Unit 4 - Notes

ECE221 7 min read

Unit 4: Operational Amplifier Configuration

1. The Ideal Op-Amp

An Operational Amplifier (Op-Amp) is a high-gain, direct-coupled, differential linear amplifier designed to amplify DC and AC signals. The "ideal" op-amp is a theoretical construct used to simplify circuit analysis.

Characteristics of an Ideal Op-Amp:

Infinite Open-Loop Voltage Gain ( $A_{OL} = \infty$ ): It will amplify any tiny differential input voltage to an infinite output voltage (practically limited by power supply voltages).
Infinite Input Impedance ( $R_{in} = \infty$ ): It draws absolutely zero current from the input signal sources ( $I_{in} = 0$ ).
Zero Output Impedance ( $R_{out} = 0$ ): It can supply any amount of current to the load without any drop in output voltage.
Infinite Bandwidth ( $BW = \infty$ ): It amplifies all frequencies equally, from DC (0 Hz) to infinity.
Infinite Common-Mode Rejection Ratio ( $CMRR = \infty$ ): It completely rejects any signal common to both input terminals.
Infinite Slew Rate ( $SR = \infty$ ): The output voltage can change instantaneously in response to changes in the input.
Zero Offset Voltage: If the differential input voltage is zero, the output voltage is exactly zero.

2. Equivalent Circuit of an Op-Amp

The equivalent circuit provides a practical model to analyze op-amp behavior under various conditions.

Input Stage: Modeled as an input resistance ( $R_i$ ) connected between the non-inverting terminal ( $v_+$ ) and the inverting terminal ( $v_-$ ). The differential input voltage is $v_d = v_+ - v_-$ .
Output Stage: Modeled as a dependent voltage source in series with an output resistance ( $R_o$ ).
Dependent Source: The magnitude of the dependent source is $A_{OL} \times v_d$ , where $A_{OL}$ is the open-loop gain.

Mathematical Representation:
$V_{out} = A_{OL} (v_+ - v_-) - I_{out} R_o$
In an ideal scenario, $R_i = \infty$ , $R_o = 0$ , making $V_{out} = A_{OL} v_d$ .

3. Ideal Voltage Transfer Curve

The voltage transfer curve is a graph plotting the output voltage ( $V_{out}$ ) against the differential input voltage ( $v_d$ ).

Linear Region: A very narrow steep line passing through the origin. Here, $V_{out} = A_{OL} \times v_d$ . Because $A_{OL}$ is extremely large (e.g., $10^5$ ), the linear region exists for only microvolts of input.
Saturation Region: Once reaches the power supply limits (typically denoted as and , which are slightly less than and ), the output cannot increase further. It clips or saturates.
- If $v_d > 0$ (even slightly), $V_{out} \rightarrow +V_{sat}$
- If $v_d < 0$ (even slightly), $V_{out} \rightarrow -V_{sat}$

4. Open Loop Op-Amp Configurations

"Open-loop" means there is no feedback path from the output back to the input. Because the open-loop gain $A_{OL}$ is immense, open-loop configurations generally act as comparators rather than linear amplifiers.

There are three basic open-loop configurations:

Differential Amplifier: Signals are applied to both inverting and non-inverting terminals. $V_{out} = A_{OL} (V_{in1} - V_{in2})$ .
Inverting Amplifier: The non-inverting terminal is grounded. The input is applied to the inverting terminal. $V_{out} = -A_{OL} V_{in}$ . The output is 180° out of phase with the input.
Non-Inverting Amplifier: The inverting terminal is grounded. The input is applied to the non-inverting terminal. $V_{out} = +A_{OL} V_{in}$ . The output is in phase with the input.

Note: Due to noise, offset voltages, and massive gain, open-loop amplifiers almost always sit in saturation, making them useless for linear amplification.

5. Op-Amp with Negative Feedback

To use an op-amp as a stable, linear amplifier, we use negative feedback. This involves returning a portion of the output signal back to the inverting input terminal.

Advantages of Negative Feedback:

Stabilizes the voltage gain (makes it independent of op-amp parameter variations like temperature or manufacturing differences).
Increases the bandwidth of the amplifier.
Decreases non-linear distortion.
Reduces the effect of noise.
Modifies input and output impedances (increases input impedance, decreases output impedance).

The trade-off is a reduction in overall voltage gain, which is highly acceptable given the massive initial open-loop gain.

6. Block Diagram Representation of Feedback Configurations

A feedback amplifier consists of two main blocks:

Basic Amplifier (Forward Path): Has a gain of $A$ (open-loop gain).
Feedback Network (Reverse Path): Has a feedback factor of $\beta$ .

