# Understanding gain crossover frequency

• evolosophy
In summary: So in summary, the gain crossover frequency for the given transfer function is approximately 14.8023.
evolosophy
Hello. This technically isn't homework, however I am studying for an upcoming final. I feel like I must be missing something. I'm trying to calculate the phase margin of a system so my first step is to find the gain crossover frequency (GCF).

## Homework Statement

Transfer function: $$H = \frac{H}{s(s^2+7s+140)}$$

## Homework Equations

GCF occurs where the magnitude of the gain = 1 , or $$|H(j \omega_{GC})|=1$$ where $$s= j \omega_{GC}$$

## The Attempt at a Solution

For the sake of brevity I'll skip most of the details of the algebra. I start with,

$$\left| \frac{20}{j \omega ((j \omega)^2 + 7 j \omega + 140)} \right| = 1$$

By definition of GCF I'm interested only in magnitude. After factoring the 2nd order term , cross-multiplying to get 20/1, and taking the norm of each root the above equation leads to,

$$20 = (\sqrt{-\omega^2}) \left( \sqrt{ \left( j \omega + j \frac{\sqrt{511}}{2} \right)^2 + \left( \frac{7}{2} \right)^2} \right) \left( \sqrt{ \left(j \omega - j \frac{\sqrt{511}}{2} \right)^2 + \left( \frac{7}{2} \right)^2 } \right)$$

At this point I just crank through the algebra (which I'll skip) and end up with,

$$400 = -\omega^6 + 280 \omega^4 - \frac{53361 \omega^2}{4}$$

Finally solving for omega I get,

$$\omega = \pm j 0.173106$$
$$\omega = \pm 7.80531$$
$$\omega = \pm 14.8023$$

Now we finally arrive at my problem. Plotting the original function shows the magnitude decreasing for all time. This makes sense as there are no zeros and 3 poles. So I should only have a single crossover frequency (also shown on a bode plot). So the question is how do I know which of the above values for omega is the actual gain crossover frequency?

I'm trying to solve this without the use of any plots.

Any help would be greatly appreciated! Thanks. :)

Edit: I should say that I do recognize that a negative frequency doesn't make sense so I only have 3 options and not 6. I think I can eliminate the option with the j term for the same reason. Yes, no?

Last edited:
I realize this is 2 years late... but better late than never right?

Your math is wrong. You should have ended up with

$\omega^{6} - 231\omega^{4}+19600\omega^{2} = 400$

There is only one real, positive solution to that equation, which is the gain crossover frequency.

## 1. What is gain crossover frequency?

Gain crossover frequency is the frequency at which the magnitude of the open-loop gain of a system is equal to 1 (0 dB) on a Bode plot. It is also known as the unity-gain frequency and is an important parameter in understanding the stability of a control system.

## 2. Why is gain crossover frequency important?

Gain crossover frequency is important because it indicates the frequency at which the feedback control system transitions from amplifying to attenuating the input signal. It is also a key factor in determining the stability and performance of a control system.

## 3. How is gain crossover frequency calculated?

Gain crossover frequency is typically calculated by plotting the open-loop transfer function of the system on a Bode plot and finding the frequency at which the magnitude is equal to 1 (0 dB). It can also be calculated using the formula: Gc(jωgc) = 1, where Gc is the open-loop transfer function and ωgc is the gain crossover frequency.

## 4. What is the relationship between gain crossover frequency and phase crossover frequency?

Gain crossover frequency and phase crossover frequency are two important parameters in understanding the stability and performance of a control system. They are related through the phase margin, which is the amount of phase difference between the actual phase of the system and -180 degrees at the gain crossover frequency. A higher phase margin indicates a more stable system.

## 5. How does gain crossover frequency affect the stability of a control system?

The gain crossover frequency has a direct impact on the stability of a control system. A higher gain crossover frequency indicates a more stable system, as it allows for a larger phase margin. On the other hand, a lower gain crossover frequency can lead to instability and oscillations in the system. It is important to design control systems with an appropriate gain crossover frequency to ensure stability and performance.

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