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The hybrid-pi model is a popular circuit model used for analyzing the small signal behavior of bipolar junction and field effect transistors. 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.
Bipolar junction (BJT) parameters
The hybrid-pi model is a linearized two-port network approximation to the BJT using the small-signal base-emitter voltage vbe and collector-emitter voltage vce as independent variables, and the small-signal base current ib and collector current ic as dependent variables.
Figure 1: Simplified, low-frequency hybrid-pi BJT model.
A basic, low-frequency hybrid-pi model for the bipolar transistor is shown in figure 1. The various parameters are as follows.
is the transconductance in siemens, evaluated in a simple mode
is the quiescent collector current (also called the collector bias or DC collector current)
is the thermal voltage, calculated from Boltzmann's constant k, the charge of an electron q, and the transistor temperature in kelvins, T. At 300 K (approximately room temperature) VT is about 26 mV (Google calculator).
is the current gain at low frequencies (commonly called hFE). Here IB is the Q-point base current. This is a parameter specific to each transistor, and can be found on a datasheet; Î² is a function of the choice of collector current.
is the output resistance due to the Early effect (VA is the Early voltage).
The reciprocal of the output resistance is named the output conductance
The reciprocal of gm is called the intrinsic resistance
Figure 2: Simplified, low-frequency hybrid-piMOSFET model.
A basic, low-frequency hybrid-pi model for the MOSFET is shown in figure 2. The various parameters are as follows.
is the transconductance in siemens, evaluated in the Shichman-Hodges model in terms of the Q-point drain current ID by (see Jaeger and Blalock):
ID is the quiescent drain current (also called the drain bias or DC drain current)
Vth = threshold voltage and VGS = gate-to-source voltage.
often is called the overdrive voltage.
is the output resistance due to channel length modulation, calculated using the Shichman-Hodges model as
using the approximation for the channel length modulation parameter Î»
Here VE is a technology-related parameter (about 4 V/Î¼m for the 65 nm technology node) and L is the length of the source-to-drain separation.
The reciprocal of the output resistance is named the drain conductance
The COMMON-EMITTER CONFIGURATION (CE) is the most frequently used configuration in practical amplifier circuits, since it provides good voltage, current, and power gain. The input to the CE is applied to the base-emitter circuit and the output is taken from the collector-emitter circuit, making the emitter the element "common" to both input and output. The CE is set apart from the other configurations, because it is the only configuration that provides a phase reversal between input and output signals
High -Frequency -pi CE transistor model
The Hybrid-Pi model is a fairly accurate description of the BJT small-signal response up to GHz range.
Since the common emitter circuit is considered the most important practical configuration , we seek a CE model suitable for high frequencies. Hybrid -pi or Giacoletto common emitter transistor model shown below. This circuit is quite simple and analysis of circuit using this model are not difficult and give result which are in excellent agreement with experiment at all frequencies for which the transistor gives reasonable amplification. Furthermore , the resisitive components in this circuit may be derived from the low frequencies H-parameters. All parameters (resistances and capacitances) in the model are assumed frequency invariant. Parameters may be vary with the quiescent operating point , but under given bias conditions they are reasonably constant for small signal variations. For high frequency analysis the transistor is replaced this high frequency hybrid PI-model and voltage gain and current gain , input impedances etc are determined.
To find current gain
Apply current divider rule to the output circuit
To find input resistance
Applying KVL to input circuit
Vs = hie ib + hre vce
Vs = ib hie + hre iL RL from equation (1)
Vs = ib hie + hre Ai ib RL ( iL =Ai ib )
Substituting in equation (2)
Ri = hie + hre Ai RL
To find voltage gain
To find output resistance
Replace RL by a voltage source. Replace independent sources by internal
impedance of the source
Applying KC L to the output circuit.
iC = hfe ib + i1
iC = hfe ib + vce hoe-----(4)
Applying KVL to input circuit
- ( hie ib + hre vce) =0
substituting for ib in equation (4)
substituting in equation (3)
To find output resistance with RL
RO1 = RO||RL
Since RL is in parallel with the voltage source, total output resistance is the parallel combination of RL and RO
Question A common emitter amplifier has the following h- parameters. hie =1KÎ©, hre = 10-4, hfe =100, hoe = 12µmho. Find current gain, Voltage gain, Ri, Ro, power gain. Take RL = 2KÎ©. Also find output power take vS = 500 mV ( rms).
HYBRID-p EQUIVALENT CIRCUIT
To obtain Hybrid-p Equivalent circuit
Consider a PNP transistor as shown above. The emitter current IE is divided in to base current IB and a component aIE of the collector current. This division of current takes place in the entire base layer at infinite number of points. For mathematical convenience, it is assumed that the division of current takes place at an imaginary terminal B1.
rb1e: It is the resistance of forward biased base to emitter junction and it is the resistance offered to the flow of the current IE.
rb1c: It is the resistance of reverse biased collector to base junction. The flow of current in this resistance represents the reverse saturation current Ico due to flow of minority charge carriers.
rbb1: It is the resistance of the base layer for the flow of the current IB. This is called base spreading resistance because the division of emitter current is spread across the entire region.
aIE: This is the current in the collector due to transistor action. When charge carriers reach the base layer from emitter, the potential gradient at the collector junction will result in the movement of the charge carriers in to the collector. This forms the current. aIE depends on the emitter current IE which inturn depends upon the voltage across base to emitter junction.
