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LECTURE 34
HIGH FREQUENCY TRANSFORMER
A. Transformer Basics
1. Geometry of Cu Wire Windings and Core Wire
Winding Window
2. Single Wire Skin Effect and Multi-wire
Proximity Effects Alter Both Primary and
Secondary R(wire coil) via Effective Acu (wire)
Changes
3. Transformer Inductance’s: Lm and L l
a. Magnetizing Inductance, Lm: Core Flux
b. Leakage inductance, L l : Air Window Flux
A crude estimate for L l is (1*Lm)/μ and
Interleaved Primary/Secondary Winding
effects occur
4. Total Core Loss: Rm
a. Hysteresis Loss due to frictional movement
b. Eddy Current Loss due to core currents
5. How is the Transformer Magnetization Current,
im ,Created?
a. im is Not effected by load current at all
b. im = ∫vLdt/Lm volt-second driven only
7. T(max) of the core of a LossyTransformer
a. Heat Balance Equation
T(core) = T(ambient) + P(core and wire
loss)*R(core and Winding Structure)
8. Comparing Inductors vs Transformers
B. Special Case for Wire Losses in Transformers
with Multiple Windings
1. V ~ Number of Wire Turns, I ~ Depends on
Load Impedance in the secondary
2. Optimum Winding Area for the k’th coil
αK = PK/PT
3. Example of PWM Converter
AK (winding) = f(PK,D), that is the winding area
required depends on the duty cycle employed in the
electrical circuit. Generally, this is a range of D
values.
Window a. Overview Lets for simplicity consider only two wire windings wound upon one magnetic core, which acts to couple the magnetic flux between the two coils with near unity transfer. The main purpose of a power transformer in Switch Mode Power Supplies is to transfer power efficiently and instantaneously from an external electrical source to external loads placed on the output windings. In doing so, the transformer also provides important additional capabilities:
- The primary to secondary turns ratio can be established to efficiently accommodate widely different input/output voltage levels.
- Multiple secondaries with different numbers of turns can be used to achieve multiple outputs at different voltage levels and different polarities.
- Separate primary and secondary windings facilitate high voltage input/output isolation, especially important for safety.
- A set of k wire windings introduces some complexity to the issues for an optimum transformer winding partition For the case of multiple windings the competing issues are more
complex as we will see at the end of this lecture. It will require Lagrangian optimum analysis using one variable for each wire coil. In the end the winding area allotted to each coil winding will vary as its power handling requirement compared to the total power level.
All the wire windings, wound on a given core must fit into its one wire winding window which we term either AW or WA in the text below. Both symbols are found in the transformer literature. Much of the actual winding area is taken up by voids between round wires, by wire insulation and any bobbin structure on which the wire turns are mounted as well as insulation between high voltage and low voltage windings. In practice, only about 50% of the window area can actually carry active conductor, This fraction is called the fill factor. In a two-winding transformer, this means that each winding can fill not more than 25% of the total wire winding window area
Total area for windings ≡ Aw=Apri+Asec Aw is split into two parts, for a two winding transformer, according to the required wire sizes (AWG#) in each coil which in turn is chosen for the expected current flow to avoid overheating of the wires. In short the primary wire winding area employed in the wire winding window is:
w
pri cu wire cu
A (prim)^ =^
N A (AWG # of primary) K (primary) Note that we use AWG tables for USA wires specified in cir mils. EUC tables for wires are specified in mm^2. Again KCu is an estimate of what % of the wire volume is actually Cu. Kcu is specified for each type of wire and spans a range: Kcu(Litz
Proximity effects, due to the collective magnetic field from many wires, also increases wire losses via cumulative MMF effects.
J flow only on the surface causes increase in Cu wire loss!
