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BJT Modeling Concepts

• Questions:

  • What is meant by BJT modeling? (2023, 2022, 2018, 2015)
  • What is the necessity of BJT modelling? (2017)

• Answer: A model is a combination of circuit elements, properly chosen, that best approximates the actual behavior of a semiconductor device under specific operating conditions. Once the ac equivalent circuit is determined, the schematic symbol for the device can be replaced by this equivalent model, permitting the use of basic methods of circuit analysis (like superposition and Thévenin’s theorem) to determine desired network quantities like voltage gain, current gain, and impedances.

• Questions:

  • Write down the procedure of obtaining ac equivalent circuit of a BJT. (2023, 2022, 2019, 2018, 2016)

• Answer: The ac equivalent of a transistor network is obtained by following these steps:

  1. Setting all dc sources to zero and replacing them by a short-circuit connection to ground.
  2. Replacing all capacitors by a short-circuit equivalent.
  3. Removing all elements bypassed by the short-circuit equivalents introduced by the first two steps.
  4. Redrawing the network in a more convenient and logical form.

• Questions:

  • Expound on the differences between small signal analysis and large signal analysis. / What are the differences between small signal analysis and large signal analysis? (2023, 2019, 2016)

• Answer: The magnitude of the input signal dictates which analysis is appropriate. Small-signal analysis is used when the applied signal is relatively small, meaning the transistor operates in a tight, linear region of its device characteristics. In contrast, large-signal analysis (often applied to power amplifiers) is used when the input signal is large enough to cause substantial oscillations that vary widely around the dc bias point, pushing the device closer to its maximum limits like the saturation or cutoff regions.

• Questions:

  • Compare $r_e$ model and hybrid model mentioning their advantages and disadvantages. / Write down the advantages and disadvantages of $r_e$ model. (2017, 2015)

• Answer:

  • The $r_e$ model utilizes the actual physical operating conditions of the transistor, using the diode equivalent resistance $r_e=26\text{ mV}/I_E$ for the base-emitter junction. Its main advantage is that it is directly sensitive to the actual dc operating conditions of the network. A disadvantage is that $r_e$ is normally not provided on a specification sheet, meaning the dc conditions must be calculated first.
  • The hybrid model uses parameters defined by a general two-port system ($h_{ie}$, $h_{fe}$, $h_{re}$, $h_{oe}$). Its primary advantage is that most manufacturer specification sheets include a list of these hybrid parameters, making them readily available. However, a major disadvantage is that these provided values are only accurate for a particular set of specified dc operating conditions and are highly sensitive to changes in collector current and temperature.

$r_e$ Model Analysis

• Questions:

  • Define $r_e$-modeling. Draw the $r_e$-models for common emitter and common collector configuration. (2016)

• Answer: $r_e$ modeling replaces the BJT in the ac domain with a linear equivalent circuit primarily consisting of a controlled current source (to represent amplification) and a forward-biased junction resistance ($r_e$) determined by the dc emitter current ($I_E$).

  • Common Emitter Model: Features a resistor $\beta r_e$ between the base and emitter, and a controlled current source $\beta I_b$ pointing from collector to emitter (often with an output resistance $r_o$ in parallel).
  • Common Collector Model: The equivalent circuit is normally the exact same model applied to the common-emitter configuration.

• Questions:

  • Derive the expression for (i) $Z_i$, (ii) $Z_o$, (iii) $A_v$, and (iv) $A_i$ of common emitter fixed biased configuration in forms of $r_e$. (2022, 2021, 2019, 2018, 2017, 2016)

• Answer: For a fixed-bias common-emitter configuration with base resistor $R_B$ and collector resistor $R_C$:

  • (i) $Z_i$: The input impedance looks into the parallel combination of the bias resistor and the transistor’s input resistance. $Z_i=R_B\parallel \beta r_e$.
  • (ii) $Z_o$: The output impedance is the parallel combination of the collector resistor and the transistor’s output resistance. If $V_i=0$, $I_b=0$, the controlled source is open, leaving $Z_o=R_C\parallel r_o$. If $r_o\ge 10R_C$, $Z_o\approx R_C$.
  • (iii) $A_v$: Output voltage is $V_o=-(\beta I_b)(R_C\parallel r_o)$, and input voltage is $V_i=I_b\beta r_e$. Dividing the two yields $A_v=-(R_C\parallel r_o)/r_e$. If $r_o\ge 10R_C$, this simplifies to $A_v=-R_C/r_e$.
  • (iv) $A_i$: Using the two-port relation $A_i=-A_v(Z_i/R_C)$ or observing the base and collector currents directly, the current gain is approximately $A_i\approx \beta$.

• Questions:

  • What is the significance of emitter resistor in emitter bias configuration? / Elucidate the significance of emitter resistor… and explain the effect of a bypass capacitor on the performance of an amplifier. (2023, 2022, 2020)

• Answer:

  • Significance of Emitter Resistor ($R_E$): In the ac domain, an unbypassed emitter resistor acts as negative feedback. It dramatically increases the input impedance of the transistor (reflecting it to the base as $Z_b\approx \beta (r_e+R_E)$). However, this trades off with a significantly reduced voltage gain (approximated as $A_v\approx -R_C/R_E$). It also makes the amplifier’s gain highly independent of variations in transistor $\beta$.
  • Effect of Bypass Capacitor ($C_E$): A bypass capacitor placed across $R_E$ acts as a short-circuit equivalent to ac signals, effectively “shorting out” the emitter resistor for ac analysis. This restores the much higher voltage gain of a fixed-bias configuration ($A_v=-R_C/r_e$) but drops the input impedance back to a lower value.

