# The Importance of Introducing the OCTC Method to Undergraduate Students as a Tool for Circuit Analysis and Amplifier Design

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. OCTC Method

#### 2.1.1. OCTC Principles

#### 2.1.2. Applying the Method

- Select the i-th capacitor, ${C}_{i}$, and remove all others. All decoupling and AC-coupling capacitors should be short-circuited.
- Set all independent sources to zero (i.e, short-circuit all independent voltage sources and open-circuit all independent current sources).
- Find the resistance ${R}_{{C}_{i}}$ seen by the ith capacitor. This can be done either by inspection or by replacing ${C}_{i}$ with a test current source ${I}_{x}$, determining the voltage ${V}_{x}$ at its terminals, and calculating ${R}_{{C}_{i}}={V}_{x}/{I}_{x}$, (or, equivalently, by applying a voltage source ${V}_{x}$ and finding the current ${I}_{x}$ drawn from it).
- Repeat steps 1–3 for $i=1,2,\dots ,n$.
- Calculate T using (4).
- Calculate the $-3$dB frequency using (5).

#### 2.2. Useful Analysis Tools

#### 2.2.1. Small-Signal Equivalent Circuit—A Frequent Form

#### 2.2.2. Miller Effect

#### 2.2.3. Knowledge of Basic Amplifier Stages

#### 2.3. Teaching Examples

- Collector–base junction capacitance$${C}_{\mu}=\frac{{C}_{jc0}}{{\left(1+\frac{{V}_{CB}}{{V}_{OC}}\right)}^{m}}$$
- Emitter–base junction capacitance$${C}_{\pi}={C}_{de}+{C}_{je}.$$

#### 2.3.1. Problem 1

- Find (try to answer) directly what impedance every node “can see” (without drawing an AC equivalent circuit).
- For the BJT 2N2222 assume: $\beta =200$, ${C}_{jc0}=8pF$, ${V}_{OC}=0.7V$, $m=0.3$, ${C}_{je0}=25pF$, ${\tau}_{F}=400ps$. Ignore the Early phenomenon (${r}_{o}\to \infty $). Applying the OCTC method estimate the high 3dB frequency ${f}_{-3dB,OCTC}$ and so the bandwidth of the amplifier.
- Using LTspice for circuit simulation, sketch the Bode plot of the amplifier for the frequency range from $1\mathit{Hz}$ to $500\mathit{MHz}$. Determine the high 3dB frequency ${f}_{-3dB}$ and so the bandwidth of the amplifier.

#### 2.3.2. Problem 2

#### 2.3.3. Solutions and Simulations

#### 2.4. Student Misconceptions

- Miller effect and open circuit time constant (OCTC) method. Students have difficulties recognizing and applying the Miller effect for a capacitance.
- Some students believe that the largest capacitor of the (transistors’ capacitances) has the greatest contribution in defining the upper cutoff frequency.
- Coupling and bypass capacitors in OCTC method. “Why don’t coupling and bypass capacitors contribute to the definition of the amplifiers’ upper cutoff frequency?” is the basic question. OCTC method approximately defines the upper cutoff frequency of an amplifier. It is used for high-frequency response of an amplifier. Upper and lower cutoff frequencies that define the bandwidth of an amplifier are often of greater interest than the complete transfer function. Coupling and bypass capacitors determine the lower cutoff frequency, whereas transistor (and stray) capacitances determine the upper cutoff frequency. Coupling and bypass capacitors are relatively large in value and their large impedances at high frequencies can be neglected. Thus coupling and bypass capacitors determine an amplifier’s low-frequency response. Transistor capacitances are relatively small in value and their large impedances at low frequencies can be neglected.
- What do we mean by “capacitors can be ignored”? Ignoring capacitors means that we operate at either a high enough or at a low enough frequency such that capacitors become either open or short circuits, leading to a “resistive” circuit. Note that the circuit is modified by the presence of the capacitors (e.g., elements may be shorted out). Capacitors typically divide into two groups: low-f capacitors (setting ${f}_{L}$) and high-f capacitors (setting ${f}_{H}$). We need to identify low-f and high-f caps. We will use absolute limits of $f=0$ (all capacitors open) and $f=\infty $ (all capacitors short) for this purpose.
- The role of bias circuits. For DC bias ($f=0$) all caps are considered as an open circuit.
- Non-symmetrical differential amplifiers. For a symmetric circuit, differential and common-mode analysis can be performed using “half-circuits”. Using “half-circuits” works only if the circuit is symmetric. Not all difference amplifiers are symmetric. Look at the load carefully. We can still use the “half-circuit” concept if the deviation from perfect symmetry is small. However, we need to solve both half-circuits.