Skin effects cause non-uniform current density profiles in all wires, BUT Non-interleaved windings allow build-up of mmf in the wire winding window. This mmf is different for each wire position and ruins the assumption that J is uniformly distributed across the wire area the same way for all wires. That is different wires have different J distributions in the real world of hf transformers
Again, the increasing MMF spatial variation makes the R(each wire turn) unique in it’s losses according to it’s position in the wire winding window. The Higher the mmf seen by the wire the more non-uniform the J. Note the interior wires, where primary and secondary meet, as we have discussed previously, have the highest mmf and the most non-uniform J in the wires causing much higher I^2 R loss.
2. Energy Storage in a Transformer Ideally a transformer stores no energy, rather all energy is transferred instantaneously from input to output coils. In practice, all transformers do store some energy in the two types of inductance’s that associated with the real transformer as compared to ideal transformers which have no inductances associated with them. There are two inductances.
- Leakage inductance, L l , represents energy stored in the wire winding
windows and area between wire windings, caused by imperfect flux coupling for a core with finite μ. In the equivalent electrical circuit
leakage inductance is in series with the wire windings. and the stored energy is proportional to wire winding current squared.
- Magnetizing inductance, LM , represents energy stored in the magnetic core and in small air gaps which arise when the separate core halves forming a closed magnetic loop core come together. In the equivalent circuit of a real transformer, mutual inductance appears in parallel with the primary windings only. The energy stored in the magnetization inductance is a function of the volt-seconds per turn applied to the primary windings and is independent of load current. 3. Various Frequencies of Interest to Magnetic Devices and Copper Loss There are several meanings to the term “frequency” in switching power supply applications, and it is easy for confusion to arise. “ Clock frequency” is the frequency of clock pulses generated in the control IC. Usually, the switching frequency is the same as the clock frequency, but not always. Occasionally, the control IC may divide the clock frequency to obtain a lower switching frequency. It is not unusual for a push-pull control IC to be used in a single-ended forward converter application, where only one of the two switch drivers used, to guarantee 50% max. duty cycle. In this case the switching frequency is half the clock frequency. “Switching Frequency” , fsw, is usually defined as the frequency at which switch drive signals are generated. It is sometimes also the frequency seen by the output filter, the frequency of the output ripple and input ripple current, and is an important concept in control loop design. In a single-ended power circuit such as the forward converter, the power switch, the transformer, and the output rectifier all operate at the switching frequency and there is no confusion. The transformer frequency and the switching frequency are the same. This is not true for the bridge converters or center tappped transformers where the current waveforms may have different frequencies, as we saw in prior lectures on transformers in switch mode topologies
Well designed transformers:L l 1 /L l 2 ≈N 12 /N 22 and ℜ (^) l 1 /ℜ (^) l 2 ≈N 12 /N 22
Both magnetizing and leakage inductance causes voltage spikes during switching transitions, resulting in EMI and possible damage or
destruction of switches and rectifiers. L l causes undesired inductive kick
in voltage @ primary/secondary windings when any rapidly changing current waveform, like a square wave, is employed. Leakage inductance also delays the transfer of current between switches and rectifiers during switching transitions. These delays, proportional to load current, are one main cause of regulation and cross regulation problems in feedback control circuits. Protective snubbers and clamps are often then required and the stored energy then ends up as loss in the snubbers or clamps. Leakage and mutual inductance energy is sometimes put to good use in zero voltage transition (ZVT) circuits. This requires caution as leakage inductance energy disappears at light load, and mutual inductance energy is often unpredictable.
Previously we showed that L l ≈Lm/μ ,but this indicated only
the effect of the choice of core material. We also saw that primary- secondary coil interweaving helped reduce proximity effects, which raised coil I^2 R losses. Now we show the effect of interleaved wire windings to reduce the leakage inductance. In particular we will
show: (^) L (^) l ~
P^2
Where P is the # of prim/sec interfaces in the winding
of the transformer. We actually get a double win from interleaving coil windings, decreased proximity effects in copper loss and reduced leakage inductance as we show below. The energy stored in leakage inductance is undesired for a transformer which aims to transfer energy from one coil to another.