Input Signal: $V_s$
Feedback Signal: $V_f = \beta V_{out}$
Error Signal (Input to Op-amp): $V_d = V_s - V_f$

The closed-loop gain ( $A_f$ ) is derived as:
$V_{out} = A \times V_d = A(V_s - \beta V_{out})$
$V_{out} (1 + A\beta) = A V_s$
$A_f = \frac{V_{out}}{V_s} = \frac{A}{1 + A\beta}$
If $A\beta \gg 1$ , then $A_f \approx \frac{1}{\beta}$ . The gain depends purely on the external feedback network.

7. Voltage Series Feedback Amplifier

This configuration is commonly known as the Non-Inverting Amplifier.

Topology: The feedback voltage is connected in series with the input signal. The output voltage is sampled in parallel (voltage sampling).
Circuit: Input signal $V_{in}$ is applied to the non-inverting terminal. A voltage divider consisting of $R_f$ (feedback resistor) and $R_1$ (ground resistor) is connected to the inverting terminal.
Virtual Short Concept: Because $A_{OL} \rightarrow \infty$ , the differential voltage $v_d \rightarrow 0$ . Therefore, $v_- = v_+$ . Since $v_+ = V_{in}$ , $v_- = V_{in}$ .
Derivation:
Current through $R_1$ is $I_1 = \frac{v_-}{R_1} = \frac{V_{in}}{R_1}$ .
Since ideal op-amp input current is zero, the same current flows through $R_f$ .
$V_{out} = v_- + I_1 R_f = V_{in} + \left(\frac{V_{in}}{R_1}\right) R_f$
Closed-Loop Gain ( $A_v$ ):
$A_v = 1 + \frac{R_f}{R_1}$
Characteristics: Very high input impedance, very low output impedance, no phase shift.

8. Voltage Shunt Feedback Amplifier

This configuration is commonly known as the Inverting Amplifier.

Topology: The feedback signal is connected in parallel (shunt) with the input signal. The output voltage is sampled in parallel.
Circuit: The non-inverting terminal is grounded. The input signal $V_{in}$ is applied through $R_1$ to the inverting terminal, and $R_f$ connects the output back to the inverting terminal.
Virtual Ground: Since the non-inverting terminal is at 0V (ground), and $v_d = 0$ , the inverting terminal is also at 0V. This is called a "virtual ground".
Derivation:
Current from input: $I_{in} = \frac{V_{in} - 0}{R_1} = \frac{V_{in}}{R_1}$
Current through feedback: $I_f = \frac{0 - V_{out}}{R_f} = -\frac{V_{out}}{R_f}$
Since op-amp input current is zero, $I_{in} = I_f$ .
$\frac{V_{in}}{R_1} = -\frac{V_{out}}{R_f}$
Closed-Loop Gain ( $A_v$ ):
$A_v = -\frac{R_f}{R_1}$
(The negative sign indicates a 180° phase shift).
Characteristics: Input impedance is exactly equal to $R_1$ , very low output impedance.

9. Differential Amplifiers

A differential amplifier amplifies the difference between two input signals and rejects any signals common to both inputs (common-mode signals).

Basic Op-Amp Differential Circuit:
Combines both inverting and non-inverting topologies.
- $V_1$ is applied to the inverting terminal via $R_1$ . Feedback resistor $R_f$ is present.
- $V_2$ is applied to the non-inverting terminal via a voltage divider ( $R_2$ and $R_g$ ).
Superposition Principle for Analysis:
1. Ground $V_2$ , calculate output due to $V_1$ (Inverting amp): $V_{out1} = -V_1 \left(\frac{R_f}{R_1}\right)$
2. Ground $V_1$ , calculate output due to $V_2$ (Non-inverting amp with attenuated input): $V_{out2} = V_2 \left(\frac{R_g}{R_2 + R_g}\right) \left(1 + \frac{R_f}{R_1}\right)$
3. Total Output: $V_{out} = V_{out1} + V_{out2}$
Special Case (Symmetrical Resistors):
If $R_1 = R_2$ and $R_f = R_g$ , the equation simplifies to:
$V_{out} = \frac{R_f}{R_1} (V_2 - V_1)$
Common Mode Rejection Ratio (CMRR):
A critical parameter for differential amplifiers. It is the ratio of the differential gain ( $A_d$ ) to the common-mode gain ( $A_c$ ).
$CMRR = 20 \log_{10} \left( \frac{|A_d|}{|A_c|} \right) \text{ in dB}$
An ideal differential amplifier has infinite CMRR, meaning it perfectly ignores noise or interference present equally on both input wires.

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