hybrid

To answer your first and most important question: Millman’s Electronic Devices and Circuits is absolutely the better book for practicing the hybrid model of BJTs.

While Boylestad is the undisputed “bible” for the $r_{e}$ model, your examiners pull their hybrid model terminology, derivation proofs, and multistage cascade numericals directly from Millman’s methodology. If you look at the PYQs from 2015 to 2023, the exact wording for cascaded hybrid problems and proofs perfectly matches Millman’s Chapters 9 and 10.

Here is your targeted practice list from Millman’s textbook, starting with a few single-stage warm-ups to grasp the concept, leading right into the exact PYQ clones for your exam.

1. Single-Stage Practice (To Grasp the Concept)

The PYQs do not ask for single-stage numericals, but doing a few will help you understand how the $h$ -parameters plug into the formulas before you start chaining them together.

Derivation Practice: Go to Millman Section 9.10. Practice writing out the exact algebraic proofs for $A_{i}$ , $Z_{i}$ , $A_{v}$ , and $Y_{o}$ . Pay special attention to deriving the input impedance ( $Z_{i}$ )—this exact mathematical proof answers the highly repeated PYQ: “The input impedance is a function of load impedance – Justify the statement”.
Numerical Warm-up (Example 9.4): This is a solved example in Millman that asks you to find the exact current gain, voltage gain, and input/output impedances for a basic Common-Emitter amplifier using the $h$ -parameters.
Numerical Warm-up (Problem 9.8): If you want to try one on your own, go to the end of Chapter 9 and solve Problem 9.8. It asks you to calculate $A_{i}, A_{v}, R_{i},$ and $R_{o}$ for a single-stage Common-Collector (CC) configuration using the exact parameters.

2. Multi-Stage / Cascaded Practice (The Exact PYQ Clones)

Almost every year (2015, 2016, 2017, 2019, 2021, and 2023), the examiners ask a 10-15 mark numerical requiring you to find the overall voltage gain, current gain, and impedances of a two-stage cascade network using the “simplified hybrid model”.

Do not practice these using exact formulas; use the simplified formulas (where $h_{re}$ and $h_{oe}$ are dropped) as instructed. Practice these specific problems from Millman Chapter 10:

The Blueprint Example (Example 10.2): This is the ultimate PYQ clone. It is a solved example of a two-stage CE-CE cascade amplifier. It asks you to compute $A_{i}, A_{v}, R_{i},$ and $R_{o}$ using the approximate (simplified) formulas. Work through this step-by-step. Remember the golden rule of cascades: Start at stage 2 and work backward, because the input impedance of stage 2 acts as the load for stage 1.
CE-CC Cascade Example: Go to the solved example summarizing Table 10.1 (Section 10.1). This walks through a cascade where the first stage is CE and the second stage is CC. This exactly matches the 2018 and 2021 PYQs which specifically asked for a CE-CC cascade numerical.
End-of-Chapter PYQ Clones: If you want to test yourself with unguided problems, go to the end of Millman Chapter 10 and do:
- Problem 10.1 (Part b): Computes input/output impedances and overall gains for a two-stage cascade “using the approximate formulas in Table 10.2”.
- Problem 10.8: “For the two-stage cascade shown, find $A_{i}, A_{v}, R_{i},$ and $R_{o}$ .“.
- Problem 10.10: Same as above but with slightly different resistor values.

3. The Darlington Pair (Highly Tested)

Derivation Practice: Go to Millman Section 10.10. This section breaks down the Darlington connection as two cascaded CC stages. Practice deriving the composite current gain ( $A_{i} \approx h_{f e 1} h_{f e 2}$ ). This exact derivation appeared in the 2015, 2017, and 2018 exams.

Your Quick Action Plan: Review the single-stage derivation in Section 9.10, do Example 10.2 to lock down the simplified cascade math, and memorize the Darlington current gain derivation in Section 10.10. Once you do those three things in Millman, you are fully prepared for the hybrid model questions on your exam!

formuales

**Based on Table 10.2 from Millman’s textbook, you only need to memorize the formulas from the Common-Emitter (CE) and Common-Collector (CC) columns to solve the cascaded amplifier numericals in your exams.

Before applying these, you must always ensure the condition $h_{oe} (R_{e} + R_{L}) \leq 0.1$ is met (which simply becomes $h_{oe} R_{L} \leq 0.1$ when the emitter resistor is grounded or absent).

