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BRIEFLY ABOUT POWER ELECTRONICS

About DC-DC Converters

Basics

A DC-DC converter is a device that accepts a DC input voltage and produces a DC output voltage. Typically the output produced is at a different voltage level than the input. The electrical components required to build an ideal DC-DC converter are: Source, load, switch, inductor and capacitor. For building a simple ideal DC-DC converter, these electrical components are more than enough. But in reality the DC-DC converters are never ideal. The aim is to reach the power losses in a real DC-DC converter. Therefore, resistors are also required as electrical components to model the losses in the real circuit.

Image:Power_Electronics_Electrical_Components.png

Image:Power_Electronics_Electrical_Components.png

The switch is the most important component of the DC-DC converter. It is either fully ON or fully OFF, with very short transition times from one of these states to the other. The switching element in DC-DC converters are normally realized by semiconductor parts. Some semiconductor parts to realize the switch are: Fast recovery diodes, bipolar junction transistors (BJT), metal oxide semiconductor field effect transistors (MOSFET), or gate turn-off thyristors (GTO). [SEV85]
The single pole double throw (SPDT) type switch, like the one shown in the figure above , is realized by a simple diode and transistor (BJT) combination. In this article the details of the semiconductor elements are not considered.

Image:Power_Electronics_Electrical_Components_1.png‎

Image:Power_Electronics_Electrical_Components_1.png‎

To describe the effect of the ideal switch in a DC-DC converter, the circuit in the figure below is drawn.

Image:Power_Electronics_ideal_switch_used.png‎

Image:Power_Electronics_ideal_switch_used.png‎

The time dependent voltage after the ideal switch is shown in the next figure.

Image:Power_Electronics_Ideal_switch_output_voltage_waveform.png

When the ideal switch in the figures above is at switch position 1 (SP1),

V s (t)

is equal to

V i n

. When the ideal switch is at SP2,

V s (t)

is equal to 0. This 2 stage process repeats itself with a very small period

T s

. The switching frequency

f s

, equal to the inverse of the switching period

T s

, generally lies in the range of 1kHz to 1MHz, depending on the switching speed of the semiconductor devices. [ERI01]
The duty ratio D is the fraction of time that the switch spends in position 1, and is a number between 0 and 1. The complement of the duty ratio, D’, is defined as (1-D).

$D=\frac{T_{position_1}}{T_s}$
The switch reduces the DC component of the voltage. The switch output voltage

V s (t)

has a DC component that is less than the converter input voltage

V i n

. From Fourier analysis, we know that the DC component of

V s (t)

is given by its average value <

V s

>:

$<V_s>=\frac{1}{T_s}\int_{0}^{T_s} V_s (t)\,dt$

Image:Power_Electronics_Determination_of_the_switch_output_voltage_Dc_component.png

As shown in the figure above, by integrating and dividing by the switching period

T s

the average value of the switch output voltage is found:
$<V_s>=\frac{1}{T_s} \left( DT_sV_{in}\right) =DV_{in}$
The switch reduces the Dc voltage by a factor of D.
It is also necessary to insert a low-pass filter after the switch. The new circuit is shown in the figure below.
Image:Power_Electronics_Insertion_of_a_low-pass_filter.png

Image:Power_Electronics_Insertion_of_a_low-pass_filter.png

The low-pass filter is designed to pass the DC component of

V s (t)

, but reject the components of

V s (t)

at the switching frequency and its harmonics. The output voltage

V o u t (t)

is then essentially equal to the DC component of

V s (t)

:

$\left( V_{out}\right) =<V_s>=DV_{in}$

The converter has been realized using lossless elements. To the extent that they are ideal, the inductor, capacitor, and switch do not dissipate power. Hence, the efficiency of the converter approaches 100%. But in real case, none of the components are ideal, therefore to reach the real efficiency of the DC-DC converter the losses of each component should be considered.
Duty ratio D is the control parameter in DC-DC converter electronics. In most cases, D is adjusted to regulate the output voltage,

V o u t

Types of Dc-Dc Converters

There are basically 3 types of DC-DC converters:

Buck Converter
Boost Converter
Buck-Boost Converter

In addition, the Cuk converter can also be counted as the forth DC-DC converter type. Since the function of the Cuk converter is very similar to the buck-boost converter, it is omitted regarding the simplicity of this article. More detailed information about complex power electronics of Cuk converters can be found in References [SEV85], [ERI01].
The converter in the figure above is a buck converter.