L 1 I =^
(^2) prim H (window) d V window volume
o
l ∫ μ^2 w
Note below how it varies with non-interleaved and interleaved wire
windings as shown below. P = 1 P = 2 P = 4
L l ≈full value = Lm/μr L l = ¼ full Lm/μ L l = 1/16full Lm/μ
One can tailor L l (leakage) to be bigger or smaller by up to an order of
magnitude from Lm/μr choices or by by choosing winding arrangements as shown.
Full Winding (L l )max Split Windings L l ↓ L l ≡ 1.0 L l → 1/4 L l → 1/
Conclusion: Split or partition windings to reduce L l and to reduce the
copper losses due to proximity effects.
5. Transformer Equivalent Circuit All of the above discussion results in a transformer model that incorporates both the ideal transformer and the two types of parasitic inductance’s as shown on the top of page 12.
seconds conditions, which determine the magnetizing current under AC drive conditions. H(core) l(core) = n im. If n im is too high, then H exceeds H(critical) or BSAT. Then the core saturates causing μr → μo. To determine B(max) for a transformer driven by AC signals we employ
the flux linkage, λ= Nφ, and vL = dλ/dt. This gives a volt-sec limit.
Find ∫vLdt = λ= Nφ= NB(core)A(core).
As long as ∫vLdt < NB(saturation)A(core) no core saturation occurs in the volt-sec limit. For a sinusoidal voltage vosinwt this means : vo/wNA < Bsat Thus the core saturation parameter sets the maximum volts per turn allowed on the windings. Any even small dc current in a transformer winding creates a different route to core saturation via amp-turn limits. The full core
anti-saturation criterion then becomes
vdt
NA
Ni
A
∫ +^ DC^ Bsat
<. The core
saturation parameter is limiting the AC voltages across the primary and the DC currents in the wire coils.
=N A A
L = N c c c
12
c c
c
12 m μ → μ
l l
Above Bsat μc → μo causing a factor 100-1000 change in Lm. In the transformer model this shorts out the primary of the ideal transformer.
a. Total Core Loss: Rm
For iL < i(sat) we get lots of core loss and copper loss occurring before core saturation occurs and L= 0. Over the past twenty years the core loss due to hysteresis and eddy currents in cores has improved primarily by the use of new core materials. Low loss cores are now available up to 5 Mhz. To briefly review the origin of the hysteresis core loss we plot below the B-H curve that the transformer typically operates under.
For im << isat we don’t have a catastrophe but we still create core loss via ∆im swings in the core. We model this combination of hysteresis and eddy current loss by an equivalent Rm in parallel | | with Lm
The equivalent transformer circuit model then has two currents
i L m for reactive current in the core and iRm which models all core losses
6. How is the magnetizing current im created? This is important enough to consider twice. The ideal transformer model will cause a primary current to flow that is soley related to the secondary current(s) drawn.
i (^) primary(from a load)^ ≈^ 2 2 1
N i (^ ) N
load due to transformer action. The
B(core) < B(sat)
∫vdt/NA + NiDC/ℜ A < B(saturation).
7. Transformer Heating Limits
Transformer losses are limited by a maximum "hot spot" temperature at the core surface or inside the center of the wire windings. As we have shown to a first approximation, temperature rise (°C) equals
core thermal resistance (°C/Watt) times total power loss (Watts). Ptotal = P(core) + P(windings)
∆T = R (^) C x PT(core plus copper) Ultimately, the appropriate core size for the application is the smallest core that will handle the required power with losses that are acceptable in terms of transformer temperature rise or power supply
efficiency. We usually cannot exceed a core temperature of 100°. Thermal radiation and convection both allow heat to escape from the core.
R (total) θ^ =^
R(conv) R(rad) R(conv) R(rad)
= Parallel Combination
Typically we find in practice for cores the thermal resistance varies over a range: 1 < (^) R (^) Q (total) < 10 C/ W°
RQ(total) depends both on core size/shape and thermal constants.
