Question

Given the following voltage transfer function: $$\begin{aligned} H(s) &=\frac{V_{o}}{V_{i}} \\ &=\frac{4 \times 10^{6}}{s^{2}+500 s+4 \times 10^{6}} \end{aligned}$$ a) At what frequencies (in radians per second) is the ratio of \(V_{o} / V_{i}\) equal to unity? b) At what frequency is the ratio maximum? c) What is the maximum value of the ratio?

Short answer

a) Unity gain at \( \omega = 2872.28 \) rad/s; b) Maximum ratio at \( \omega = 2000 \) rad/s; c) Maximum value is 4.

Step-by-step solution

Understand H(s)

The voltage transfer function is given by \( H(s) = \frac{4 \times 10^{6}}{s^{2} + 500s + 4 \times 10^{6}} \). This can be interpreted as a standard second-order system, where we need to find its response characteristics.

Find the Unity Gain Frequencies

To find the frequencies where the gain \( |H(j\omega)| = 1 \), set the magnitude of \( H(j\omega) \) equal to 1:\[|H(j\omega)| = \frac{4 \times 10^{6}}{\sqrt{(4 \times 10^{6} - \omega^2)^2 + (500\omega)^2}} = 1\]Solve for \( \omega \): \[4 \times 10^{6} = \sqrt{(4 \times 10^{6} - \omega^2)^2 + 250000\omega^2}\]Squaring both sides gives \[(4 \times 10^{6})^2 = (4 \times 10^{6} - \omega^2)^2 + 250000\omega^2\]Simplifying this will yield the frequencies where the gain is unity.

Simplify for Unity Gain Frequencies

After expanding and simplifying the square, we have:\[(4 \times 10^{6})^2 = (4 \times 10^{6} - \omega^2)^2 + 250000\omega^2\]\[16 \times 10^{12} = 16 \times 10^{12} - 8 \times 10^{6} \omega^2 + \omega^4 + 250000\omega^2\]\[0 = \omega^4 - 8 \times 10^{6} \omega^2 + 250000\omega^2\]This becomes:\[\omega^4 - 8250000 \omega^2 = 0\]\[\omega^2(\omega^2 - 8250000) = 0\]Therefore, \( \omega = 0 \) and \( \omega^2 = 8250000 \), which gives \( \omega = \sqrt{8250000} \approx 2872.28 \) rad/s.

Find the Frequency of Maximum Ratio

The maximum gain in a second-order system occurs at the undamped natural frequency \( \omega_n \), where \( \omega_n = \sqrt{k} \), with \( k = 4 \times 10^6 \). Thus, \( \omega_n = \sqrt{4 \times 10^6} = 2000 \) rad/s.

Calculate Maximum Value of the Ratio

The maximum value of the ratio is found using the peak gain of the transfer function. Evaluating the transfer function at the resonance frequency:\[|H(j \omega_n)| = \frac{4 \times 10^6}{\sqrt{(4 \times 10^6 - 2000^2)^2 + (2000 \times 500)^2}}\]Substitute \( \omega_n = 2000 \):\[|H(j2000)| = \frac{4 \times 10^6}{\sqrt{(4 \times 10^6 - 4 \times 10^6)^2 + (10^6)^2}} = 4\]Thus, the maximum value is 4.

Key concepts

Unity Gain Frequency

The concept of unity gain frequency, or 0 dB gain, in a voltage transfer function is a critical point for understanding system behaviors and characteristics. In simple terms, it's the frequency at which the magnitude of the transfer function is equal to one. This occurs when the output voltage equals the input voltage. For a given transfer function, such as the one in our exercise, the unity gain frequency is determined by setting the magnitude of the function equal to one and solving for the frequency. The formula for the magnitude can be expressed as:\[ |H(j\omega)| = \frac{4 \times 10^{6}}{\sqrt{(4 \times 10^{6} - \omega^2)^2 + (500\omega)^2}} \]By setting this expression equal to one and solving, we find the frequencies where gain is unity. Solving the equation involves breaking down the expression and simplifying mathematically to find that the unity gain occurs at specific frequencies including \(\omega = 0\) and \(\omega = \sqrt{8250000} \approx 2872.28\) rad/s. Unity gain is especially important in control systems and signal processing, indicating the point where amplification transitions to attenuation.

Resonant Frequency

The resonant frequency is defined as the frequency at which a system naturally oscillates with maximum amplitude. It's a significant concept in the context of second-order systems because it indicates the frequency at which energy input to the system is most efficient. In the provided exercise, resonant frequency (\(\omega_n\)) is calculated as the square root of the constant term from the characteristic equation of the transfer function:\[ \omega_n = \sqrt{4 \times 10^6} = 2000 \text{ rad/s} \]At resonant frequency, the gain of the system is at its peak. This is the frequency where the physical components of the system, like inductors and capacitors in an electrical circuit, align to boost the system's response. Understanding and identifying the resonant frequency enable engineers to predict how a system might behave in response to various inputs, and it can be critical for ensuring stability and avoiding unwanted oscillations.

Second Order System Analysis

Analyzing a second-order system involves exploring its standard form, which is characterized by a second-order differential equation. The general form of a second-order system transfer function is:\[ H(s) = \frac{k}{s^2 + 2\zeta\omega_n s + \omega_n^2} \]Where \(\zeta\) is the damping ratio and \(\omega_n\) is the undamped natural frequency. In the given exercise, the transfer function outlined fits this model, allowing for insights into its behavior in both time and frequency domains.Key characteristics such as unity gain frequency, resonant frequency, and maximum gain are derived from second-order system analyses. These features help in predicting how the system will respond to different inputs and in identifying potential issues such as resonance-induced instability.Parameters like \(\zeta\) dictate how quickly the system can return to equilibrium after a disturbance. A low \(\zeta\) might lead to prolonged oscillations, while a high value results in a faster settling time. Comprehensive analysis includes calculating these parameters and assessing their impact on system dynamics.