#### 2.5. A Sampling Test

- Question 1: the MOS amplifier circuit displayed in Figure 3 is given. What is the computation relation, according to the circuit elements, of ${R}_{Cgd}$?
- Question 2: the equivalent signal circuit of a two stage BJT amplifier, displayed in Figure 4a is given:
- (a)
- Find ${R}_{C\mu 1}$.
- (b)
- Which of the capacitances ${C}_{\mu 1}$, ${C}_{\pi 1}$, ${C}_{\mu 2}$ and ${C}_{\pi 2}$ do you think has the greatest impact on limiting the amplifier bandwidth and why?
- (c)
- What circuit modification can you propose to reduce the impact of the capacitance you selected in question (c)?

## 3. Results

#### 3.1. Students’ Performance—Homework Problems

#### 3.2. Students’ Performance—Sampling Test

#### 3.3. Students’ Performance—Final Exam

#### 3.4. Quantitative Evaluation

- Q[1.1] The course will increase my affinity to OCTC method.
- Q[1.2] I am interested in the course.
- Q[1.3] The effort imposed by the course is worthwhile because of the abilities and knowledge that I will acquire.
- Q[1.4] The use of simulation tools will increase my affinity to the OCTC method.
- Q[1.5] The use of simulation tools will help me to improve my academic results.

- Q[2.1] The course has increased my affinity to the OCTC method.
- Q[2.2] The course was interesting.
- Q[2.3] The effort imposed by the course was worthwhile because of the abilities and knowledge acquired.
- Q[2.4] The use of simulation tools increased my affinity to OCTC method
- Q[2.5] The use of simulation tools has helped me improve my academic results.

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Academic Year in Greece

## References

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**Figure 4.**AC equivalent circuits of the amplification stages used in the two problems: (

**a**): Problem 1. (

**b**): Problem 2.

Q${}_{\mathit{i}}$ | ${\mathit{C}}_{\mathit{\mu}\mathit{i}}$ (pF) | ${\mathit{C}}_{\mathit{\pi}\mathit{i}}$ (pF) | ${\mathit{R}}_{{\mathit{C}}_{\mathit{\mu}\mathit{i}}}$ ($\mathbf{\Omega}$) | ${\mathit{R}}_{{\mathit{C}}_{\mathit{\pi}\mathit{i}}}$ ($\mathbf{\Omega})$ | ${\mathit{R}}_{{\mathit{C}}_{\mathit{\mu}\mathit{i}}}{\mathit{C}}_{\mathit{\mu}\mathit{i}}$ (ns) | ${\mathit{R}}_{{\mathit{C}}_{\mathit{\pi}\mathit{i}}}{\mathit{C}}_{\mathit{\pi}\mathit{i}}$ (ns) |
---|---|---|---|---|---|---|

1 | $4.25$ | $65.62$ | $109.41\times {10}^{3}$ | $691.95$ | $465.11$ | $45.41$ |

2 | $4.52$ | $114.70$ | $3.88\times {10}^{3}$ | $42.69$ | $17.55$ | $4.90$ |

Q_{i} | ${\mathit{C}}_{\mathit{\mu}\mathit{i}}$ (pF) | ${\mathit{C}}_{\mathit{\pi}\mathit{i}}$ (pF) | ${\mathit{R}}_{{\mathit{C}}_{\mathit{\mu}\mathit{i}}}$ ($\mathbf{\Omega}$) | ${\mathit{R}}_{{\mathit{C}}_{\mathit{\pi}\mathit{i}}}$ ($\mathbf{\Omega})$ | ${\mathit{R}}_{{\mathit{C}}_{\mathit{\mu}\mathit{i}}}{\mathit{C}}_{\mathit{\mu}\mathit{i}}$ (ns) | ${\mathit{R}}_{{\mathit{C}}_{\mathit{\pi}\mathit{i}}}{\mathit{C}}_{\mathit{\pi}\mathit{i}}$ (ns) |
---|---|---|---|---|---|---|

1 | $6.38$ | $65.37$ | $1.41\times {10}^{3}$ | $693.40$ | $9.01$ | $45.33$ |

2 | $4.54$ | $115.09$ | $3.88\times {10}^{3}$ | $42.45$ | $17.60$ | $4.89$ |

3 | $4.62$ | $65.30$ | $3.88\times {10}^{3}$ | $26.02$ | $17.95$ | $1.70$ |

**Table 3.**Open-circuit-time-constant (OCTC) question performance distribution (%) for students taking the final exam, from February of the first academic year to September of the second academic year

