Circuit Theory/Bode Plots

Decibel

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. There are 2 ways to calculate decibels, depending on whether we want to analyze the voltage gain or the power gain. Considering that decibels are just a ratio between other quantities, decibels are generally followed by other units, such as "dBW" (decibels, watts), "dBmV" (decibels, millivolts), etc. In this book, if no units are supplied with the decibels, we will assume we mean "dBV" (decibels, volts).

Voltage

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

This is to describe the difference between the input voltage to a system, and the output voltage from the system. 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.

Power Gain

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

This is used to compare the input power of a system to the output power of a system. Notice that the only difference between the equations for power gain and voltage gain is the coefficient out front.

Notes on Decibels

Template:SideBox 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 place where "gain" is used in an equation, decibels can not be used, because decibels are not numbers. Many people get this confused, so don't worry if it takes a while for this to sink in.

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 catagory 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.

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:

$Z(j\omega )=0$

$P(j\omega )=0$

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:

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

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 $({\omega }_N+j\omega )$. 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 ${\omega }_N$ 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 then the break frequency, the term becomes much greater then 1. When the radial Frequency is much smaller then 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 then" and ">>" to mean "Is much greater then". 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:

$Magnitude=20{\log }_{10}(DCGain)$

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:

$Gain=(1+0)=1$

$Magnitude=20{\log }_{10}(1)=0db/decade$

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

$Gain=(1+1)=2$

$Magnitude=20{\log }_{10}(2)=3db/decade$

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

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

$Magnitude=20{\log }_{10}(10)=20db/decade$

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!