V o u t

is always less than V_{in} for a buck converter. The voltage ratio is D.
A boost converter is drawn in the figure below.
Image:Power_Electronics_Boost_converter_circuit.png‎

Image:Power_Electronics_Boost_converter_circuit.png‎

The boost converter always increases the input voltage. The voltage ratio is found through a similar process shown above as:

$\frac{V_{out}}{V_{in}}=\frac{1}{1-D}$

Finally, a buck-boost converter is drawn.
Image:Power_Electronics_Buck-boost_converter_circuit.png

Image:Power_Electronics_Buck-boost_converter_circuit.png

The buck-boost converter can either increase or decrease the input voltage. An important difference of buck-boost converters from the others is that the voltage polarity is changed. The voltage ratio is:

$\frac{V_{out}}{V_{in}}= \frac{-D}{1-D}$

Approximations and Assumptions

Small Ripple Approximation

It is impossible to build a perfect low-pass filter that allows the DC component to pass but completely removes the components at the switching frequency and its harmonics. So the low-pass filter must allow at least some small amount of the high-frequency harmonics generated by the switch to reach the output. [ERI01] Hence, in practice the output voltage waveform

V o u t (t)

can be expressed as:

$V_{out}\left(t\right)=V_{out}+V_{ripple}\left( t\right)$

So the actual output voltage

V o u t (t)

consists of the desired DC component

V o u t

, plus a small undesired AC component

V r i p p l e (t)

arising from the incomplete attenuation of the switching harmonics by the low-pass filter. The figure below shows the DC component and also the actual waveform in which the magnitude of

V r i p p l e (t)

has been exaggerated.
Image:Power_Electronics_Output_voltage_waveform.png

The output voltage switching ripple should be small in any well-designed converter, since the object is to produce a DC output. Generally the switching ripple is less than 1% of the output voltage. So it is nearly always a good approximation to assume that the magnitude of the switching ripple is much smaller than the DC component: [ERI01]

| V r i p p l e | < < V o u t

Therefore, the output voltage

V o u t (t)

is well approximated by its DC component

V o u t

, with the small ripple term

V r i p p l e (t)

neglected: [ERI01]

$V_{out}\left( t\right) \approx= V_{out}$

This approximation, known as the small-ripple approximation, greatly simplifies the analysis of the converter waveforms.
The inductor current can also be analyzed in a similar manner to reach a simplifying approximation. For this analysis the buck converter circuit presented above is re-drawn, but this time switching positions are shown in two different figures. The first figure shows the buck converter in position SP 1, the second figure shows it in position SP 2.
Image:Power_Electronics_Buck_converter_circuit.png‎

Image:Power_Electronics_Buck_converter_circuit.png‎

With switch position 1 , the inductor voltage

V L (t)

is:

$V_L=V_{in}-V_{out}\left( t\right)$

As described above, the output voltage

V o u t (t)

consists of the Dc component Vout, plus the ripple term

V r i p p l e (t)

. Making the small ripple approximation, the equation reduces to:

V L = V i n - V o u t

So with the switch in position 1, the inductor voltage is essentially constant and equal to

V i n - V o u t

. By knowledge of the inductor voltage waveform, the inductor current can be found by use of the inductor equation:
$V_L\left( t\right) =L\frac{di_L(t)}{dt}$
Thus:

$\frac{di_L(T)}{dt}=\frac{V_L(t)}{L}$

During the first interval, when

V L (t)

is approximately

(V i n - V o u t)

, the slope of the inductor current waveform is:

$\frac{di_L(T)}{dt}=\frac{V_L(t)}{L}=\frac{V_{in}-V_{out}}{L}$

Therefore, when the switch is in position 1, the inductor current slope is constant and the inductor current increases linearly.
Similar arguments apply during the second interval, when the switch is in position 2 (Figure 2-10(b)):
$V_L\left( t\right) =-V_{out}\left( t\right) \approx =-V_{out}$

$\frac{di_L(t)}{dt}\approx = -\frac{V_{out}}{L}$

Hence, during the second interval the inductor current changes with a negative and essentially constant slope. The steady-state inductor voltage and current waveforms are drawn in the next to figures.
Image:Power_Electronics_Steady-state_inductor_voltage_waveform_of_the_buck_converter.png

Image:Power_Electronics_Steady-state_inductor_voltage_waveform_of_the_buck_converter.png

Image:Power_Electronics_Steady-state_inductor_current_waveform_of_the_buck_converter.png

