Why is my zero and pole frequency of this transfer function wrong?

Question

So I have this transfer function: $$\begin{align} \dfrac{g_m-sC_{gd}}{s(C_{gs}+C_{gd})}=\dfrac{-C_{gd}\left(s-\dfrac{g_m}{C_{gd}}\right)}{(C_{gd}+C_{gs})\left(s-\dfrac{0}{C_{gd}+C_{gs}}\right)}=-\dfrac{C_{gd}}{C_{gd}+C_{gs}}\dfrac{s-\dfrac{g_m}{C_{gd}}}{s} \end{align}$$

Then im supposed to find the zero and pole frequency expression based from my transfer function and this is what I got: $$ ω_{z} = \frac{-gm}{C_{gd}} $$ $$ ω_{p} = \frac{1}{C_{gd} + C_{gs}} $$

But then that two is wrong. But I dont understand why. Then we are asked for the gain to when ω = ω_z and this is my formula:

$$ A_{v},dB = 20\log\left|\frac{C_{gd}*\sqrt{1+(\frac{\omega_{z}}{\omega_{z}})^2}}{ \sqrt{(-\frac{ω_{z}}{w_{p}})^2}}\right| $$

The above formula for my gain is also wrong but i guess thats because my formula for the zero and pole frequency is also wrong. So where are my mistakes here? Help would be much appreaciated!

Answer 1

When you write a transfer function, it is important to respect a so-called low-entropy format where the poles and the zeroes clearly appear. In your case, for instance, the correct expression should be in the form of: \$H(s)=\frac{1-\frac{s}{\omega_z}}{\frac{s}{\omega_{po}}}\$.

The numerator hosts a right-half-plane zero (a RHPZ) while the denominator shows a pole located at the origin for \$s=0\$. By factoring \$g_m\$ in the numerator, you can determine the zero and what I call the 0-dB crossover pole. Rearranging, you have \$\omega_z=\frac{g_m}{C_{gd}}\$ and \$\omega_{po}=\frac{g_m}{C_{gd}+C_{gs}}\$. Please note that you always need to verify the homogeneity of your formulas: a pole or a zero is the inverse of a time constant expressed in seconds [s]. As \$g_{m}\$ is expressed in [A]/[V], we are ok.

Now, this expression is not the best we can do because it brings no insight in what is going on with this formula. The most compact formula uses what is called an inverted zero: from your first expression, format \$sC_{gd}\$, simplify by \$s\$ and you now have \$H(s)=H_{inf}(1-\frac{\omega_z}{s})\$ naturally highlighting the gain of this expression when \$s\$ approaches infinity: \$H_{inf}=\frac{C_{gd}}{C_{gd}+C{gs}}\$.

I have gathered these different expressions in the below Mathcad sheet and they are identical:

Many times, I have seen people confused with transfer functions that are not optimally written. Properly rearranging a transfer function with a leading term (when it exists) followed by a dimensionless quotient is the way to go.