left( \exp\frac{V_{BC}}{U_{T}}-1\right) .\end{split}\]

Small-signal dynamic model#

In most cases, the small-signal dynamic behavior of the bipolar junction transistor in the active forward operating region can sufficiently accurate be described with the simplified hybrid-\(\pi\) equivalent circuit from Fig. 87. The bulk resistances \(R_{e}\) and \(R_{c}\) have been omitted. If XJC~\(\leq0.5\), the base-collector capacitances \(c_{bc}\) and \(c_{bx}\) can both be connected to the internal node (b’), if not, they should both be connected to the external node (b). For both cases we may then use

\[c_{\mu}=c_{bx}+c_{bc}.\]

The small-signal parameters of the simplified hybrid-\(\pi\) equivalent circuit can be estimated from the Gummel-poon model parameters and the operating point, thereby ignoring the current dependency of the current gain, of the base resistance and of the forward transit time. In this way we obtain, for operation in the active forward region

\(r_{b}=\text{RB;}\)

\(r_{o}=\frac{\text{VAF}+V_{CE}}{I_{C} };\)

\(c_{\pi}=\text{CJE}+\text{TF}\frac{I_{C}}{V_{T}};\)

\(r_{\pi}=\text{BF}\frac{V_{T}}{I_{C}};\)

\(g_{m}=\frac{I_{C}}{V_{T} };\)

\(c_{\mu}=\text{CJC.}\)

\(f_{T}\) and \(f_{\max}\)#

The cut-off frequency \(f_{T}\) is defined as the unity-gain frequency of \(\beta_{AC}\). It can be obtained as

\[f_{T}=\frac{g_{m}}{2\pi\left( c_{\pi}+c_{\mu}\right) }.\]

The cut-off frequency can be seen as a figure of merit for the amplifying capabilities of a transistor.

A better figure of merit for the amplifying capabilities of an active device, is the frequency at which the power gain of the transistor equals unity, when is driven from and terminated with the complex conjugate of its input and output impedances, respectively. Unity power gain of the BJT is found at the frequency \(f_{\max}\), were

\[f_{\max}=\sqrt{\frac{f_{T}}{8\pi r_{b}c_{\mu}}}.\]

Noise model#

The simplified noise model is shown in Fig. 88. The noise contributions of the bulk resistances \(R_{e}\) and \(R_{c}\) are omitted, while the remaining noise sources of the complete noise model from Fig. 76 have been added to the simplified hybrid-\(\pi\) equivalent circuit from Fig. 87.

../_images/Hy-pi-simple-noise.svg

Fig. 88 Simplified hybrid-\(\pi \) small-signal equivalent circuit with stationary noise sources.#

Determination of hybrid-\(\pi\) parameters from simulation#

The small-signal parameters can also be obtained from simulation. Fig. 89 shows the simulation test bench for determination of the small-signal hybrid-\(\pi\) parameters.

The procedure is as follows:

  1. Bias the transistor in the required operating point with the aid of the \(DC\) voltage sources \(V_{BE}\) and \(V_{CE}\). The value of \(V_{BE}\) can be obtained from an operating point simulation with the test bench from Fig. 85.

  2. Make \(V_{AC1}=1,0\) (magnitude \(1\), phase \(0\)) and \(V_{AC2}=0\) and perform an AC analysis over the frequency range of interest

  3. Obtain approximations the values of the following parameters

    \[\begin{split}r_{b}+r_{\pi} & =\frac{1}{\operatorname{Re}\{-I(V_{AC1})\}},\\ g_{m}\frac{r_{\pi}}{r_{b}+r_{\pi}} & =\operatorname{Re}\{-I(V_{AC2})\},\\ c_{\pi}+c_{\mu} & =\frac{\operatorname{Im}\{-I(V_{AC1})\}}{2\pi f}.\end{split}\]
  4. Make \(V_{AC1}=0\) and \(V_{AC2}=1,0\) (magnitude \(1\), phase \(0\))

  5. Obtain approximations for the values of the following parameters

    \[\begin{split}r_{o} & =\frac{1}{\operatorname{Re}\{-I(V_{AC2})\}},\\ c_{\mu} & =\frac{\operatorname{Im}\{I(V_{AC1})\}}{2\pi f},\\ r_{\mu} & =\frac{r_{\pi}+r_{b}}{r_{\pi}}\frac{1}{\operatorname{Re} \{-I(V_{AC1})\}}.\end{split}\]
  6. The base resistance \(r_{b}\) can at best be determined from noise measurements. This will be explained later.