Here are the exact approximate (simplified) formulas you must memorize:

1. Common-Emitter (CE) Formulas

(Use these for standard CE-CE cascades or the first stage of a CE-CC cascade)

Current Gain ( $A_{I}$ ): $- h_{f e}$
Input Resistance ( $R_{i}$ ): $h_{i e}$
Voltage Gain ( $A_{V}$ ): $- \frac{h _{f e} R _{L}}{h _{i e}}$

2. Common-Collector (CC / Emitter-Follower) Formulas

(Use these specifically for the final stage in your CE-CC cascade PYQs)

Current Gain ( $A_{I}$ ): $1 + h_{f e}$
Input Resistance ( $R_{i}$ ): $h_{i e} + (1 + h_{f e}) R_{L}$
Voltage Gain ( $A_{V}$ ): $1 - \frac{h _{i e}}{R _{i}}$
Output Resistance ( $R_{o}$ ): $\frac{R _{s} + h _{i e}}{1 + h _{f e}}$ (where $R_{s}$ is the source resistance or the output resistance of the previous stage)
Loaded Output Resistance ( $R_{o}^{'}$ ): $R_{o} ∥ R_{L}$

What you can skip from Table 10.2: The table also includes columns for “CE with $R_{e}$ ” and “CB” (Common Base). Based on your PYQs, the examiners do not ask for cascaded numericals using these specific configurations, so you can safely skip memorizing those two columns and focus your energy entirely on the CE and CC equations above._

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processs

To solve a double-stage (cascaded) hybrid parameters problem, you must rely on the “simplified hybrid model” and analyze the circuit in reverse.

Here are the core concepts and the exact step-by-step process you need to remember based on Millman’s methodology:

1. The Core Concepts to Remember

The Simplified Hybrid Approximation: Whenever the load resistance $R_{L}$ is small enough to satisfy the condition $h_{oe} R_{L} \leq 0.1$ , you can safely ignore the reverse voltage gain ( $h_{re}$ ) and output admittance ( $h_{oe}$ ) parameters. This simplifies your calculations significantly, requiring you to use only $h_{i e}$ and $h_{f e}$ for a Common-Emitter (CE) stage.
The “Loading” Effect: In a cascaded amplifier, the collector resistance of one stage is shunted by the input impedance of the next stage. Therefore, the input impedance of the second stage acts as the effective load for the first stage.
The Golden Rule (Backward Analysis): Because of the loading effect, you cannot calculate the first stage without knowing the parameters of the second stage. You must always start your calculations at the final stage (Stage 2) and work backward toward the input stage (Stage 1).

2. The Step-by-Step Solving Process

Assume you have a standard two-stage Common-Emitter (CE-CE) cascade. Follow these steps exactly:

Step 1: Analyze Stage 2 (The Final Stage) Start by determining the parameters for the second transistor, using its specific collector resistor $R_{C 2}$ as its load ( $R_{L 2}$ ).

Input Impedance ( $R_{i 2}$ ): $R_{i 2} = h_{i e}$ .
Current Gain ( $A_{I 2}$ ): $A_{I 2} = - h_{f e}$ .
Loaded Voltage Gain ( $A_{V 2}$ ): $A_{V 2} = \frac{- h _{f e} R _{L 2}}{h _{i e}}$ .

Step 2: Calculate the Effective Load of Stage 1 ( $R_{L 1}$ ) This is the most critical step. The load for Stage 1 ( $R_{L 1}$ ) is the parallel combination of its own collector resistor ( $R_{C 1}$ ), the biasing resistors of Stage 2 (if present), and the input impedance of Stage 2 ( $R_{i 2}$ ) that you just calculated.

Formula: $R_{L 1} = R_{C 1} ∥ R_{biasing of Stage 2} ∥ R_{i 2}$ .

Step 3: Analyze Stage 1 (The Input Stage) Now, analyze the first transistor using the effective load ( $R_{L 1}$ ) you found in Step 2.

Input Impedance ( $R_{i 1}$ ): $R_{i 1} = h_{i e}$ .
Current Gain ( $A_{I 1}$ ): $A_{I 1} = - h_{f e}$ .
Loaded Voltage Gain ( $A_{V 1}$ ): $A_{V 1} = \frac{- h _{f e} R _{L 1}}{h _{i e}}$ .

Step 4: Calculate the Overall System Parameters Once you have the individual stage parameters, combine them for the final system answers:

Overall Voltage Gain ( $A_{V}$ ): The total voltage gain of a cascaded system is simply the product of the individual loaded voltage gains: $A_{V} = A_{V 1} \times A_{V 2}$ .
Overall Input Impedance ( $Z_{i}$ ): This is the parallel combination of the first stage’s input impedance ( $R_{i 1}$ ) and any biasing resistors present at the very beginning of the circuit.
Overall Current Gain ( $A_{I}$ ): You can calculate the total current gain easily using the overall voltage gain, total input impedance, and the final load resistance: $A_{I} = - A_{V} \times (\frac{Z _{i}}{R _{L 2}})$ .

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