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LECTURE 33
Inductor Design
A. Overview of Copper versus Core Loss in Inductors
1. Core Material Limitations
2. Core Materials Compared
3.“Filter” Inductor Design via Erickson’s Four
Step Design Rules
4. Ten Commandments For Inductor Design
5. Summary
B. Inclusion of Core Losses and Relation to Wire
Winding Losses
1. Sinusoidal Flux Density B(wt) Driving the Core
2. Bpeak ~
∫ vdt
NA c (core) NAc
THIS ALSO SAYS N~ 1/B that is as B
decreases the number of turns will increase
3. Tailoring Bopt for Minimizing Total Losses:
a. Trading Fe vs Cu to Achieve Minimum Total
Loss
b. N↑ makes for larger I^2 R(wire) losses
c. N↓ makes for higher Bpeak and more core
losses
The various core geometry’s are shown below
For a term paper on Integrated Inductor Design See as
a starting point the article “LAYOUT OF
INTEGRATED RF SPIRAL INDUCTOR” in
Circuits and Devices March 1998 pgs 9-
B^2 (SAT) l(core) A(core) /2μ. A convoluted inter-dependence of the wire current and the magnetic field in a transformer core is as follows. The volt-sec limit says that B(core)= (^) λp =∫V (^) Ldt /N AC. Hence, we find that N~1/B and later we will
employ N^2 ~ 1/B^2 in relations for the copper loss to show PCu ~1/B^2. As regards the inductor voltage applied without considering the equivalent series resistance rL, iL on a DC basis will continue to increase with a DC VL ,since all of the applied voltage appears across vL. With rL the inductor current increases until iLrL equals the input voltage. Again the B-H curve of the inductor core looks like a ∫ELdt versus IL curve as shown below. At high ∫Edt,I → ∞ where the core saturates and L → 0. For a flux or B change to occur a time interval is required over which the applied inductor voltage is applied.
In high frequency PWM applications, the major magnetic core material limitations are: 1. Flux Density Saturation and 2. Core losses, both of which depend upon flux swing and applied frequency of the flux. In these applications inductor windings are usually driven with rectangular voltage waveforms derived from low impedance sources. Since the voltage, pulse width, and number of turns are quite accurately known, it is easy to apply Faraday’s Law to determine the maximum flux swing and appropriately limit it.. In the case of a sinusoidal voltage, Faraday’s law gives V= N dϕ / dt =Nω AC BSAT. Hence V/ωNAc must be less than BSAT.
A flyback transformer is actually an inductor with multiple windings. It stores energy taken from the input in its mutual inductance during one portion of the switching period, then delivers energy to the output during a subsequent interval. In a flyback transformer, the magnetizing current is virtually important, because it represents the energy storage required by the application. In this case, the magnetizing current can be calculated quite accurately using Ampere’s Law, because it depends on the very predictable characteristics of the gap in series with the core, and the uncertain core contribution to energy storage is negligible. Magnetizing inductance is an essential element in a flyback transformer.
2. Core Materials Compared There are three major core materials of interest. Metal Alloy Tape Powdered Metal Ferrites
- Bsat = 1 Tesla 1. Bsat = ½ Tesla 1. Bsat = 0.2 Tesla
- μr = 60,000 2. μr = 100 2. μr = 1000
- ρc is low, high eddy current losses
- Highest Loss Core (^) 3. ρc is high, low eddy current losses
- Sharp Saturation suddenly L → 0
- Soft Saturation L(1/4 tesla) = ½L(B=0) Hence Im increaes gradually
- Sharp Saturation suddenly L → 0
For a slotted core inductor the “ winding your own L magnetic core” looks like that below. Most energy storage occurs in the air gap region as we showed in the prior lecture.
W(air) = ½B^2 l g/Agμo >> W(core) = ½B^2 l c/Acμc
r (^) L L (^) t
I (^) ac dominates
Ptotal loss = Pcu(loss) + Pcore(loss) Herein we will learn to trade Fe vs Cu in order to minimize the total loss. That is we can find an optimum number of turns of wire that provide minimum total loss. If N is too high I^2 R wire losses dominate. While if N is too low Bpeak = ∫Edt/NAc will rise and core losses dominate. The question is now how do we rationally trade off copper losses and core losses to achieve the desired inductor for the specific circuit use. While the graph below illustrates the balance needed to achieve minimum total loss, we will not fully understand this plot till lectures 34 –.
