Bipolar_Transistor.md

Introduction to Bipolar Transistor

The Bipolar Transistor (BJT) is made of three doped silicon regions. The outer two regions are doped with the same polarity while the middle region is doped with opposite polarity. Therefore, there is a PNP and NPN bipolar transistor. In this document, we will abstract away the physical behaviors of the BJT and model it with a small and large signal model when it's in the region of operation.

1. Bipolar Transistor Summary

NPN Bipolar Transistor

Forward-Active Region:

$$V_{BE} > 0$$

$$V_{CE} > V_{BE}$$

Large Signal Models

$$I_B = \frac{1}{\beta} I_C$$

$$I_C = I_s e^{\frac{V_{BE}}{V_T}}$$

$$I_E = \frac{\beta + 1}{\beta} I_C$$

$$\beta \in [50, 200]$$

$$V_T = 26 \space mV$$

PNP Bipolar Transistor

Forward-Active Region:

$$V_{EB} > 0$$

$$V_{EC} > V_{EB}$$

Large Signal Model

$$I_B = \frac{1}{\beta} I_C$$

$$I_C = I_s e^{\frac{V_{EB}}{V_T}}$$

$$I_E = \frac{\beta + 1}{\beta} I_C$$

$$\beta \in [50, 200]$$

Small Signal Model: Same for NPN and PNP

$$g_m = \frac{I_C}{V_T}$$

$$r_{\pi} = \frac{\beta}{g_m}$$

$$r_o = \frac{V_A}{I_C}$$

2. NPN Regions of Operation - Forward Active Region

NPN BJT Requirement

• Base-emitter junction is forward-biased ($V_{BE} > 0$)
• Base-collector junction is reverse-biased ($V_{BC} < 0$)

$$V_B - V_C < 0$$

$$\implies V_B < V_C$$

$$\implies V_B - V_E < V_C - V_E$$

$$\implies V_{BE} < V_{CE}$$

In summary,

$$V_{BE} < V_{CE}$$ $$V_{BE} > 0$$

Collector Current

$$I_C \approx I_s e^{\frac{V_{BE}}{V_T}}$$

where $V_T = \frac{kT}{q} \approx 26 \space mV$ at 300 K

which means,

$$V_{BE} \approx V_T \ln\Big(\frac{I_C}{I_s}\Big)$$

Ideally, the collector current is not a function of $V_{CE}$. We want it to be a voltage-dependent current source that depends on $V_{BE}$.

Large Signal Model

Since $V_{BE} &gt; 0$, the base-emitter junction is forward-biased and resembles a diode. Also, the collector current is modeled by the $I_C$ equation, which depends on $V_{BE}$.

Therefore, a diode is placed between the base and emitter and a voltage-controlled current source is placed between the collector and emitter.

The cause and effect relationship: $$V_{BE} \to I_C \to I_B \to I_E$$

The base-emitter voltage generates a collector current, which requires a proportional base current, and they combines to the emitter current.

$$I_C = \beta I_B$$

$$I_E = \frac{\beta + 1}{\beta} I_C$$

$\beta$ is the current gain of the transistor because it shows how much the current gain is amplified. The $\beta$ of npn transistor typically ranges from 50 to 200.

The base-emitter junction is modelled as a diode because the collector current equation is approximately the same as the diode equation.

$$I_B = \frac{I_C}{\beta} = \frac{I_S}{\beta}e^{\frac{V_{BE}}{V_T}}$$

Small-Signal Model

We want to work with the BJT without working with the $I_C = I_se^{\frac{V_{BE}}{V_T}}$ nonlinear equation.

Assume that we have a DC voltage with some ripple: $V_{BE} + \Delta V_{BE}$ going into the BJT's base. The collector current $I_C$, will be:

$$I_C = I_Se^{\frac{V_{BE} + \Delta V_{BE}}{V_T}}$$

$$= I_S e^{\frac{V_{BE}}{V_T}} e^{\frac{\Delta V_{BE}}{V_T}}$$

If $\frac{\Delta V_{BE}}{V_T}$ is small, we could approximate $e^x$ as $1 + x$ via the Taylor's expansion of e.

$$I_C = I_Se^{\frac{V_{BE}}{V_T}} (1 + \frac{\Delta V_{BE}}{V_T})$$

$$= I_{C, DC} + I_{C, DC} \frac{\Delta V_{BE}}{V_T}$$

$$= I_{C, DC} + g_m \Delta V_{BE}$$

$$= I_{C, DC} + I_{C, AC}$$

where $I_{C, DC}$ and $I_{C, AC}$ is the current caused by the DC and AC voltage, and $g_m$ is the small signal transconductance.

Therefore, we can analyze the BJT by separating the $V_{BE}$ into DC (b) and small perturbation (a) components, then add the results.

Now,

$$I_B = \frac{I_C}{\beta} = \frac{I_{C, DC}}{\beta} + \frac{g_m \Delta V_{BE}}{\beta} = I_{B, DC} + I_{B, AC}$$

The resistance between the base and emitter in the small signal analysis is

$$r_{\pi} = \frac{\Delta V_{BE}}{I_{B, AC}} = \frac{\beta}{g_m}$$

The small signal model describes the behavior of the transistor when a small perturbation of voltages (EX: Sinusoidal perturbation) is applied to the transistor nodes.

$g_m$ is the Transconductance, as it measures how well the transistor converts voltage to current. Therefore, it's the slope in the $I_C$ vs $V_{BE}$ graph.

We say the transistor is biased at a collector current $I_C$, meaning that the device carries a bias current of $I_C$ in the absence of signals.

$r_{\pi}$ is the resistance between the base and emitter.

$r_o$ is the early effect resistance between the collector and emitter.

$$r_o = \frac{d V_{CE}}{d I_C} = \frac{V_A}{I_C}$$

Early Effect

The collector current becomes larger than usual and a function of $V_{CE}$

In the large signal model, a correction factor is added to account for $I_C$'s dependence on $V_{CE}$. $V_A$ is called the early voltage.

Now,

$$I_C = \Big(I_s e^{\frac{V_{BE}}{V_T}}\Big)\Big( 1 + \frac{V_{CE}}{V_A}\Big)$$

As $V_A \longrightarrow \infty$, there's no early effect.

Input/Output Impedance

3. NPN Regions of Operation - Soft Saturation

• Base-emitter junction is forward biased ($V_{BE} &gt; 0$)
• Base-collector junction is weakly forward-biased ($-200\space mV \leq V_C - V_B \leq 0$)

4. NPN Regions of Operation - Deep Saturation

• Base-emitter junction is forward biased ($V_{BE} &gt; 0$).
• Base-collector junction is heavily forward-biased ($V_C - V_B \leq -200 \space mV $).

The transistor loses its voltage-controlled current capability and $V_{CE}$ becomes constant.