# Are cutoff frequencies of bandpass and bandstop filters equal?

Consider the two filter in picture

simulate this circuit – Schematic created using CircuitLab

Are the cutoff frequencies (i.e. the two frequencies at which $V_{out}=\frac{1}{\sqrt{2}} V_{in}$) the same for the two filters? (And therefore also the bandwidth is the same for the two filters?)

I think they should be the same since in both cases the condition $V_{out}=\frac{1}{\sqrt{2}} V_{in}$ reduces to

$$R^2=(\omega L-\frac{1}{\omega C})^2$$

Nevetheless it is strange that there is a frequency for which both $V_{out}$s taken at resistor and at LC series have the same value $\frac{1}{\sqrt{2}} V_{in}$.

Doesn't this violates Kirckoff voltage law? (At any time it should be $V_{in}=V_{R}+V_{LC\ series}$ but if $V_{R}=V_{LC\ series}=\frac{1}{\sqrt{2}} V_{in}$ then this is not true).

• When - at a certain frequency - the voltage across R is 70.7% of the total voltage (bandpass cut-off), you can calculate by yourself which voltage will be across the L-C combination at the same time. It is up to you to decide if it makes sense to declare this frequency as the band-stop cut-off. – LvW Mar 20 '17 at 15:46
• @LvW Thanks for the comment, the voltage across LC would be $\sim 30 \%$ but isn't the definition of cutoff frequenciies for band stop filter "the frequencies when $V_{out}=\frac{1}{\sqrt{2}} V_{in}$"? (as in the picuture I added in the question) – Sørën Mar 20 '17 at 15:55

We can apply the fast analytical techniques here (FACTs) and they show that if you reduce the excitation voltage $V_{in}$ to 0 V in your example (replace the source by a short circuit), the structure remains unchanged regardless where you observe $V_{out}$. Therefore, the denominator $D(s)$ is the same between the two schematics you have drawn. The principle is always the same, observe the circuit for $s=0$ first and then determine the time constants combining the various energy-storing elements to form the denominator $D(s)$. Look at the below schematic, it is quite easy to follow:

The denominator is obtained by assembling the time constants the following way:

$D(s)=1+s(\tau_1+\tau_2)+s^2(\tau_2\tau_{21})$

You can then rearrange it under the classical canonical form

$D(s)=1+\frac{s}{\omega_0Q}+(\frac{s}{\omega_0})^2$

The zero are obtained by observing what impedance combination could become a transformed short circuit nulling $V_{out}$ when $s=s_z$? Obviously this is when the series impedance made of $C_1$ and $L_2$ becomes a transformed short circuit. If you solve $Z_1(s)=0$, then $1+s^2L_1C_1=0$ and you have immediately $\omega_{0N}=\frac{1}{\sqrt{L_1C_2}}$. The final transfer function is given in the below Mathcad sheet for the bandstop filter:

For the bandpass filter, we already have $D(s)$ so no need to re-derive it. It is easier to slide resistor $R_1$ and redraw the schematic. This time, capacitor $C_2$ places a pole at the origin and blocks dc. To obtain the zero, we can apply the generalized transfer function expression

$N(s)=H_0+s(H^1\tau_1+H^2\tau_2)+s^2H^{12}\tau_2\tau_{21}$

In our case, $H_0$ is 0 since $C_2$ blocks the dc. Therefore, we can calculate the remaining gains quite easily as shown below:

Once this is done, assemble the calculated terms as shown in the Mathcad screenshot below

I even rearranged the transfer function in a low-entropy form showing a 0-dB gain at the peak.

The FACTs are truly an excellent way of deriving transfer functions in a swift and efficient manner. Very often, in particular with passive circuits, the polynomial expressions can be formed by inspection without writing a single line of algebra: just draw small sketches and determine the $a_i$ and $b_i$ terms for $N$ or $D$ individually. If you see a mistake, just correct the guilty term without re-starting from scratch. Of course, when you tackle circuits featuring active sources like current- or voltage-controlled sources, you often need to resort to KVL and KCL but the obtained result is always expressed in a meaningful polynomial form and it is easy to correct in case a typo is detected. If you want to know more about FACTs, have a look at the seminar taught at APEC 2016