B B (^) opt
P (^) T
More Cu loss P (^) cu ~ 1/B 2
Surprise! result only true for transformers
As B we find More Core losses P (^) core ~ B 2 or B x as expected
for minumum loss
3. “Filter” Inductor Design via Ericksons Four Step Design Rules
We tailor the inductor design to the switched mode converter circuit topology. Design considerations for inductors include: n core size and permeability n effects of air gaps on reluctance and L linearity as well as the sensitivity of L to changes in the core n core loss limits n saturation values of flux density in the chosen material n core window size (for allowing all needed wire windings) n current density allowed in wire windings
n maximum values of ac voltage, dc volt-sec, and dc current n operating frequency 1 st^ Select core via the emperical Kg factor Both hysteresis and eddy current core loss effects in inductors are caused by time-varying flux. If an inductor carries a constant dc current below its saturation limit, the core flux will be constant, and the hysteresis and eddy current losses of the core will be zero. In practice any switch mode converter has large current ripple in the inductors, at the switching frequency. The flux variation will be follow the AC current. Magnetic material can handle only a limited loss per unit volume without getting too hot. Losses of 1 W/cm^3 are usually considered high for ferrites, and values in the range of 0.2 W/cm^3 are more common in well designed cores. At any rate we can translate circuit variations into a single parameter which helps to specify the core needed as shown in Lecture 32
g [ ]
(^2) max 2
pk
(^2) u K >^ L I^ (wire)^8 B R K
ρ
Get L, R, Imax from PWM converter electrical specs Get Bpk, Ku from manufacturers core and wire specs respectively. The core geometry gives us both Ac(core) and WA(wire winding area) as well as the mean length of turn wound on the core.
2 nd^ Spec the air gap to be cut in the inductor core
g o^
(^2) max
pk c
L I
B A
l [m]
μ
As an alternative to the gap size, l g, we can specify the AL, or specific
inductance in mH per turn^2 , factor for an inductor core. Nw is the number of wire turns and ℜ (^) c(core) is core reluctance including the airgap.
ℜ ℜ
≡ turn A L =^1 =^1 mH L N (^2) w c c^ - We arbitrarily fix N @ 1000 turns as core
manufacturers do so in their AL specifications. 1 < AL < 1000; AL is typically 100 for N = 1000 turns.
and the wire will overheat before the saturation amp-tum limit of the magnetic core is reached. If the wire-winding window is too large, then core saturation will be reached prematurely and the copper winding might be underutilized. Core size sets the mean length of turn required to encircle the core and hence the length per turn of the wire.
Rwire = (resistance per meter of the wire) x (length per turn) X N Or alternatively R=ρN (MLT)/AW(wire). MLT depends on core geometry. Cores made of sectioned structures tend to be easier to wind with automated wire winding machines than toroids. For toroid cores, windings are often designed to form a single layer of copper material around the inside of the core. This keeps the device small and minimizes flux leakage. For other core shapes, windings often use the largest wire that will fit conveniently into the window. This minimizes losses, and maximizes the power rating.
3. General Inductor Design via Ten Design R ules(SKIP THIS SECTION IF YOU ARE AN UNDERGRADUATE)
a. Overview of ten step L design sequence
Start L Irms and f required by PWM converter circuit.
- T(core surface) }In steps 1-2 the }Energy and heat
- Stored energy required}flow requirements ~1/2 Li^2 }help chooses both }material type and
- From tables of cores vs}geometric shape 1/2 Li^2 }of the required }core to wind wire
- Thermal Analysis }upon. T(surface), T(ambient)
Steps 5 and 6 quantify: Bac waveform in the core from core data specifies eddy current core loss
Peak B is crucial to hysteresis loss B(peak) < Bsat
Winding parameters for copper wire
Maximum Lmax = N^2 /ℜ (core)
}Ldesired < Lmax
}L=N^2 /(ℜ (core)+ ℜ (gap))
}8-10 set }the “L” }value by }μ(core) }and
} l g(gap)
b. Detailed Approach for Each of 10 Steps
For HW#1 verify the specific quantities in the example
below on the left hand side.
Assemble Inductor design inputs
STEP 1
Six Design Inputs Specific (1) L values desired by circuit: (1) L = 300 μH with 4A rms (2-3) Irms and Ipeak: Ipeak = found from
- to match to the core core size - Bmax < Bsat core shape - ease of putting N windings of wire
- “on the iron” (core cost as well as wire cost) is also an issue in some low cost commodity circuits. STEP 3 ↓ Choose core material, shape and size
We need a manufacturers core data base to guide our possible core choices. Below we list just the pertinent data for this 100 kHz example
Nφ= Li = flux linkage |
Where Nw = Kcu
Awindow Acu
| (^) rms RMS cu
J =^
I
A
φ= B Acore | Acopper = Acu KCu can vary from 0.3(Litz wire) to 0.9 for foil. We will need a larger core for the choice of Litz wire ands a smaller core for foil wire. Wire type with a copper fill factor will need to be balanced with core size. Now we reconcile electrical circuit specs and the wire and core specifications. Circuit Spec’s Core and Winding Specifications LIrmsI(peak) ← → Five Parameters Kcu, Jrms, Bpk, Aw, Ac Design specs Two Materials Issues: Jrms, Bpk
from circuit Three Geometry Choices: Aw, Ac, Kcu Kcu is derived from wire size,the number of turns and the core wire winding area and is not an independent variable. It varies by a factor of 3 from Litz wire Kcu ≈0.3 to Foil Kcu ≈0. STEP 4 Thermal Resistance, Rθ , and Power loss Assume the total inductor power: PL ≡ Pcore + Pwinding. As a first guess assume that that core and wiring contribute equal parts.
