“Bode plots are a very useful tool when designing stable op amp circuits and should definitely be added to your arsenal. When you start working on multi-pole and multi-zero circuits, you realize the great potential of Bode plots. The stability of the circuit can be quickly determined by the cutoff ratio between the amplifier open-loop gain and the feedback network.

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Author: Bonnie Baker

In the 1930s, Hendrik Wade Bode created an intuitive gain/phase method with circuit stability as the sole goal. This is what is now called a Bode plot, an intuitive way to graphically represent the frequency of a circuit or amplifier’s gain, phase, and feedback system.

Given its usefulness and importance, let’s take a moment to understand the technique of Bode stability analysis, looking at the magnitude of open-loop amplifier and circuit feedback coefficients in decibels (dB) and phase response (degrees). This blog will explore the above concepts and give advice on how to avoid designing a “dithering” circuit when your main goal is frequency stability.

To practice this technique, you can download a printable version of the Bode plot from the resources in the online Digi-Key Innovation Brochure.

**Single Pole Bode Plot**

The configuration of the single-pole circuit allows the dc VIN signal to go directly to VOUT, and at higher input frequencies, VOUT equals zero decibels (dB). The construction of a Bode plot is simple. The units for the Y-axis are logarithms of frequency, and the units for the X-axis are linear unit gain in decibels or phase in degrees. There are quite a few formulas that can already be applied when designing a Bode plot, but we’re going to experience this quick fix firsthand.

The simplicity of a Bode plot is that all you need to draw is a ruler tool and knowing some rules (Figure 1).

Figure 1: A single-pole Bode plot showing magnitude and phase shift uses straight lines to show the frequency and phase response of a circuit. (Image credit: Bonnie Baker)

The two graphs in Figure 1 show frequency versus gain and phase for a single-pole resistor/capacitor pair. The X-axis frequency range for the upper and lower graphs is 1 hertz (Hz) to 10 megahertz (MHz). The Y-axis of the graph above ranges from 0 decibels (dB) to 100 dB, where a signal value at 1 Hz equals 100 dB. This value corresponds to a gain factor of 100,000 x VIN. The blue curve is the single-pole gain response at fP or 100 Hz, where R equals 159 kiloohms (kΩ) and C equals 10 nanofarads (nF).

The blue curve decreases at a rate of -20dB/decade or -6dB/octave as the frequency increases beyond the pole frequency (fP). This decay rate is the first Bode plot rule of thumb to remember: every pole in the circuit drops at a rate of -20dB/decade from the pole frequency. Therefore, if both poles attenuate VOUT in the same frequency range, the attenuation is -40dB/decade.

The lower graph in Figure 1 shows the phase of this single-pole circuit. At 1Hz, the phase of the R/C network is 0 degrees (°). One decade before fP, or 10 Hz in this example, the single-pole phase begins to drop toward its -90° target at a rate of -45°/decade.

Several rules apply to the phase response of the poles. The second Bode plot rule of thumb for pole circuits is that the phase at fP equals -45°. The third and fourth Bode plot rules describe the phase points for decay and completion. The single-pole phase begins to drop one decade before the pole frequency (fP) and settles at -90° one decade after fP.

**Single Zero Bode Plot**

A single-zero Bode plot reflects the opposite rule of a single-pole Bode plot. Position switching, with the same value for R and C, to block the DC VIN voltage while allowing higher frequencies to pass through the capacitor (Figure 2).

Figure 2: Single-zero Bode plot showing amplitude and phase shift. (Image credit: Bonnie Baker)

The blue curve rises at a rate of +20dB/decade as the frequency increases beyond fZ. The lower graph in Figure 2 shows the phase of this single-zero circuit. One decade before fZ, the single-zero phase begins to rise towards its +90° target at +45°/decade. At fZ, the zero circuit phase is equal to +45°.

Summarizing the values in Figure 1, the pole location is fP, and the gain magnitude after fP has a slope of -20dB/decade. The phase has a -45°/decade slope through fP, the phase starts to decay at 0.1x fP and settles to -90° at 10x fP. Summarizing the values in Figure 2, the zero position is fZ, and the gain magnitude after fZ has a +20dB/decade slope. The phase has a +45°/decade slope through fZ, the phase starts to decay at 0.1x fZ and settles to +90° at 10x fZ.

**Amplifier Open Loop Bode Plot**

Standard op amp products have multiple poles and zeros in their transfer function for frequency operation from subhertz to zero dB cutoff frequency. There is nothing magical about an amplifier Bode plot, just follow the rules (Figure 3).

Figure 3: A possible Bode plot of an op amp showing amplitude and phase shift. (Image credit: Bonnie Baker)

Figure 3 shows an example of an op amp with two poles (f1 and f2) in its transfer function. With these two poles, the gain drops -20dB/decade each time, and the phase drops a total of -180 degrees.

So far we have had a good start on how to build a Bode plot, but when it comes to actual projects, there is also a feedback system in it.

**Stability of closed-loop amplifier systems**

If you take a moment to look at the op amp circuit, you will see the presence of the feedback network. The classic op amp feedback network has a gain forward element (AOL(jω)) and a feedback element (β(jω)).