Therefore, the voltage VB1E controls aIE. VB1E is the independent variable. This depends on charge carrier concentration and temperature.
cb1e and cb1c: This is the stray capacitance across the two P-N junction. The reactance of the capacitor is very high at mid-frequency. Hence approximately, capacitors are replaced by open circuit (not considered). But for high frequency, the reactance becomes finite. Hence considered in the analysis.
All the above terms are called Hybrid-p parameters. These parameters can be represented by the following circuit and it is called Hybrid-p equivalent circuit or Giacollette equivalent circuit.
gm vb1e is the component of collector current(aie) expressed as a function of independent variable vb1e. gm is the Transconductance of the transistor. This represents ability of the transistor in transforming the input voltage vb1e in to output current.rce: rce is the internal resistance of the current source.
To find Hybrid-p parameters
Hybrid -p equivalent circuit
Let the output terminals be short circuited .
Considering mid- frequency, reactance of all capacitors becomes infinite. Therefore, all capacitors can be replaced by open circuit.
rb1c is the resistance of reverse biased collector junction whose value is very high. Therefore it can be approximated to open circuit.
rce is short circuited, becomes redundant. Hence can be removed
To find gm
where Î”IC and Î”VB1E are the changes in the currents and voltages around quiescent condition.
We know that
IC = aIE + ICO
Since ICO is very small and a is very close to unity,
Differentiating with respect to VB1E
If t = 27oC
substituting in (1)
In the above equation, IC represents the dc collector current or quiescent current. Its value can be found graphically by drawing the dc load line, locating the Q point on the load line and then measuring IC. OR if know the biasing arrangement of the transistor, then the circuit can be solved using biasing technique and then IC can be calculated.
To find rb1e
From the two port network theory, we know that
vse = ib hie + hre vce -----(4)
ic = ibhfe + hoe vce -----------(5)
From equation -(5)
In the hybrid p equation circuit, VCE is already 0. Therefore obtain the ratio
From hybrid p equation circuit and equate it to equation (6).
Equating to equation (6).
hfe =gm rb1e
To find rbb1
From equation (4)
From the hybrid-p equivalent circuit, applying KVL to input circuit.
Vs = ib(rbb1 + rb1e)
To find rb1c
Rewriting the hybrid p equivalent circuit by neglecting all capacitances( open circuit)
From equation (4)
Taking ib = 0 in the hybrid-p equivalent circuit, since there is no voltage drop across rbb1, vs = vb1e.
Substituting in equation (8).
From the hybrid -p equivalent circuit. Applying voltage divider rule to circuit(2).
rb1e is the resistance of the forward biased junction and rb1c is the resistance of the reverse biased junction.
Therefore rb1e can be neglected in the denominator.
To find rce
Applying KCL at the output terminal
ic = i1 + gm vb1e + i2
substituting in the above equation
Since rb1e << rb1c, rb1e + rb1crb1c
To find Cb1C
Cb1C is the junction capacitance of reverse biased collector to base junction. When a PN junction is reverse biased, the width of the depletion layer increases and capacitance decreases. Therefore Cb1C is very low of the order of few pico farads.
To find Cb1e
This is the capacitance of forward biased PN junction. When a PN junction is forward biased, width of the depletion layer decreases and capacitance increases.
Cb1e + Cb1C =
Where fT is called the transition frequency.
fT = hfe fb
fb is called upper cutoff frequency.
A transistor amplifier is operating with a dc condition of (10V,10mA). The operating temperature is 300C. The H-parameters of the transistor are hie =1Ko, hre =2.5X10-4, hfe=100, hoe=25X10-5mho. Calculate hybrid-p parameters given that CC=3PF. Take fT=1MHz.
References and notes
^ R.C. Jaeger and T.N. Blalock (2004). Microelectronic Circuit Design (Second Edition ed.). New York: McGraw-Hill. pp. Section 13.5, esp. Eqs. 13.19. ISBN 0-07-232099-0.
^ R.C. Jaeger and T.N. Blalock. Eq. 5.45 pp. 242 and Eq. 13.25 p. 682. ISBN 0-07-232099-0.
^ R.C. Jaeger and T.N. Blalock. Eq. 4.20 pp. 155 and Eq. 13.74 p. 702. ISBN 0-07-232099-0.
^ a b W. M. C. Sansen (2006). Analog Design Essentials. DordrechtÎ¼: Springer. p. 0124, p. 13. ISBN 0-387-25746-2.