(Editor's Note: There are links to the previous parts of this series at the end, below the author's biography.)

AC parameters which describe the amplifier's performance over frequency can appear very confusing. One tool used in many descriptions and analysis is the Bode plot, Reference 1. This graphical presentation shows the gain and phase relationship of a network or system. Due to its scaling, calculations can be accomplished with graphical techniques.

The total gain for a series of stages is calculated by taking the product of the individual stages. The Bode plot displays gain as a function of frequency in decibels (dB). The dB units are based on the log of the transfer ratio. In a mathematical statement this becomes (Equation 1):

Multiplication is accomplished when logarithms are added so total system gain can be calculated by graphical addition of the individual gain curves. Phase-shift is a linear variable, and the phase-shift through stages in series simply adds algebraically. Many of the Bode-plot features can be illustrated by analyzing a simple R-C low-pass filter (Figure 1).

Figure 1: R-C low-pass filter

The corner frequency in the response curve for this circuit is at the point where (Equation 2):

With equal impedances in each leg of the voltage divider, one might make the mistake that Vout is one-half the input voltage. The error is that the capacitive impedance is on the imaginary axis, versus the real axis resistance (R1). The magnitude of Vout is then 0.707 Vin. This calculates as a gain of -3.01 dB. The voltage across C1 is phase-shifted 90° from the voltage across R1. Since the real and imaginary voltage components are equal, it gives a phase-shift through the circuit of 45°.

Figure 2 reflects the Bode plot for this circuit response:

Figure 2: Bode plot for a single pole, low-pass filter
(Click on image to enlarge)

While attempting to accurately determine the corner frequency from the gain plot can be difficult, the 45° phase-shift frequency is much easier to identify.

The curves in Figure 2 are calculated from the circuit analysis. Good first-order results can be obtained with straight line approximations, Figure 3.

Figure 3: Straight line approximations of gain and phase
(Click on image to enlarge)

The straight line gain approximation is accomplished by drawing a line parallel to the X-axis from the Y-axis to the corner frequency (fc), then a straight line with a downward slope of 20 dB per frequency decade. The phase plot starts from 0° at fc/10, through 45° at fc, to 10 times fc.

The line slopes are just reversed for a zero. As the slope of the phase-shift plot for a pole is a negative 45° per frequency decade, the slope associated with a zero is a positive 45° per frequency decade.

This all applies to the signal chain because the operational amplifier's (op amp) open-loop response shows a single pole response up to a frequency where the gain crosses 0 dB (gain of one). These techniques and approximations will be used in future articles to help predict stability and total frequency response of op amp circuits.

Reference
1. Hendrik W. Bode, Ph.D. Research Mathematician, Bell Telephone Laboratories, Inc.

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 11): Introducing voltage- and power-conditioning circuits", www.planetanalog.com/features/showArticle.jhtml;?articleID=207001505, click here
• "SIGNAL CHAIN BASIC Series (Part 10): Exploring the Delta-Sigma Converter", www.planetanalog.com/features/showArticle.jhtml;?articleID=206903892, click here
• "SIGNAL CHAIN BASIC Series (Part 9): SAR Converter Operation Explored", www.planetanalog.com/features/showArticle.jhtml;?articleID=206901015, click here
• "SIGNAL CHAIN BASIC Series (Part 8): Flash- and Pipeline-Converter Operation Explored", www.planetanalog.com/features/showArticle.jhtml;?articleID=206504089, click here
• "SIGNAL CHAIN BASIC Series (Part 7): Op Amp Performance Specification--Bias Current", www.planetanalog.com/features/showArticle.jhtml;?articleID=206101908, click here
• "SIGNAL CHAIN BASIC Series (Part 6): Op Amp Input Voltage Offset", www.planetanalog.com/features/showArticle.jhtml;?articleID=205901111, click here
• "SIGNAL CHAIN BASICS Series (Part 5): Introduction to the Instrumentation Amplifier", www.planetanalog.com/features/showArticle.jhtml;?articleID=205208593, click here
• "SIGNAL CHAIN BASICS Series (Part 4): Introduction to analog/digital converter (ADC) types", www.planetanalog.com/features/showArticle.jhtml;?articleID=204803631, click here
• "SIGNAL CHAIN BASICS Series (Part 3): Analog and the digital world", www.planetanalog.com/features/showArticle.jhtml;?articleID=204400376, click here
• "SIGNAL CHAIN BASICS Series (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