Realize that PL(max) is limited via thermal heat: T - Ts^ a R(core total)
We cannot allow the core to reach 100°C from the balance of heat versus heat taken away. LIrmsIpeak ~ Kcu[JrmsBpkAw,Ac] Note that the type of wire must be chosen at his point but not wire size. A specific choice is Litz wire with Kcu = 0.3. Note the trends set in motion by Kcu choice. Kcu ↑ then required core size ↓ ↓ then required core size ↑ This is classic Trading “core” for “copper” in the inductor design.
General Specific
P T - T R (core)
≡ s^ a θ
From core data base or by
calculating Rθ from geometry P = PmVT Typically for a core V = Core windings plus Core Volume, Rθ ~ 10o/W VT = Vc + Vw
In General Core loss Pm varies with the flux density to some power and Pm increases with frequency as shown below.
Acu, N)
General Specific
- Kcu = N cu window
A
A
- Each type of wire has
We use WA for core wire window and Kcu is set from choice of wire type. We previously chose 0.3 Litz wire Kcu = 0.3 for Litz Wire This allows us to set WA for the core when the number of wire turns is known
- Acu from rms rms
I
J
wire database 2. Acu =
4 A
Jrms Irms comes from circuit waveforms Jrms from wire data sheet Jrms comes from wire data base and 4A from circuit spec. We utilize: KCu =0.3, and find first JRMS and then the required wire area at the given current level, 4A. Acu as determined form 4A/ Acu = 6A/mm^2 This is shown on the next page.
General Specific The wire winding losses in Watts per cm^3 is next sought
- PM = Pwindings = 22 kcu J^2 rms^ ^ mWcm 3 ^
^
- (^) rms cu
J =^
k
6A/mm^2 Given Pm we find Jrms. This sets the wire area, ACu=4A/6A/mm^2 =0.7mm^2 We also know from the core data A(window) =140 mm^2 so we know
N = K^ A A
cu window cu
N =
Note the importance of Awindow = 140 mm^2 from core data base for the number of turns of copper wire.
Required # of copper wire turns results! 8 ↓ Find maximum inductance of selected magnetic core
General Specific Ac=1.5 * 10-4m^2 from chosen magnetic core data base
max
c pk peak
L
N A B
I
≡ N, Bpk and Ip are all given
Lmax must > Lspec to insure low iripple (^) max
L =^
But not too much bigger because = 290 μH bigger core is more costly.
Double E core has Pbm of one big gap is three distributed that B(fringe) from the core goes out air gaps -3 air gaps ⇒ further into copper windings increasing skin are better than one Why? effect and eddy current loss in the wires.
The effective air gap with an E core geometry is made in a cross-section and neglecting fringing.
Σ gaps = 3 * E core spacing ↓ The area of the gap Ag = Make 1/3 of size for less fringing Ag (a+g)(d+g) gap size g << d,a - core sizes if flux extends out additional g/2 from core edge
Ag ≈ad + g(a+d) + 0 second order terms
ℜ (^) total = ℜ (^) c(core) + ℜ (^) gaps ↓ ↓ c c Ac
l μ
3g μ o Ag
Usually assume ℜ (^) c is small compared to ℜ (air gap). We tailor 3 g to give desired Bpk(core) when Ipk is applied so that we do not exceed B(critical). NI = ℜ (^) gφ
pk c pk (^) o g
NI
A B
3g μ A
Tune g for desired Bpk so Bpk < Bsat (core)
Set L to design value
if Lspec < Lmax We have two routes to reduce Lmax to reach Lspec.
L = N
2
ℜg
We could increase ℜ (^) g alone but this both reduces Bpk and Bac which is
good only for core loss. Since ℜ (^) g =
3g μ o (^) Ag
by reducing Ag we could use
a smaller core, thereby reducing L and lowering core cost.
If we reduce N^2 to reach Lspec from Lmax, then we save copper. But lower N increases Bpk and hence core loss again we trade: “Cu” for “iron”.
