AC Model of BJT#
Author: Reza Farasati
Contact: rezafarasati2004@gmail.com
Introduction#
In electronic circuit analysis, modeling components for specific operating conditions is critical. For Bipolar Junction Transistors (BJTs), AC analysis requires a model that incorporates the transistor’s behavior under small signal variations. Just as DC models simplify the analysis of biasing circuits, AC models ease the study of signal amplification.
The AC BJT model is based on its DC counterpart, with some adjustments to account for the dynamic resistance of the base-emitter junction. This resistance plays a crucial role in determining the performance and stability of small-signal amplifiers.
find AC model of the cercuit below:#
as shown in the cercuit, we have: $\( \begin{aligned} i_{B}=I_{B}+i_{b} \\ i_{E}=I_{E}+i_{e} \\ i_{C}=I_{C}+i_{C} \\ \end{aligned} \)$
which
Similarly, $\( V_{B E}, V_{C E} \text{ are DC Voltages}. \)\( \)\( \mathrm{V}_{\mathrm{be}}, \mathrm{V}_{\mathrm{ce}} \text{ are AC Voltages.} \)\( \)\( \mathrm{V}_{\mathrm{BE}}, \mathrm{V}_{\mathrm{CE}} \text{ are DC + AC Voltages.} \)$
Large Signal BJT Model#
from the Shockley equation, we know that:
Obviously, if we apply \( sin \) input voltage \( V_{BE} \), we will have \( exponential \) output current \( i_C \). so the circuit is not linear.
BJT with small ac input signal#
Small ac signal refers to the input signal ( \(\mathrm{v}_{\mathrm{be}}\) ) whose magnitude is much small than thermal voltage \(\left(\mathrm{V}_{\mathrm{T}}\right)\) i.e. \(\mathrm{v}_{\mathrm{be}} \ll \mathrm{V}_{\mathrm{T}}\)
Magnitude of the ac signal applied for amplification must be small so that
the transistor operates in the linear region for the whole cycle of input (called as a linear amplifier)
the transistor is never driven into saturation or cut-off region
On the other hand, if the input signal is too large. The fluctuations along the load line will drive the transistor into either saturation or cut off. This clips the peaks of the input and the amplifier is no longer linear.
small Signal Analysis#
If we have an \(ac+dc\) input signal, the total \(\mathrm{v}_{\mathrm{BE}}\) becomes $\( v_{B E}=v_{b e}+V_{B E} \)$
And the collector current becomes:
Using the Taylor series expansion: $\( e^x \simeq 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \quad \)\( \)\( \text{for } (x \ll 1) \quad \Rightarrow \quad e^x \simeq 1 + x \)$
If \( (v_{be} \ll 1) \) (small signal approximation):
Let \( ( I_{C} = I_S e^{\frac{V_{BE}}{V_T}} ) \), then:
Now this is a linear expression, because for \(sin\) input voltage \( v_{be} \) we will get \(sin\) output current \( i_C \).
\( g_m \) is called the small signal transconductance: $\( g_{m}=\frac{i_{c}}{v_{b e}}=\frac{I_{C}}{V_{T}} \)\( It represents the slope of \)\mathrm{i}{\mathrm{C}}-\mathrm{v}{\mathrm{BE}}$ curve at the Q point.
Small-signal Transconductance#
The small signal approximation implies that signal is so small that operation is restricted to an almost linear segment of the \(\mathrm{i}_{\mathrm{C}}-\mathrm{v}_{\mathrm{BE}}\) exponential curve. $\( g_{m}=\left(\frac{i_{c}}{v_{b e}}\right)_{v_{b e}=0}=\left(\frac{\partial i_{C}}{\partial v_{B E}}\right)_{i_{C}=I_{C}} \)\( \)\( g_{m}=\frac{i_{c}}{v_{b e}}=\frac{I_{C}}{V_{T}} \)\( \)\( i_{c}=g_{m} v_{b e} \)$
The small signal analysis suggests that for a small signal, transistor behaves as a voltage controlled current source. The input port of the controlled current source is between base and emitter and output port is in between collector and emitter.