• Questions:

  • Write the merits of emitter follower configuration. Using $r_e$ modeling concept, derive the expression for input impedance ($Z_i$) and output impedance ($Z_o$) for emitter follower configuration. (2015)

• Answer:

  • Merits: The emitter-follower (common-collector) configuration has a relatively high input impedance, low output impedance, and a voltage gain of approximately 1 (it is a noninverting buffer circuit).
  • $Z_i$ Derivation: The input voltage is $V_i=I_b\beta r_e+I_e{R}_E$. Since $I_e=(\beta +1)I_b$, the impedance looking into the base is $Z_b=V_i/I_b=\beta r_e+(\beta +1)R_E\approx \beta (r_e+R_E)$. The total input impedance is $R_B$ in parallel with $Z_b$, yielding $Z_i=R_B\parallel Z_b$.
  • $Z_o$ Derivation: Setting $V_s$ to 0, analyzing the output looking back into the emitter terminal yields an output impedance approximately equal to the internal dynamic emitter resistance, $Z_o\approx r_e$.

Hybrid Model Analysis

• Questions:

  • Define hybrid model and hybrid parameters. Draw the hybrid models of three different configurations of transistor. (2018)
  • Define h-parameters. How can you determine h-parameters from transistor characteristic curves? (2020, 2015)

• Answer:

  • Definition: The hybrid model is an ac equivalent circuit based on the general two-port system parameters. The “hybrid” (h) parameters are a mixture of units (ohms, siemens, and unitless ratios) that relate the input voltage and current to the output voltage and current ($V_i,I_i,V_o,I_o$).
  • Determination: The h-parameters are determined graphically by taking the ratio of small changes ($\Delta$) around the specific quiescent (Q) point of operation on the transistor characteristic curves, holding either $V_{CE}$ or $I_B$ constant. For example, $h_{ie}=\Delta v_{be}/\Delta i_b$ (with $V_{CE}$ constant), and $h_{fe}=\Delta i_c/\Delta i_b$ (with $V_{CE}$ constant).
  • Drawing Models: The general hybrid equivalent model features an input resistance ($h_i$), a reverse-voltage controlled source ($h_r{V}_o$), a forward-current controlled source ($h_f{I}_i$), and an output conductance ($h_o$). The exact models for the three configurations (CE, CB, CC) look structurally identical, simply substituting the appropriate variables (e.g., $V_{be},I_b$ for CE and $V_{eb},I_e$ for CB) and adding the correct subscript letter to the parameters ($e,b,\text{ or }c$).

• Questions:

  • “The input impedance is a function of load impedance” – Justify the statement using the hybrid modeling concept. (2022, 2021, 2017)

• Answer: In the complete hybrid equivalent model, a controlled voltage source is present on the input side defined by $h_r{V}_o$ (where $V_o$ is the output voltage). Applying Kirchhoff’s voltage law to the input circuit yields $V_i=I_i{h}_i+h_r{V}_o$. Because the output voltage is dependent on the load ($V_o=-I_o{R}_L$), substituting this into the input equation links the input voltage/current relationship directly to $R_L$. Thus, the reverse voltage feedback mathematically ensures that $Z_i$ is a function of $R_L$.

• Questions:

  • Derive the expression for $A_i,Z_i,A_v$, and $Y_o$ for a basic transistor amplifier circuits using h-parameters model. (2015)

• Answer: Using the complete hybrid model:

  • $A_i$ (Current Gain): At the output, $I_o=h_f{I}_i+h_o{V}_o$. Substituting $V_o=-I_o{R}_L$ yields $I_o=h_f{I}_i-h_o{I}_o{R}_L$. Factoring gives $A_i=I_o/I_i=h_f/(1+h_o{R}_L)$.
  • $Z_i$ (Input Impedance): From the input loop, $V_i=I_i{h}_i+h_r{V}_o$. Substituting $V_o=-I_o{R}_L$ and $I_o=A_i{I}_i$ yields $Z_i=V_i/I_i=h_i-h_r{A}_i{R}_L$.
  • $A_v$ (Voltage Gain): Using $V_o=-I_o{R}_L$ and $I_o=A_i{I}_i$, we get $V_o=-A_i{I}_i{R}_L$. Since $I_i=V_i/Z_i$, substitution yields $A_v=V_o/V_i=-A_i{R}_L/Z_i$.

Darlington Pair & Multistage

• Questions:
  • Discuss the working principle of Darlington pair circuit using necessary diagram. (2023, 2019)
  • What is Darlington pair circuit? Derive the expression for current gain ($A_i$) of Darlington pair using h-parameters and comments on the obtained expression. (2018, 2017, 2015)
• Answer:
  • Working Principle: A Darlington pair consists of two interconnected transistors where the emitter of the first transistor is tied directly to the base of the second. This cascades the amplification of both stages because the amplified emitter current of the first transistor serves as the base current for the second, resulting in a system that acts as a single transistor with an exceptionally high current gain.
  • Derivation of Current Gain ($A_i$): Ignoring output impedances, the current gain is derived by tracing the input to output. The input current to the second transistor is $I_{b2}=I_{e1}=({\beta }_1+1)I_{b1}$. The total output emitter current is $I_o=I_{e2}=({\beta }_2+1)I_{b2}$. Substituting $I_{b2}$ yields $I_o=({\beta }_2+1)({\beta }_1+1)I_{b1}$. Therefore, the current gain is roughly the product of the individual gains: $A_i=I_o/I_i\approx {\beta }_1{\beta }_2$ (or $h_{fe1}{h}_{fe2}$).
  • Comments on Expression: The resulting expression demonstrates that the configuration acts like a single equivalent transistor whose current gain is the multiplication of the individual beta values, easily allowing for current gains in the thousands (e.g., $10,000$).