T(core) - T(ambient) = RT (W/°C) PT (total power loss)
Practically, T(core), is never to exceed 100oC since core μ(T) and wire insulation degrades. Typically, T(ambient) = 40oC, RT = 10oC/W, and PT = Typically 2-20W
T(core) = RTPT + 40oC = 90oC
8. Compare an Inductor versus a Transformer for the Same sinusoidal excitation: f, Max B, specific core
FOR HW #1 GO THROUGH THIS SECTION AND VERIFY THE
ARGUMENTS STEP BY STEP
Inductor Transformer
- Power rating in V-A 1. Power rating in V-A
L I I S 2.2 f rms peak
≈ T
ST =^ V Ip p rating
ST = 2.2f LIrmsIpeak For a sinusoidal excitation
V p =^ N Ap c c
d dt
[B (max) sinwt]
V (^) p =^ N (^) p A (^) c w B (max)c Ip = Jp A(Cu wire)
Given L and Ipeak & (^) T p c S =^ c c
N A
w B (max) * J H
Irms we can then say for a sinusoid current
I (^) peak =^2 I (^) rms cu cu w p
A =^
K A
2 N
The V-A rating The V-A rating ST(inductor)= πfL I^2 rms
ST(transformer)=2.2KcufAcAwJpΒ(max) This ST(V-A) rating we determine the required core size. We next relate L size to that of a transformer rated at a particular value of ST.
iL ↑ Bmax(L) ↑ | ip and isec do not affect im for an inductor only! | for a transformer.
| (^) i (^) m =
Vdt L
Bmax < B(sat) | im(max) < im(max for saturation)
For both inductors and transformers ∫vdt/N(Cu)A(core) < B(sat)
dc: Vdc∆t < BsatN(Cu)A(core) ac: V(peak)/N < BsatwA(core)
must be wrapped around the same core in the one available wire winding window. Each winding has a resistance, which could be minimized by using the biggest diameter wire. But this increases the resistance of the other windings. The resistance of the j’th winding will be proportional to N 2 (for the j’th winding) rather than N because the optimum area of the wire in the j’th winding will go as 1`/NJ as we will show below. Each winding takes a fraction of the available window winding area of the core.
k
k A
A
W
α for K th (^) winding
AK is the winding area for the K’th winding only.
WA is the full core window area for all windings as shown below.
The total resistance of the Kth^ winding which employs wire of length K and area Ak. Clearly, the length of the wire winding depends on the core it is wound around but also the number of prior wire turns that were wound the core before this winding was started. For a single turn in the k’th winding we can say:
k
k wk
R =^
l A
ρ for Kth^ winding
What about Nk turns in the k’th winding? Will the total wire resistance of this winding vary as ~ Nk or Nk^2? Can you guess why one rather than the other? There is a hidden NK variable here as the choice of Awk for the wire in the k’th winding was set by its fractional area of the total
winding area or the parameter αk. That is Awk = WA Kcuαk /Nk. Later we will calculate the optimum values of α for each winding. For now just realize that Awk~1/ Nk. The length of wire is then:
l (^) k ≡ N (^) k * (MLT )k Where MLT = mean length per turn and NK = total # of turns in Kth winding
ink winding
employed
areaofwire Awk
from AWG # gauge tables
The total area of copper in the Kth^ winding is given by # turns times area per each wire. (^) A (^) wk N (^) k = (^) WA α (^) k Ku(wire fill factor)
As always Ku or KCu depends on wire type chosen for windings: Ku = 0.3 for Litz wire and Ku = 0.9 for foil. This varies RK by a factor three. The real surprise however is in the dependence on the number of turns NK.
k
k
2 k A u k
R =^
N (MLT )
W K
ρ α
Note: Rk ~ Nk^2 not Nk.