Small-signal Analysis: Current and Input Resistance#
The total base current:
$\(
\mathbf{i}_{\mathbf{B}} = \mathbf{I}_{\mathbf{B}} + \mathbf{i}_{\mathbf{b}}
\)$
So:
$\(
\quad i_{B} = \frac{i_{c}}{\beta} = \frac{I_{C}}{\beta} + \frac{1}{\beta} \frac{I_{C}}{V_{T}} v_{b c}
\)$
Signal component of base current:
$\(
\quad i_{b} = \frac{1}{\beta} \frac{I_{C}}{V_{T}} v_{b e} = \frac{g_{m}}{\beta} v_{b e}
\)$
\(\mathbf{r}_{\pi}\) is the small-signal input resistance between base and emitter, looking into the base:
$\(
\mathrm{r}_{\pi} = \frac{v_{b e}}{i_{b}} = \frac{\beta}{g_{m}} = \frac{V_{T}}{I_{B}}
\)$
Knowing that the total emitter current:
$\(
\mathbf{i}_{\mathrm{E}} = \mathrm{I}_{\mathrm{E}} + \mathbf{i}_{\mathrm{e}}
\)$
So:
$\(
\quad i_{E} = \frac{i_{C}}{\alpha} = \frac{I_{C}}{\alpha} + \frac{i_{c}}{\alpha}
\)$
And:
$\(
i_{e} = \frac{i_{c}}{\alpha} = \frac{I_{C}}{\alpha V_{T}} v_{b e} = \frac{I_{E}}{V_{T}} v_{b e}
\)$
\(r_{e}\) is the small-signal input resistance between base and emitter, looking into the emitter:
$\(
\quad \mathrm{r}_{e} = \frac{v_{b e}}{i_{e}} = \frac{\alpha}{g_{m}} = \frac{V_{T}}{I_{E}}
\)$
Therefore:
$\(
v_{b e} = i_{b} r_{\pi} = i_{e} r_{e}
\)$
And:
$\(
\quad r_{\pi} = (\beta + 1) r_{e}
\)$
Hybrid-\(\pi\) small signal model of BJT#
This model represents that transistor as a voltage controlled current source with control voltage \(\mathrm{v}_{\mathrm{be}}\) and include the input resistance looking into the base. $\( \mathrm{r}_{\pi}=\frac{v_{b e}}{i_{b}}=\frac{\beta}{g_{m}}=\frac{V_{T}}{I_{B}} \)\( \)\( \mathrm{r}_{e}=\frac{v_{b e}}{i_{e}}=\frac{\alpha}{g_{m}}=\frac{V_{T}}{I_{E}} \)\( \)\( g_{m}=\frac{i_{c}}{v_{b e}}=\frac{I_{C}}{V_{T}} \)$
Small signal T model of BJT#
This model represents that transistor as a voltage controlled current source with control voltage \(\mathrm{v}_{\mathrm{be}}\) and include the input resistance looking into the emitter.
Conclusion#
In coclusion, the algorithm of BJT circuit analysis using small signal model is:
BJT Circuit Analysis using Small Signal Model:#
Determine the DC operating point of the BJT and in particular, the collector current \(\mathrm{I}_{\mathrm{C}}\)
Calculate small-signal model parameters \(g_{m}, r_{π},\) and \(r_{e}\) for this DC operating point
Eliminate DC sources:
Replace DC voltage sources with short circuits
Replace DC current sources with open circuits
Replacing all capacitors by a short circuit equivalent and remove all elements bypassed by the short circuit equivalents.
Replace BJT with an equivalent small-signal model
Analyze the resulting circuit to determine the required quantities e.g. voltage gain, input resistance…etc.