The first article in this series, published November 5, 2007 (link below) left a gain expression in a form that could be confusing to some readers. To clarify the math, start with the last sequence of expressions from the sidebar:

Redraw the circuit as here in Figure 1 with the definition of the voltage β.

Figure 1: Op amp as a gain stage
(Click on image to enlarge)

Treating the gain setting resistors as a voltage divider yields:

The ideal or requested gain is 1/β and the transfer function becomes:

It should be noted that Aol and β are functions of frequency. The results of this relationship are seen in the Bode plot as shown in Figure 2 (Note: Aol is shown in blue, 1/β is shown in orange).

Figure 2: Combined Bode plot
(Click on image to enlarge)

This example demonstrates the power of the Y-axis log scale. Subtraction of logs is the arithmetic operation of division. Therefore, the distance between the two curves in Figure 2 is the division in the transfer function. The accuracy of the closed loop gain can be seen in the distance between the two curves. As the frequency increases, the distance between the curves decreases. Hence, the second term in the transfer function is no longer one. The approximation that assumes an infinite open loop gain is no longer valid.

Another operational amplifier (op amp) performance key that is easily displayed on the Bode plot is the difference between the Gain-Bandwidth-Product (GBP) and the Unity-Gain Bandwidth (UGB). All op amps have a pole in their response at a low frequency. They also have a high frequency pole.

This pole occurs at a frequency near the frequency where the open loop gain curve crosses zero dB (Unity Gain). Consider the gain curves for the two op amps shown in Figure 3.

Figure 3: Unity-gain-bandwidth vs. gain-bandwidth product
(Click on image to enlarge)

Unit 1 shows a UGB of 1 MHz, and unit 2 shows a UGB of approximately 300 kHz due to the pole at 100 kHz.

The GBP for unit 1 in Figure 3 is the same as its UGB of 1 MHz. Unit 2 has a UGB of 300 kHz, but the GBP is also 1 MHz. For the single-pole system moving one decade in frequency results in a gain change of 10× (20 dB).

To calculate the GBP for unit 2, note the last point on the gain curve that is still on the 20 dB/decade line. In this case that is 20 dB at 100 kHz. Twenty dB is a gain of 10. Multiply the gain times the frequency to calculate the GBP (10 × 100 kHz = 1 MHz). The value of the GBP is constant for any point on the response curve at frequencies between the two poles (i.e., gain of 100 at 10 kHz equals a gain of 1,000 at 1 kHz).

The GBP and. the Bode plot will be revisited many times in the future for frequency related performance, and especially to aid in understanding stability issues.

About the author



William P. (Bill) Klein is a Senior Applications Engineer with the High Performance Analog group at Texas Instruments. Bill joined TI through its acquisition of Burr-Brown in August 2000. His experience as an analog circuit designer covers over 40 years in fields ranging from mineral exploration to medical nuclear imaging. One current role Bill has is hosting the Analog e-LAB Web Cast, presenting real world solutions to real world problems in analog circuit design. In addition to a BSEE from Arizona State University and registration as a Professional Engineer in the State of Arizona, he has authored numerous magazine articles, application notes and conference papers.

Previous installments of this series:

• "SIGNAL CHAIN BASICS (Part 12): The Bode plot, an essential ac-parameter display tool", www.planetanalog.com/features/showArticle.jhtml;?articleID=207400431, click here
• "SIGNAL CHAIN BASICS (Part 11): Introducing voltage- and power-conditioning circuits", www.planetanalog.com/features/showArticle.jhtml;?articleID=207001505, click here
• "SIGNAL CHAIN BASICS (Part 10): Exploring the Delta-Sigma Converter", www.planetanalog.com/features/showArticle.jhtml;?articleID=206903892, click here
• "SIGNAL CHAIN BASICS (Part 9): SAR Converter Operation Explored", www.planetanalog.com/features/showArticle.jhtml;?articleID=206901015, click here
• "SIGNAL CHAIN BASICS (Part 8): Flash- and Pipeline-Converter Operation Explored", www.planetanalog.com/features/showArticle.jhtml;?articleID=206504089, click here
• "SIGNAL CHAIN BASICS (Part 7): Op Amp Performance Specification--Bias Current", www.planetanalog.com/features/showArticle.jhtml;?articleID=206101908, click here
• "SIGNAL CHAIN BASICS (Part 6): Op Amp Input Voltage Offset", www.planetanalog.com/features/showArticle.jhtml;?articleID=205901111, click here
• "SIGNAL CHAIN BASICS (Part 5): Introduction to the Instrumentation Amplifier", www.planetanalog.com/features/showArticle.jhtml;?articleID=205208593, click here
• "SIGNAL CHAIN BASICS (Part 4): Introduction to analog/digital converter (ADC) types", www.planetanalog.com/features/showArticle.jhtml;?articleID=204803631, click here
• "SIGNAL CHAIN BASICS (Part 3): Analog and the digital world", www.planetanalog.com/features/showArticle.jhtml;?articleID=204400376, click here
• "SIGNAL CHAIN BASICS (Part 2): Op Amp--Basic operations", www.planetanalog.com/features/showArticle.jhtml;?articleID=203101699, click here
• "SIGNAL CHAIN BASICS: Operational Amplifier--The Basic Building Block", www.planetanalog.com/features/showArticle.jhtml;?articleID=202801320, click here