Circuit Theory/Bode Plots

Decibel

More information about Decibels can be found in:
The Appendix

Decibels (abbreviated "dB") are not units per se. Instead, a Decibel is simply a logarithmic ratio of the input to the output of a circuit. Decibels represent a power ratio unless they are clearly tagged differently. Most db tags are NOT indicative of the ratio units, but rather, they indicate a comparison of a variable with an accepted standard. For example, dbm means 10 times the log of a power variable in ratio to 1 milliwatt. This is very popular in dealing with telephone lines. Decibels were invented by power weenies and belong to them, so it's power unless someone indicates otherwise.

Voltage

[Decibel relation]

$dB = 20\log{\frac{V_{out}}{V_{in}}}$

This is the power ratio ( out / in ) that exists for a voltage ratio of V_(out)to V_(in). Power goes as the square of the voltage, therefore the 10 log becomes 20 log. Bels, the precursors to decibels, were invented to describe power ratios; therefore, it is customary to describe a voltage ratio in terms of the power ratio that voltage ratio represents. This is used even when the input and output impedance differ. This makes little sense, but it is common practice. This "voltage-ratio-yields-power-ratio" comparison only really works when the impedance is the same for In and Out. This is true in such cases as telephone and video systems. The terms "Vin" and "Vout" can either be the time-domain values of the voltage input and voltage output, or they can be the magnitudes of the respective phasors. The original definition was bels = log (Power-out / Power-in), where log is the common (base 10) log, but the bel is a very large unit, hence the birth of the decibel, one-tenth of a bel. You can take a log ratio of any two quantities, but they must be the same unit. Oranges over oranges = unit-less.

Power Gain

$dB = 10\log{\frac{P_{out}}{P_{in}}}$

This is used to compare the output power of a system to the input power of a system. Notice that since power is being compared to power, the log prefix is 10. For voltage comparisons the prefix was 20 (due to power being proportional to voltage squared).

Notes on Decibels

Remember:
"Decibels are not numbers, they are ratios"

Decibels are plotted on a "log10" graph, where each hash-mark on the axis is a successive power of 10. Also, the values on the X-axis are plotted on a log-10 scale as well. Each successive power-of-10 on the frequency axis is known as a "decade". It is important to mention that decibels are simply a convenient way to represent a scaling factor, and that decibels are not numbers: they are ratios. Any quantity ratio can be expressed in decibels. The decibel is really just a common (base ten) log expressed in 0.1 log points; a log with enhanced resolution. It is customary to use a letter to warn the reader that a non-typical quantity is being compared by the log of a ratio. For voltage, it is usually dbv. This is a voltage comparison, NOT a power-due-to-the-voltage comparison. So 100 volts out for 1 volt in is 20 dbv. Impedance is not relevant for dbv. But this usage is fraught with peril, because dbv is also used to indicate that a variable is being ratioed to 1 volt. Decibels has been converted to an absolute. Beware and make yourself clear. Early telephony usage converted decibels to an absolute power level by setting the denominator to a fixed agreed power value: 6 milliwatts at 500 or 600 ohms impedance. The ratio of a variable to this fixed quantity was called dbm. Modern practice is to use 1 milliwatt as the reference for dbm.

Bode Plots

Bode plots can be broken down into 2 separate graphs: the magnitude graph, and the phase graph. Both graphs represent the circuit response in each category to sinusoids of different frequencies.

Magnitude Graph

The Bode Magnitude Graph is a graph where the radial frequency is plotted along the X-axis, and the gain of the circuit at that frequency is plotted (in Decibels) on the Y-axis. The bode magnitude graph most frequently plots the power gain against the frequency, although they may also be used to graph the voltage gain against the frequency. Also, the frequency axis may be in terms of hertz or radians, so the person drawing a bode plot should make sure to label their axes correctly.

Phase Graph

The Bode Phase Plot is a graph where the radial frequency is plotted along the X axis, and phase shift of the circuit at that frequency is plotted on the Y-axis. The phase change is almost always represented in terms of radians, although it is not unheard of to express them in terms of degrees. Likewise, the frequency axis may be in units of hertz or radians per second, so the axes need to be labeled correctly.

Bode Plots by Different Methods

Bode plots can be used both with Phasors (Network Functions), and with the Fourier Transform (Frequency Response). However, there are slightly different methods to doing it each way, and those methods will be examined in the following chapters. The Laplace Transform can be used to construct a bode plot by transposing from the s-domain to the fourier-domain. However, this is rarely done in practice and the Laplace Transform is instead used with other graphical methods that are unfortunately, outside the scope of this wikibook.

Bode Analysis

The pages in this section will talk about how to analyze a bode plot of a given circuit, and draw conclusions from that plot.

This section of the Circuit Theory wikibook is a stub. You can help by expanding this section.

Log Magnitude Graph

Using a network function (remember phasors?), we can find the log magnitude and the phase bode plots of a circuit. This page will discuss how to find the bode polt from the network function of a circuit.

Poles and Zeros

The topic of "Poles and Zeros" are discussed in excruciating detail in advanced texts in Electrical Engineering. We will introduce the concepts of what a pole and a zero are in this chapter.

Transfer Polynomials

Let's say that we have a given frequency response:

$H(j\omega) = \frac{Z(j\omega)}{P(j\omega)}$

Where both Z and P are polynomials. We then set each of these equations to zero, and solve:

Z(jω) = 0

P(jω) = 0

The solutions to the equation Z = 0 are called the "Zeros" of H. The solutions to the equation P = 0 are called the "Poles" of H.