The inductor current begins at some initial value

i L (0)

. During the first interval it increases with the constant slope calculated above. At time

t = D T s

the switch changes to position 2. The current then decreases with the constant slope calculated above for the second interval. At time

t = T s

, the switch changes back to position 1, and the process repeats.
Typical values of

i L

lie in the range of 10% to 20% of

I L

. It is undesirable to allow

i L

to become too large; doing so would increase the peak currents of the inductor and of the semiconductor switching devices, and would increase their size and cost. So by design the inductor current ripple is also usually small compared to the DC component

I L

. The small ripple approximation

i L (t) = I L

is usually justified for the inductor current. [ERI01]
It is entirely possible to solve converters exactly, without use of small ripple approximation. One could use the Laplace transform to write the expressions for the waveforms, then invert the transforms, match boundary conditions, and find the periodic steady-state solution of the circuit. Having done so, one could then find the DC component of the waveforms and the peak values. But this is a great deal of work, and the results are nearly always intractable. Besides, the extra work involved in writing equations that exactly describe the ripple is a waste of time, since the ripple is small. The small ripple approximation is easy to apply, and quickly yields simple expressions for the DC components of the converter waveforms.

Inductor Volt-Second Balance

Steady-state conditions leads to the principle of inductor volt-second balance. It is simply the requirement that, in equilibrium, the net change in inductor current over one switching period be zero. Integrating both sides of the equation from above.
$i_L\left( T_s\right) -i_L(0) = \frac{1}{L}\int_{0}^{T_s} V_L (t)\,dt$

In steady-state, the initial and final values of the inductor current are equal, and hence the left-hand side of the equation is zero. Therefore, in steady-state the integral of the applied inductor voltage must be zero:

$\int_{0}^{T_s} V_L (t)\,dt=0$

Dividing both sides of this equation to

T s

, and remembering the second equation of this article:

$\frac{1}{T_s}\int_{0}^{T_s} V_L (t)\,dt =<V_L>=0$

Therefore, the average value, or the Dc component, of

V L (t)

is zero. This also means that the area below the inductor voltage curve for one period

T s

is zero:

Image:Power_Electronics_principle_of_inductor_volt-second_balance_for_the_buck_converter.png‎

$<V_L>=\frac{A}{T_s}= D\left( V_{in}-V_{out}\right) +\left( 1-D\right) \left( -V_{out}\right) =0$

Using the equation above,

V o u t

can be calculated with the known values of D and

V i n

. Or the control parameter D can be calculated using

V i n

and

V o u t

Capacitor Charge Balance

Similar arguments described above can be applied to capacitors. The defining equation of a capacitor is:

$i_c\left( t\right)=C\frac{dV_c(t)}{dt}$

Integration of this equation over one switching period yields:

$V_c(T_s)-V_c(0)=\frac{1}{C}\int_{0}^{T_s} i_c (t)\,dt$

In steady-state, the net change over one switching period of the capacitor voltage must be zero, so that the left-hand side of the equation is equal to zero. Therefore, the integral of the capacitor current over one switching period should be zero.

$\frac{1}{T_s} \int_{0}^{T_s} i_c (t)\,dt=<i_c>=0$

The average value, or DC component, of the capacitor current must be zero in equilibrium. The capacitor charge balance can be used to find the steady-state currents in a switching converter.

Real Converters

In the discussions above, the DC-DC converters are assumed as ideal. So the efficiency was 100%. For ideal case:

P i n = P o u t

V i n I i n = V o u t I o u t

But there is no ideal converter in reality. There are a number of sources of losses. To calculate the real case efficiency, these losses must also be considered. The most significant sources of losses are:

Inductor Copper Loss
Semiconductor Conduction Losses
Switching Losses

Inductor copper loss (1) is the loss due to the non-ideal inductor. Ideally the inductors have no internal resistance, but in reality there is some small resistance, which causes a loss of useful energy. This loss will be realized by adding a resistance just after the inductor in the circuit.
Semiconductor conduction losses (2) are the losses due to non-ideal semiconductor components. Transistors and diodes have ideally no internal resistance in their ON state, but in reality they have. Also, the diode has some small internal opposing voltage. The semiconductor conduction losses will be realized by adding one resistance just after the transistor; one resistance and one small opposing voltage just after the diode.
The turn-on and turn-off transitions of semiconductor devices require times like tens of nanoseconds to microseconds.[ERI01] The losses occur during these transitions are called switching losses (3). Some simple methods to estimate switching losses are given in