Op-Amp Circuit Analysis

Hello! Welcome to your third lesson.

In our last session, we established the fundamental characteristics of bioelectric signals. We learned that signals like EEG are incredibly faint (in the microvolt range) and exist within specific frequency bands. This directly sets up our current topic: how do we build the electronics to amplify these weak signals to a usable level?

Today, we'll dive into the workhorse of analog electronics: the operational amplifier (op-amp). This lesson directly addresses the learning outcome: Analyze operational amplifier (op-amp) circuits to determine gain, input impedance, and bandwidth. As someone with a prior ECE degree, this should be a good refresher, but we'll focus specifically on the aspects most critical for biomedical instrumentation. For your work at Neuraease, mastering op-amps is non-negotiable; they are the essential first stage in any device that measures physiological signals.

The Ideal Op-Amp: A Perfect Amplifier

An op-amp is a differential amplifier with extremely high gain. It amplifies the voltage difference between its two inputs. In its raw, "open-loop" state, its gain is so massive (often >100,000) that it's practically unusable. The key to taming it is negative feedback, where a portion of the output signal is fed back to the negative (inverting) input. This allows us to create stable amplifiers with precisely controlled characteristics.

For most initial design and analysis, we use the ideal op-amp model. To learn about the properties of an ideal op-amp, please read the following section from the MIT OpenCourseWare notes.

From that reading, we can summarize the two "golden rules" for analyzing op-amp circuits with negative feedback, assuming the output is not saturated:

1. The input voltage difference is zero. The voltage at the non-inverting input ($V_{+}$) is equal to the voltage at the inverting input ($V_{-}$). This is because any tiny difference would be multiplied by the infinite open-loop gain, saturating the output. The feedback loop adjusts the output to force this condition.
2. No current flows into the input terminals. The input impedance is infinite, so the op-amp draws no current from the source circuit.

With these two rules, we can analyze the most common amplifier configurations.

Core Op-Amp Configurations

Let's explore the fundamental building blocks you'll use constantly in instrumentation design.

1. The Inverting Amplifier

In this configuration, the input signal is applied to the inverting (-) terminal through a resistor $R_{in}$, and the non-inverting (+) terminal is connected to ground.

To get a quick overview of this circuit and its non-inverting counterpart, watch the first few minutes of this video.

Now, let's analyze it with our golden rules:

A standard inverting amplifier configuration. The output is an amplified and inverted version of the input.

1. From Rule 1, since $V_{+}$ is at ground (0V), $V_{-}$ must also be at 0V. This point is called a virtual ground.
2. From Rule 2, no current flows into the $V_{-}$ terminal. Therefore, the current flowing through $R_{in}$ must be equal to the current flowing through the feedback resistor, $R_f$.

Let's call this current $i$. Using Ohm's law:
$i=\frac{V_{in}-V_{-}}{R_{in}}=\frac{V_{in}-0}{R_{in}}=\frac{V_{in}}{R_{in}}$
This same current $i$ flows through $R_f$. The voltage drop across $R_f$ is:
$V_{out}-V_{-}=V_{out}-0=V_{out}$
And the current is flowing from $V_{-}$ to $V_{out}$, so:
$i=\frac{V_{-}-V_{out}}{R_f}=\frac{0-V_{out}}{R_f}=-\frac{V_{out}}{R_f}$
Setting the two expressions for $i$ equal:
$\frac{V_{in}}{R_{in}}=-\frac{V_{out}}{R_f}$
Rearranging for the voltage gain ($G$), we get the classic formula:
$G=\frac{V_{out}}{V_{in}}=-\frac{R_f}{R_{in}}$
The negative sign indicates that the output signal is 180° out of phase with the input—it's an inverted signal.

2. The Non-Inverting Amplifier

Here, the input signal is applied directly to the non-inverting (+) terminal. The feedback network looks similar, but $R_i$ now connects the inverting (-) terminal to ground.

Let's return to the video for a quick look at this circuit.

A standard non-inverting amplifier. The output is an amplified, in-phase version of the input.

