Useful Symbols: 
α β Ψ Ω π Γ Φ ρ ζ • ρ ∞ ° ω ℓ ℒ σ ℛ μ_{r} μ_{0} ε_{r} ε_{0} ± ∓ ≈ η& ≠ Φ ÷ × √ Δ ∂ ∫ Ʃ θ « » Σ Φ ε κ ² ³ ½ ¼ ¾ ⅐ ⅑ ⅓ ⅔ ⅒ ⅕ ⅖ ⅗ ⅘ ⅙ ⅚ ⅛ ⅜ ⅝ ⅞ ≡ Ξ ∑ Π © & ® δ λ ∮ ∴ ⇔ ⇒ ∝ ∞ ∠ ∡ ∴ ⊕ ⊖ ⊗ ⊙ ⦾ ✓ 
2^{nd} ORDER LOW PASS FILTER DESIGN  
\(H(s)=\dfrac{\omega_n^2}{s^2+2\zeta\omega_ns+\omega_n^2}\) To find the poles, equate the denominator \(s = \dfrac{2\zeta\omega_n\pm\sqrt{4\zeta^2\omega_n^24\omega_n^2}}{2}\) \(s = \zeta\omega_n\pm\sqrt{\zeta^2\omega_n^2\omega_n^2}\) \(s=\omega_n(\zeta\pm\sqrt{\zeta^21})\) 


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POLEZERO DIAGRAM


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THREE DIMENSIONAL VIEW


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PEAKING FREQ AND AMPLITUDE \(H(j\omega)=\dfrac{1}{1\dfrac{\omega^2}{\omega_n^2}+\dfrac{j2\zeta\omega}{\omega_n}}\)

This is the peaking frequency. To find the peak
magnitude, replace the peaking frequency in the formula for \(H(j\omega)\): 
\(H(j\omega) = \dfrac{1}{1(\sqrt{12\zeta^2})^2+j2\zeta\sqrt{12\zeta^2}}=\dfrac{1}{2\zeta^2+j2\zeta\sqrt{12\zeta^2}}\)
Then convert to a magnitude value by squaring the terms: 
\(H(j\omega)= \dfrac{1}{\sqrt{(2\zeta^2)^2+4\zeta^2(12\zeta^2)}}\) =
\(\dfrac{1}{\sqrt{4\zeta^4+4\zeta^2 8\zeta^4}}\) = \(=\dfrac{1}{2\zeta\sqrt{1\zeta^2}}\)
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QUALITY FACTOR, AMPLITUDE AND PHASE AT \(\omega_n\)
\(H(j\omega)=\dfrac{1} {1\dfrac{\omega^2}{\omega_n^2}+\dfrac{j2\zeta\omega}{\omega_n}}\) and if \(\omega = \omega_n\), then \(H(j\omega)=\dfrac{1}{11+j2\zeta} = jQ\) \(=Q\angle{90}\)
Consider the picture below. It has two identical windings that are closely magnetically connected.

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With the secondary unloaded, the current that flows into the primary is the magnetization current.
This current
\(I_{MAG} = \dfrac{230}{2\pi\times 50\times 10}\) = 73.2 mA RMS
The magnetization current is in addition to any primaryreferredsecondaryload current.
However, it is only the magnetization current that produces magnetic flux in the transformer core.
If we looked
The rate at which primary current rises is dictated by \(V = L\dfrac{di}{dt}\Longrightarrow\dfrac{di}{dt} = \dfrac{V}{L}\)
So, if V is 1 volt and L = 10 henries, the primary current would rise at 100 mA per second and eventually start rising 
________________________________________________________________________________ Magnetization current is the "main" current that flows into the primary when the secondary is unloaded but eddy currents also can flow. Eddy currents circulate in laminates that make up the core. If the core was one solid metal block it would be like a shorted turn so insulated laminates are used to reduce the magnitude of eddy currents.
If the core is ferrite, the ferrite material is designed to be nonconducting hence it exhibits very small eddy current loss and can be used in quite high frequency applications. 
________________________________________________________________________________ LEAKAGE INDUCTANCE AND COPPER LOSSES When coils are wound in close proximity inevitably the flux produced is not 100% coupled to the secondary winding. Coupling (k) might be about 98% so, if the primary inductance is 10 henries only 9.8 henries will couple magnetic flux to the secondary. In effect 200 mH can be regarded as an external series component that will drop voltage under loaded conditions and lead to reduced regulation performance.