OCTC Grade (%)/Exam | Feb. Year 1 | Sep. Year 1 | Feb. Year 2 | Sep. Year 2 |
---|---|---|---|---|

85–100 | $21.6$ | $4.8$ | $27.0$ | $45.4$ |

70–84 | $15.7$ | $14.3$ | $38.1$ | $18.2$ |

55–69 | $13.7$ | $19.1$ | $17.4$ | $18.2$ |

40–54 | $15.7$ | $19.1$ | $7.9$ | $18.2$ |

<40 | $33.3$ | $8.6$ | $9.5$ | $0.0$ |

Mean OCTC grade | $52.4$ | $46.5$ | $72.6$ | $80.4$ |

**Table 4.**Final exam student’s performance distribution (%), from February of the first academic year to September of the second academic year.

Overall Exam Grade (%)/Exam | Feb. Year 1 | Sep. Year 1 | Feb. Year 2 | Sep. Year 2 |
---|---|---|---|---|

85–100 | $2.0$ | $9.5$ | $7.9$ | $27.3$ |

70–84 | $11.8$ | $0.0$ | $15.9$ | $45.5$ |

55–69 | $25.5$ | $9.5$ | $25.4$ | $9.1$ |

40–54 | $23.5$ | $28.6$ | $19.0$ | $9.1$ |

<40 | $37.2$ | $52.4$ | $31.7$ | $9.1$ |

Mean Exam grade | $44.9$ | $41.8$ | $53.7$ | $72.3$ |

Rating/Question | Q[1.1] | Q[1.2] | Q[1.3] | Q[1.4] | Q[1.5] |
---|---|---|---|---|---|

1 | $13.9$ | $18.0$ | $22.9$ | $9.0$ | $8.3$ |

2 | $24.3$ | $16.7$ | $27.8$ | $18.7$ | $8.3$ |

3 | $38.2$ | $31.2$ | $23.6$ | $29.9$ | $25.7$ |

4 | $15.3$ | $28.5$ | $20.1$ | $25.0$ | $27.1$ |

5 | $8.3$ | $5.6$ | $5.6$ | $17.4$ | $30.6$ |

Mean Rating Value | $2.80$ | $2.87$ | $2.58$ | $3.23$ | $3.63$ |

Rating/Question | Q[2.1] | Q[2.2] | Q[2.3] | Q[2.4] | Q[2.5] |
---|---|---|---|---|---|

1 | $4.2$ | $8.3$ | $10.4$ | $2.8$ | $3.5$ |

2 | $20.1$ | $10.4$ | $11.1$ | $7.6$ | $11.1$ |

3 | $38.9$ | $25.0$ | $27.8$ | $10.4$ | $26.4$ |

4 | $21.5$ | $35.4$ | $26.4$ | $43.1$ | $31.2$ |

5 | $15.3$ | $20.9$ | $24.3$ | $36.1$ | $27.8$ |

Mean Rating Value | $3.24$ | $3.50$ | $3.43$ | $4.02$ | $3.69$ |

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## Share and Cite

**MDPI and ACS Style**

Voudoukis, N.; Dimas, C.; Asimakopoulos, K.; Baxevanakis, D.; Papafotis, K.; Oustoglou, K.; Sotiriadis, P.P.
The Importance of Introducing the OCTC Method to Undergraduate Students as a Tool for Circuit Analysis and Amplifier Design. *Technologies* **2020**, *8*, 7.
https://doi.org/10.3390/technologies8010007

**AMA Style**

Voudoukis N, Dimas C, Asimakopoulos K, Baxevanakis D, Papafotis K, Oustoglou K, Sotiriadis PP.
The Importance of Introducing the OCTC Method to Undergraduate Students as a Tool for Circuit Analysis and Amplifier Design. *Technologies*. 2020; 8(1):7.
https://doi.org/10.3390/technologies8010007

**Chicago/Turabian Style**

Voudoukis, Nikolaos, Christos Dimas, Konstantinos Asimakopoulos, Dimitrios Baxevanakis, Konstantinos Papafotis, Konstantinos Oustoglou, and Paul Peter Sotiriadis.
2020. "The Importance of Introducing the OCTC Method to Undergraduate Students as a Tool for Circuit Analysis and Amplifier Design" *Technologies* 8, no. 1: 7.
https://doi.org/10.3390/technologies8010007