Give step-by-step solution with explanation and final answer:Question # 4
Operational Amplifier — Frequency Response
25 Marks
Consider an internally compensated op-amp with a frequency
response as

Question

Give step-by-step solution with explanation and final answer:Question # 4
Operational Amplifier — Frequency Response
25 Marks
Consider an internally compensated op-amp with a frequency
response as depicted below.
141 dB)
tf, = 100 Re
| \. ~20 dB/decade
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Modeling the open-loop gain of this particular op-amp with a STC
transfer function, i.e.,
A
As) = —
1+ Py
determine the values of A, and ws. For the amplifier circuit depicted
below, find an expression for the closed-loop gain, vo/vi, as a function
of s, assuming that the op-amp operates ideally in every sense except
for the open-loop gain, A(s), which is as described above. Identify the
corresponding closed-loop dc gain and the closed-loop 3-dB corner
frequency from this closed-loop gain expression. For the specific case
of R1=1-kQ and R2=100-kQ, plot the resultant Bode plot indicating the
key frequencies.
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VIPSolutions · Accepted Answer

Let's break down the solution step by step:

---

# **Step 1: Extract Open-Loop Gain Parameters ($A_$, $\omega_b$)**

The open-loop transfer function is:

$$
A(s) = \frac{A_}{1 + \frac{s}{\omega_b}}
$$

From the Bode plot:
- $A_$ (DC gain) is the low-frequency gain (at $f = $):
  - From the plot, $ |A| = 100 $ dB at low frequencies.
  - $A_ (\text{linear}) = 10^{100/20} = 10^5$
- The break frequency ($f_b$) where the gain starts to fall at $-20~\text{dB/decade}$ is at $10^2~\text{Hz}$.
  - $\omega_b = 2\pi f_b = 2\pi \times 10^2 = 628.3~\text{rad/s}$

---

# **Step 2: Closed-Loop Gain Expression ($v_o/v_i$)**

For a non-inverting amplifier:

$$
\text{Closed-loop gain (ideal)} = 1 + \frac{R_2}{R_1}
$$

With finite op-amp gain $A(s)$:

$$
\frac{v_o}{v_i} = \frac{A(s)}{1 + A(s)\beta}
$$
where $\beta = \frac{R_1}{R_1 + R_2}$

Substituting $A(s)$:

$$
\frac{v_o}{v_i} = \frac{\frac{A_}{1 + \frac{s}{\omega_b}}}{1 + \frac{A_}{1 + \frac{s}{\omega_b}} \beta}
$$

Multiply numerator and denominator by $1 + \frac{s}{\omega_b}$:

$$
\frac{v_o}{v_i} = \frac{A_}{1 + A_\beta + \frac{s}{\omega_b}}
$$

---

# **Step 3: DC Closed-Loop Gain**

As $s \to $:

$$
\left(\frac{v_o}{v_i}\right)_{DC} = \frac{A_}{1 + A_\beta}
$$

For large $A_$, $\frac{A_}{1 + A_\beta} \approx \frac{1}{\beta} = 1 + \frac{R_2}{R_1}$.

---

# **Step 4: Closed-Loop 3-dB Corner Frequency**

The denominator is $1 + A_\beta + \frac{s}{\omega_b}$:

$$
\Rightarrow f_{cl} = \frac{\omega_{b,cl}}{2\pi}
$$

where

$$
\omega_{b,cl} = \omega_b (1 + A_\beta)
$$

---

# **Step 5: Substitute Values (Given $R_1 = 1~k\Omega, R_2 = 100~k\Omega$)**

$$
\beta = \frac{R_1}{R_1 + R_2} = \frac{1}{1 + 100} = \frac{1}{101}
$$

$$
A_\beta = 10^5 \times \frac{1}{101} \approx 990
$$

$$
\omega_{b,cl} = 628.3 \times (1 + 990) \approx 628.3 \times 991 \approx 622,259~\text{rad/s}
$$
$$
f_{cl} = \frac{622,259}{2\pi} \approx 99,085~\text{Hz} \approx 99~\text{kHz}
$$

DC closed-loop gain:

$$
\frac{A_}{1 + A_\beta} = \frac{10^5}{1 + 990} \approx 101
$$
or in dB: $20\log_{10}(101) \approx 40~\text{dB}$

---

# **Step 6: Bode Plot**

- **Low-frequency gain:** $40~\text{dB}$ (closed-loop)
- **Low-frequency gain (open-loop):** $100~\text{dB}$
- **Open-loop 3-dB frequency:** $f_b = 100~\text{Hz}$
- **Closed-loop 3-dB frequency:** $f_{cl} \approx 99~\text{kHz}$

## **Key Points for Bode Plot:**
- $100~\text{dB}$ flat until $100~\text{Hz}$ (open-loop)
- Then, slope of $-20~\text{dB/decade}$ (open-loop)
- Closed-loop gain: flat at $40~\text{dB}$ until $99~\text{kHz}$
- At $99~\text{kHz}$, slope drops $-20~\text{dB/decade}$ (closed-loop)

---

## **Summary Table**

| Parameter        | Value                      |
|------------------|---------------------------|
| $A_$          | $10^5$ ($100~\text{dB}$) |
| $\omega_b$     | $628.3~\text{rad/s}$ ($100~\text{Hz}$)   |
| $\beta$        | $.0099$                |
| Closed-loop DC gain | $101$ ($40~\text{dB}$) |
| $f_{cl}$ (3-dB) | $99~\text{kHz}$         |

---

## **Final Answers**

**1. $A_ = 10^5$, $\omega_b = 628.3~\text{rad/s}$ ($100~\text{Hz}$)**

**2. Closed-loop gain:**
$$
\frac{v_o}{v_i} = \frac{A_}{1 + A_\beta + \frac{s}{\omega_b}}
$$

**3. DC closed-loop gain:** $101$ ($40~\text{dB}$)

**4. Closed-loop 3-dB frequency:** $99~\text{kHz}$

**5. Bode plot:**
- Flat at $40~\text{dB}$ up to $99~\text{kHz}$, then rolls off at $-20~\text{dB/decade}$.

---

### **Bode Plot (Sketch)**

![Bode Plot](https://i.imgur.com/ItLzv8F.png)

*Alt text: Bode plot showing open-loop gain at 100 dB up to 100 Hz, then -20 dB/decade; closed-loop gain at 40 dB up to 99 kHz, then -20 dB/decade.*

---

Let me know if you need a hand-drawn or more detailed Bode plot!

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