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TRANSFORMER EQUIVALENT CIRCUIT

When you apply electrical current to a coil that surrounds a closed magnetic core, you get a certain flux density in that core. However, if you apply too much current through too many turns on a core with inadequate geometry you will get magnetic saturation. It is advisable to work well within saturation limits such as those shown by the blue BH curve below: 
Flux densities of 200 mT, 300 mT and 400 mT are projected to the middle of the 25°C curves to obtain a Hfield values of 23, 46 and 125 At/m. As you can see, to produce 300 mT requires twice the Hfield than at 200 mT. To double the flux to 400 mT requires a Hfield that is over 5 times larger. In other words, saturation effects are happening even at 200 mT.

____________________________________________________________________________________________________ RELUCTANCE, MAGNETO MOTIVE FORCE, H, FLUX, B AND CORE DIMENSIONS In the previous section it was shown that B and H are interrelated. This relationship is founded in the following basic equations: 
\(A_e\) is the effective cross sectional area of the magnetic core \(\mu\) is the magnetic permeability (absolute not relative) of the magnetic core
The driving force behind magnetism is current and the term magneto motive force (MMF or \(F_M\)) is the product of amps
and turns. The total flux produced in a magnetic core is simply this: 
\(\Phi\) = \(\dfrac{F_M}{R_M}\) Using the formula for reluctance we can say: 

____________________________________________________________________________________________________ If you require a stable inductor, a gap can reduce the flux density considerably. For instance, if the initial permeability is 2000 and the core length is 100 mm (with a 1 mm gap), the resulting effective permeability is reduced to a value of 95.24: 
\(\ell_e\) is the effective length of the magnetic core, \(\mu_i\) and \(\mu_e\) are initial and effective permeabilities.
For 3F3 material with a 1.1 mm gap, \(\mu_e\) has fallen to 45 from 1520.
For the same 200 mT flux density, comparing 23 At/m (11 turns) and 777 At/m (64 turns) means the gapped inductor realizes a peak
current that is about 5.8 times higher than the ungapped inductor. However, more turns means more copper loss so there is give and take.

____________________________________________________________________________________________________ SIMULATING THE GAPPED AND UNGAPPED 3F3 E38/8/25 CORESET
Below is MicroCap 12's BH simulation of two inductors placed in series.
The black trace is the ungapped 3F3, E28/8/25 coreplate set (11 turns) and the green trace is the 1.1 mm gapped coreplate set (64 turns): 

____________________________________________________________________________________________________ TEMPERATURE EFFECTS ON MAGNETIC PERMEABILITY Consider the example of Ferroxcube's 3F3 material and how permeability changes with temperature: 
At 25°C the permeability is 2000 but will rise to about 3300 at 100°C. Clearly this is undesirable when designing an inductor because inductance will rise proportionately. Gapping however comes to the rescue because temperature dependence is reduced by the factor:  \(\dfrac{\mu_e}{\mu_i}\) Using the example of the planar 3F3 E38/8/25 coreset discussed previously: 
This is very important to achieving a stable inductor value. 
The Ferroxcube E38/8/25 3F3 coreset is considered: 
The core could surround a primary of 8 PCB layers with 8 turns on each layer. A ninth layer would be required for the secondary
winding (assuming that it can be accommodated on one layer). It's tight (and a little hypothetical) but doable: 

still under construction