Circuits with magnetic coupling

1. Coils and magnetic flux

Ampere’s (circuital) law relates current to the magnetic field in the space around a closed path around the current. This law works both ways: (a) knowing the current gives the magnetic field, and (b) knowing the magnetic field lets us know the total current.

We need to be careful because the total current is the current you know from circuit analysis — that which flows through wires and devices — and bound current. Bound current itself has two parts to it:

Magnetization current, from the net motion of electrons still bound to their atoms in a certain direction.
Polarization current, from the separation of electrons within a material from a changing external electric field.

ALL of these types of current are from the motion of electrons, so in one sense there is no microscopic distincition.

References

2. Magnetic induction

Faraday’s law of induction relates a changing magnetic field to an induced (read caused) voltage around a loop whose area intercepts the field.

This caused electromotive force (EMF), measured in volts, has a sign that opposed the change. In other words, if a resistor was placed across this voltage, the current that flows will generate a magnetic field (Ampere’s law) that is in the opposite direction of the original field.

References

3. Inductance

By now, you know that inductance is the main property of an inductor and it relates the change in current through an inductor to the voltage across it. This is a direct consequence of the combination of Ampere’s and Faraday’s laws when the voltage and the current (and magnetic flux) are all related to the same coil.

\[v_L(t) = L \dfrac{\mathrm{d}}{\mathrm{dt}} \bigl[ I_L(t) \bigr]\]

4. 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.
\(f\)	\(\!\rightarrow\! 0\)	\( \!\rightarrow\! \infty\)
inductor \(sL\)	short	open
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

Replacing with a short- or open-circuit usually makes the resulting circuit easy to analyze. Let’s quickly go over these approximations by adding some spaced-repetition prompts. Remember, login with your Valpo email and you will receive a few email reminders to review the questions after several days.

5. First-order circuits

RC and RL in all possible series-shunt configurations.

Figure 2. First-order circuit combinations.

Refer 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.^[2]

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 0 V 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 0 V?

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 inf 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 half-power
voltage ratio of \(\frac{1}{\sqrt{2}} \approx 0.707\)

6. Second-order circuits

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

\(Z = \dfrac{V}{I}\)

\(Y = \dfrac{I}{V}\)

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

6.1. First example

Consider operation at DC or very low frequencies:

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

What about very high frequencies?

The series inductor becomes a
while the shunt capacitor becomes a .
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”

6.2. 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

1. Or synonomoysly: very rapid changes.

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