Figure 4: A classic op amp feedback network has a feedforward element (AOL(jω)) and a feedback element (β(jω)). (Image credit: Bonnie Baker)

In Figure 4, the open-loop gain of the operational amplifier (AOL) is relatively large, while the feedback coefficient is relatively small. This configuration sends the output back to the inverting terminal, creating a negative feedback condition that keeps the output under control. We will use the inverse of β or 1/β to determine the stability of the op amp circuit.

The easiest way to calculate 1/β is to place a voltage source called VSTABILITY at the non-inverting input of the op amp. This computational strategy will provide a good path to determine the stability of the circuit (Figure 5).

Figure 5: Non-inverting op amp circuit a.) and inverting op amp circuit b.) both have an imaginary VSTABILITY voltage source at their non-inverting input in order to accurately calculate the circuit’s 1/beta factor, or noise gain . (Image credit: Bonnie Baker)

If you examine the circuit in Figure 5, the feedback circuit from the non-inverting terminal to the output terminal is the same. The location of the VSTABILITY voltage source allows an accurate calculation of the circuit’s 1/beta factor, or noise gain.

1/β stability analysis using VSTABILTIY. If you assume that the open loop gain of the op amp is infinite, then the transfer function of the two circuits is equal to:

Equation 1

Equation 2

Equation 3

When the frequency component jω of Equation 3 is equal to zero:

Equation 4

When jω approaches infinity in Equation 3:

Equation 5

The frequencies of the zeros (fZ) and poles (fP) of 1/β are:

Equation 6

Equation 7

The Bode plot of the stability analysis curve for 1/β that conforms to the above rules is shown in Figure 6 .

Figure 6: The 1/β frequency responses of Figure 5 a.) and b.) are the same. Zeros appear at lower frequencies, while poles appear at higher frequencies. (Image credit: Bonnie Baker)

Figure 6 depicts the 1/β frequency and phase response or noise gain of an op amp circuit. This figure summarizes Equations 4 to 7 graphically. Equations 4 and 5 define the DC gain and Y gain. Equations 6 and 7 determine the zeros and poles of the circuit. The information in Figures 3 and 6 provides the first step in establishing the stability of an op amp circuit by defining the transfer function of the system and the locations of poles and zeros. The final step is to overlay Figures 3 and 6 into one graph.

**Determination of System Stability**

The intersection, or cutoff, of the open-loop and closed-loop gains defines the phase shift of the circuit. In general, a cutoff ratio less than or equal to 30 dB indicates that the circuit is stable. Cutoff ratios greater than 30dB lead to unstable circuit conditions (Figure 7).

Figure 7: AOL gain and phase response of a superimposed op amp versus 1/beta gain and phase response. (Image credit: Bonnie Baker)

In Figure 7, the cutoff between the AOL and 1/beta gain curves is equal to 40 dB. A 40dB cutoff indicates a phase shift greater than 135°, which indicates an unstable circuit. In this configuration, the 180° cutoff produces an oscillating circuit.

There are many solutions to the above problems. Resistor or capacitor values can be changed by moving the pole and zero frequencies. Another option is to choose a different op amp (Figure 8).

Figure 8: Using an op amp with a higher bandwidth than the one in Figure 7 without changing the zero and pole frequencies. (Image credit: Bonnie Baker)

In Figure 8, the bandwidth of the op amp is about two decade higher without changing the 1/β network. The green dashed line reflects the actual calculation and provides a more realistic Bode plot. The increase in amplifier bandwidth changes the cutoff from 40 dB to 20 dB. The resulting phase shift is now ~105°, indicating that the circuit is stable.

The green dashed line in Figure 8 goes beyond the Bode plot drawn with a ruler and pencil to include the real-world response.

**Measure the gain and phase of a circuit**

Measuring the gain and phase of an amplifier circuit requires a function generator to provide the input signal, and a network analyzer (Figure 9). Representative is the Tabor Electronics LS3081B 3 GHz RF analog swept-frequency function generator.

Figure 9: Gain and phase measurement configuration for the inverting amplifier circuit of Figure 5 b.). (Image credit: Bonnie Baker)

In Figure 9, the application of the function generator’s input signal occurs from port 1 to the VSTABILITY node. The signal propagates through the amplifier circuit to the output of the circuit (VOUT), where the network analyzer captures the signal at port 2 and compares it with the function generator’s port 1 signal.

**Epilogue**

Bode plots are a very useful tool when designing stable op amp circuits and should definitely be added to your arsenal. When you start working on multi-pole and multi-zero circuits, you realize the great potential of Bode plots. The stability of the circuit can be quickly determined by the cutoff ratio between the amplifier open-loop gain and the feedback network.

While this blog can help you master the use of Bode plots, showing the simple use of a ruler on a graph paper to estimate the gain versus phase relationship of a first-order pole and zero circuit, the best way to learn is practice. Additionally, you can get started by downloading a printable version of the Bode plot from the resources in the online Digi-Key Innovation Brochure.

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