1. From Rule 1, the voltage at the inverting terminal, $V_{-}$, must be equal to the voltage at the non-inverting terminal, $V_{+}$. Since $V_{in}$ is applied to $V_{+}$, we have $V_{-}=V_{in}$.
2. From Rule 2, no current flows into the op-amp inputs. The resistors $R_i$ and $R_f$ form a simple voltage divider with $V_{out}$ as the input and $V_{-}$ as the output.

The voltage at $V_{-}$ can be expressed using the voltage divider formula:
$V_{-}=V_{out}\left(\frac{R_i}{R_i+R_f}\right)$
Substituting $V_{-}=V_{in}$:
$V_{in}=V_{out}\left(\frac{R_i}{R_i+R_f}\right)$
Rearranging for the gain ($G$):
$G=\frac{V_{out}}{V_{in}}=\frac{R_i+R_f}{R_i}=1+\frac{R_f}{R_i}$
This gain is always positive (non-inverting) and greater than or equal to 1. A special case where $R_f=0$ and $R_i\to \infty$ (an open circuit) gives a gain of 1. This is called a voltage follower or buffer, and it's extremely useful for impedance matching, which we'll discuss next.

Input Impedance: A Critical Parameter for Bio-signals

Input impedance is the effective resistance seen by the signal source when it's connected to the amplifier. For measuring bio-potentials, high input impedance is crucial. The electrode-skin interface has its own impedance (which can be high and variable), and if the amplifier's input impedance is too low, it will act like a voltage divider, attenuating the already tiny signal before it's even amplified.

This next video provides an excellent and intuitive explanation of how to determine the input impedance for our two main configurations.

To summarize the key findings from the video:

• Inverting Amplifier: Because the inverting terminal is a virtual ground (0V), the input impedance seen by the source $V_{in}$ is simply $R_{in}$. This means there is a trade-off: to get high input impedance, you need a large $R_{in}$, which may require an impractically large $R_f$ to achieve the desired gain.
• Non-Inverting Amplifier: The signal source connects directly to the non-inverting input. Since an ideal op-amp draws no current (Rule 2), the input impedance is infinite.

This makes the non-inverting configuration highly preferable for the input stage of a biopotential amplifier like an EEG or ECG. It can sense the voltage from the electrodes without loading them down.

Bandwidth and the Gain-Bandwidth Product

An op-amp cannot provide high gain across all frequencies. There is an inherent trade-off between gain and bandwidth. This relationship is characterized by the Gain-Bandwidth Product (GBWP), a value found on the op-amp's datasheet.

For most op-amps, the GBWP is constant. This leads to a simple, powerful relationship for circuits with negative feedback:

$\text{Closed-Loop Gain}\times \text{Bandwidth}\approx \text{GBWP}$

This means if you configure an op-amp with a GBWP of 1 MHz for a gain of 100, its bandwidth will be approximately:
$\text{Bandwidth}\approx \frac{1\text{ MHz}}{100}=10\text{ kHz}$
If you increase the gain to 1000, the bandwidth drops to 1 kHz.

To understand this concept more deeply, please review the following resource.

This concept is vital. If you need to amplify a 50 Hz ECG signal, an amplifier with a 1 kHz bandwidth is fine. But if you need to amplify a high-frequency surface EMG signal (up to 10 kHz), you must ensure your amplifier has enough bandwidth at the required gain.

Conclusion

Today we've refreshed the fundamentals of op-amp analysis, focusing on what matters for medical instrumentation. We've seen how a few simple rules allow us to analyze circuits and make critical design choices.

Key Takeaways:

• Op-amps are controlled using negative feedback to create amplifiers with predictable gain.
• The gain of an inverting amplifier is $G=-R_f/R_{in}$, and its input impedance is $R_{in}$.
• The gain of a non-inverting amplifier is $G=1+R_f/R_i$, and its input impedance is ideally infinite, making it ideal for biopotential front-ends.
• An op-amp's performance is limited by its Gain-Bandwidth Product (GBWP), creating a trade-off: higher gain results in lower bandwidth.

Preview of the Next Lesson:
We now know how to build an amplifier with a specific gain. But what about all the noise and artifacts outside our signal's frequency range? In the next lesson, we will combine op-amps with capacitors to design active filters. You'll learn how to build circuits that not only amplify the signal but also selectively pass the frequencies we want while rejecting those we don't.