Estimating the frequency response of L and C circuits

1. Impedance estimates

You recall the common approximations for how to deal with inductors and capacitors at DC and also very large frequencies.^[1]

Table 1. Impedances of elements at extreme frequencies.
\(f\)	\(\!\rightarrow\! 0\)	\( \!\rightarrow\! \infty\)
Z_inductor = \(sL\)	short	open
Z_capacitor = \(\dfrac{1}{sC}\)	open	short

Look at each of the four places in Table 1 and think about what each one means and where (in frequency) it is located. See in your mind’s eye a wire representing a short-circuit and a nothing for the open-circuit, like in Figure 1.

Figure 1. Open or short approximations

Replacing a device with an open- or short-circuit usually makes the simplified circuit easy to analyze.

Let’s quickly review these approximations by adding some spaced-repetition prompts. Remember, login with your Valpo email and you will receive an email reminder to review these questions after several days.

Read the prompt, think of the answer, then click Show Answer. Click Forgotten or Remembered as appropriate. (The system will show the prompt earlier or later according to your response.)

2. First-order circuits

Figure 2 shows R·C and R·L circuits in all possible series-shunt configurations.^[2]

Figure 2. First-order circuit combinations.

Refer back to Figure 2 for the following questions.

First answer the question in your head; only after you’ve answered should you click to reveal the answer.^[3]

Which circuit(s) have out shorted to in at DC? (Refer to them like graph quadrants)

Quadrant I

Which circuit(s) have out shorted to the lower node at DC?

Quadrant IV

Which circuit(s) shorts out to in at very large frequencies?

Quadrant III

Which circuit(s) blocks very high frequencies from appearing at out by short-circuiting out to the lower node?

Quadrant II

Which circuits are appropriately named low-pass filters?

Quadrants I and II

Which circuits are appropriately named high-cut filters?

Trick question! Also quadrants I and II. Sometimes circuits for audio applications use this terminology.

Which circuits should be called high-pass filters?

Quadrants III and IV

The time constants for these circuits are \(\tau = RC\) and \(\tau = L/R\), corresponding to special frequencies (−3 dB frequencies here) of \(\frac{1}{2\pi RC}\) and \(\frac{1}{2\pi\, \left. ^L \!/\! _R \right.}\), respectively. Remember that the −3 dB point is where the output magnitude is 3 dB lower than the “flat” portion of the response.

−3 dB is a power ratio of \(1/2\),
this is a voltage ratio of \(\frac{1}{\sqrt{2}} \approx 0.707\)

3. Second-order circuits

Relationships (between V and I) can be complex(-valued)

Here is a summary of all of the terminology around the relationship between V and I.

\(Z = \dfrac{V}{I}\), in units of ohms (Ω)

\(Y = \dfrac{I}{V}\), in units of siemens (S) or mhos (℧)

\(G = \dfrac{1}{R}\) only when \(X=B=0\)

3.1. First example

Consider operation at DC or very low frequencies:

The series inductor becomes a -circuit.
while the shunt capacitor becomes a -circuit.
Therefore, low frequencies (do | do not)^[4] pass through this filter.

What about very high frequencies?

The series inductor becomes a -circuit.
while the shunt capacitor becomes a -circuit.
Therefore, high frequencies (do | do not) pass through this filter.

Do you notice how the passing or blocking of signals is double-reinforced with the arrangement of open- and short-circuits?

What is an appropriate name for a filter of this behavior?

“low-pass or high-cut filter”

3.2. Second example

Describe what happens at DC or low frequencies. Finish your mental description before revealing!

series capacitor is an open-circuit
shunt inductor is a short-circuit

→ output is near zero

Describe what happens at high frequencies.

series capacitor is a short-circuit
shunt inductor is an open-circuit

→ output is same as v_in

Because the low’s get cut off, this is a low-cut filter.

Because the high’s pass through, this is a high-pass filter.

…same thing.

3.3. Parallel LC

We most frequently deal with the relationship between V and I for an element by its impedance (V/I), but here it is easier to deal with admittance.

Parallel elements have an equivalent impedance of \(Z_{eq} = \left( \dfrac{1}{Z_1} + \dfrac{1}{Z_2} \right)^{-1}\) and an equivalent admittance of simply \(Y_{eq} = Y_1 + Y_2\). If you remember the impedances of reactive elements, the admittances are simply their recipricals:^[5]

\(Y_C = j\omega C\) and \(Y_L = \dfrac{-j}{\omega L}\)

So, an L-C in parallel has an admittance of \(Y_{L\parallel C}=j\omega C - \dfrac{j}{\omega L}\).

Both low and high frequencies yield a short-circuit at the output node to 0 V, so this is neither a low-pass nor a high-pass filter. What happens at middle frequencies?

Look back carefully at \(Y_{L\parallel C}\) and notice that the term can be zero if ω is the right value.

What value of ω makes \(Y_{L\parallel C}\) ? Do the algebra yourself first!

\(w_0 = \pm \dfrac{1}{\sqrt{LC}}\)

Yes, the precise mathematical solutions give two possibilites. Negative frequency is a thing, expecially in digital signal processing (DSP). If your equations come from real circuits (that you could build), the frequencies always come in pairs that are mirror images about the real axis. Because of this, we never worry about negative frequencies in physical circuits.

Remember that admittance is \(\frac{I}{V}\) and a value of zero would mean that zero current flows no matter the voltage difference. This is simply the description of an open-circuit.

So, at some middle frequency, the parallel LC behaves like an open-circuit, allowing the input signal to pass through the circuit at full-scale.

Now your turn. Use what you know to sketch the frequency response of this filter.

3.4. Series LC

Use what you know to sketch the frequency response of the following two RLC circuits.

Will you use impedance or admittance to find the LC series equivalent value?

impedance, of course, because Z's add in series.

What happens at the special frequency?

Elsie is a program from Tonne Software that designs, analyzes, and plots lumped-element (L-C) filters. Combine engineers' tendency to name things plainly with a dry sense of humor and this is what you get :)

1. Or synonomoysly: very rapid changes.

2. The vertical part on the left side with the horizontal part hanging out to the right would be called a “shunt-series” circuit.

3. Clicking first without thinking reduces the chance you can answer something similar on an exam later by about 30% — don’t be lazy!

4. There is no try.

5. Only because the real part is zero.