We often encounter a wide range of frequencies when plotting the frequency response of a circuit. This makes plotting the response via the transfer function with a linear scale for the frequency axis a rather difficult task. As a result, it is standard practice to plot the transfer function on a pair of semi-logarithmic plots. These are known as "Bode plots". A Bode plot has two separate plots depicting the following:
- Magnitude (in decibels) plotted against the logarithm of the frequency.
- Phase (in degrees) plotted against the logarithm of the frequency
These two plots contain the same information as a non-logarithmic plot but are easier to construct.
Bode Plots and the Transfer Function
Consider the transfer function as expressed in its polar and exponential form: $$ \mathbb{H} = H\angle \phi = He^{j\phi} $$ If we take the natural log of both sides we get: $$ \ln \mathbb{H} = \ln He^{j\phi} $$ $$\quad \; \; \; = \ln H + \ln e^{j\phi} \quad (recall: \; \ln e^x = x)$$ $$ \ln \mathbb{H} = \ln H + j\phi $$ We see that the above expression has both a real and imaginary part where: $$ I_m \{ \mathbb{H} \} = phase $$ $$ R_e \{ \mathbb{H} \} = gain = H_{dB} = 20 \log_{10}H $$ For the two plots that make up a Bode plot, this gain and phase is plotted against frequency.
Plotting the factors of a transfer function
Consider the transfer function represented in the following form: $$ \mathbb{H}(\omega) = \frac{\mathbb{N}(\omega)}{\mathbb{D}(\omega)} $$ where: $$ \mathbb{N}(\omega) = numerator \; polynomial $$ $$ \mathbb{D}(\omega) = denominator \; polynomial $$ This expression may be written in the following standard form (in terms of factors having real and imaginary parts):
Transfer function standard form:
$$ \mathbb{H}(\omega) = \frac{K(j\omega)^{\pm1} \Big(1+\frac{j\omega}{z_1}\Big) \Big[ 1+j2\zeta \frac{\omega}{\omega_k}+ \Big( \frac{j\omega}{\omega_k} \Big)^2 \Big]} {\Big( 1+\frac{j\omega}{p_1} \Big) \Big[1+j2\zeta_2 \frac{\omega}{\omega_n}+ \Big( \frac{j\omega}{\omega_n} \Big)^2 ... } \qquad, (Eqn\;1) $$ While equation 1 is considered "standard form" it is but one of several different representations that contain some number of the following types of factors:
Transfer Function Factors:
| $$ \qquad \qquad K = $$ | $$ Gain \; (constant) $$ |
| $$ \qquad \; (j\omega)^{-1} = \frac{1}{j\omega} = $$ | $$ pole \; at \; origin $$ |
| $$ \qquad \qquad j\omega = $$ | $$ zero \; at \; origin $$ |
| $$ \qquad \frac{1}{1+\frac{j\omega}{p_1}} = $$ | $$ Real/Simple \; pole $$ $$ (not \; at \; origin) $$ |
| $$ \qquad 1+\frac{j\omega}{z_1} = $$ | $$ Real/Simple \; zero $$ $$ (not \; at \; origin) $$ |
| $$ \frac{1}{1+j2\zeta_2(\frac{\omega}{\omega_n})+(\frac{j\omega}{\omega_n})^2} = $$ | $$ Complex(Quadratic) \; pole $$ |
| $$ 1+j2\zeta_1\Big(\frac{\omega}{\omega_k}\Big)+\Big(\frac{j\omega}{\omega_k}\Big)^2 = $$ | $$ Complex(Quadratic) \; zero $$ |
A transfer function may contain one or more of these terms. The goal is to take the frequency response of each factor and use the properties of logarithms to add them together in a Bode Plot. In the next several sections we will look at each of these Transfer Function factors more closely.
Continue on to Bode Plots and constant factors...