Properties of poles and zeros

Let's say that we have a frequency response that has a zero at N, and a pole at M. We then plug in these values to our frequency response:

$H(j\omega)|_{\omega \to N} = \frac{Z(j\omega)}{P(j\omega)} = \frac{0}{P(j\omega)} = 0$

And:

$H(j\omega)|_{\omega \to M} = \frac{Z(j\omega)}{P(j\omega)} = \frac{Z(j\omega)}{0} = \infty$

Now, some of the purists will immediately say "but you arent allowed to divide by zero", and to those people I say: you can write in a limit, if you really want to.

Bode Equation Format

let us say that we have a generic transfer function with poles and zeros:

$H(j\omega) = \frac{(\omega_A + j\omega)(\omega_B + j\omega)}{(\omega_C+ j\omega)(\omega_D + j\omega)}$

Each term, on top and bottom of the equation, is of the form (ω_N + jω). However, we can rearrange our numbers to look like the following:

$\omega_N(1 + \frac{j\omega}{\omega_N})$

Now, if we do this for every term in the equation, we get the following:

$H_{bode}(j\omega) = \frac{\omega_A \omega_B}{\omega_C \omega_D} \frac{(1 + \frac{j\omega}{\omega_A})(1 + \frac{j\omega}{\omega_B})} {(1 + \frac{j\omega}{\omega_C})(1 + \frac{j\omega}{\omega_D})}$

This is the format that we are calling "Bode Equations", although they are simply another way of writing an ordinary frequency response equation.

DC Gain

The constant term out front:

$\frac{\omega_A \omega_B}{\omega_C \omega_D}$

is called the "DC Gain" of the function. If we set $\omega \to 0$, we can see that everything in the equation cancels out, and the value of H is simply our DC gain. DC then is simply the input with a frequency of zero.

Break Frequencies

in each term:

$(1 + \frac{j\omega}{\omega_N})$

the quantity ω_N is called the "Break Frequency". When the radial frequency of the circuit equals a break frequency, that term becomes (1 + 1) = 2. When the radial frequency is much higher than the break frequency, the term becomes much greater than 1. When the radial Frequency is much smaller than the break frequency, the value of that term becomes approximately 1.

Much Greater and Much Less

We use the term "much" as a synonym for the term "At least 10 times". So "Much Greater" becomes "At least 10 times greater" and "Much less" becomes "At least 10 times less". We also use the symbol "<<" to mean "is much less than" and ">>" to mean "Is much greater than". Here are some examples:

• 1 << 10
• 10 << 1000
• 2 << 20 Right!
• 2 << 10 WRONG!

Engineering Approximations

For a number of reasons, Electrical Engineers find it appropriate to approximate and round some values very heavily. For instance, manufacturing technology will never create electrical circuits that perfectly conform to mathematical calculations. When we combine this with the << and >> operators, we can come to some important conclusions that help us to simplify our work:

If A << B:

• A + B = B
• A - B = -B
• A / B = 0

All other mathematical operations need to be performed, but these 3 forms can be approximated away. This point will come important for later work on bode plots.

Using our knowledge of the Bode Equation form, the DC gain value, Decibels, and the "much greater, much less" inequalities, we can come up with a fast way to approximate a bode magnitude plot. Also, it is important to remember that these gain values are not constants, but rely instead on changing frequency values. Therefore, the gains that we find are all slopes of the bode plot. Our slope values all have units of "decibel per decade", or "db/decade", for short.

At Zero Radial Frequency

At zero radial frequency, the value of the bode plot is simply the DC gain value in decibels. Remember, bode plots have a log-10 magnitude Y-axis, so we need to convert our gain to decibels:

Magnitude = 20log ₁₀(DCGain)

At a Break Point

We can notice that each given term changes it's effect as the radial frequency goes from below the break point, to above the break point. Let's show an example:

$(1 + \frac{j\omega}{5})$

Our breakpoint occurs at 5 radians per second. When our radial frequency is much less than the break point, we have the following:

Gain = (1 + 0) = 1

Magnitude = 20log ₁₀(1) = 0db / decade

When our radial frequency is equal to our break point we have the following:

$Gain = |(1 + j)| = \sqrt{2}$

$Magnitude = 20\log_{10}(\sqrt{2}) = 3db/decade$

And when our radial frequency is much higher (10 times) our break point we get:

$Gain = |(1 + 10 j)| \approx 10$

Magnitude = 20log ₁₀(10) = 20db / decade

However, we need to remember that some of our terms are "Poles" and some of them are "Zeros".

Zeros

Zeros have a positive effect on the magnitude plot. The contributions of a zero are all positive:

Radial Frequency << Break Point 

0db/decade gain.

Radial Frequency = Break Point 

3db/decade gain.

Radial Frequency >> Break Point 

20db/decade gain.

Poles

Poles have a negative effect on the magnitude plot. The contributions of the poles are as follows:

Radial Frequency << Break Point 

0db/decade gain.

Radial Frequency = Break Point 

-3db/decade gain.

Radial Frequency >> Break Point 

-20db/decade gain.

Conclusions

To draw a bode plot effectively, follow these simple steps:

1. Put the frequency response equation into bode equation form.
2. identify the DC gain value, and mark this as a horizontal line coming in from the far left (where the radial frequency conceptually is zero).
3. At every "zero" break point, increase the slope of the line upwards by 20db/decade.
4. At every "pole" break point, decrease the slope of the line downwards by 20db/decade.
5. at every breakpoint, note that the "actual value" is 3db off from the value graphed.

And then you are done!

See Also

• Bode Plots on ControlTheoryPro.com