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The inductor volt-second balance. The Buck Converter – Chapter three

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Let us more closely examine the inductor and capacitor waveforms in the buck converter illustrated in Figure 2-6.

Actually, the output voltage appears as illustrated in Figure 2-7.

This approximation is known as The small ripple approximation or the linear ripple approximation.

With this approximation, let us analyze the inductor current waveform. Since the inductor voltage is essentially constant during the first interval (switch in position 1), the inductor current slope of equation (6) is also essentially constant and the inductor current increase linearly, as in Figure 2-10:

Solution for L yields:

Equations (1) to (3) and Figure 2.10 are derived under steady-state conditions. Let us consider next what happens to the inductor current when the converter is first turned on. Suppose that the inductor current and output voltage are initially zero, and an input voltage Vg is the applied. As shown in Figure 2.11, during the first subinterval, with the switch position in 1, we know that the inductor current will increase.

Since the inductor current iL(t) flows to the output, the output capacitor will charge slightly, and will v(t) increase slightly too. The process repeats during the second and succeeding switching periods, with inductor current iL(t) increasing during each subinterval 1 and decreasing during each subinterval 2.

As the output capacitor continues to charge and v(t) increases, the slope during each subinterval 1 decreases and the slope during each subinterval 2 becomes more negative. There is no change in the inductor current over a complete switching period and the converter reachs the steady state condition.

The requirement that, in equilibrium, the net change in inductor current over one switching period be zero lead us to a way to find steady-state condition in any switching converter. That is what we call the inductor volt-second balance.

Given the defining relation of an inductor:

Integration over one complete switching period, say from t=0 to t=Ts   yields:

In steady-state condition, the initial and final value of the inductor current is equal, so:

The right hand of equation (6) has the units of volt-second or flux-linkages. It states that the total area, or net volt-seconds, under the vL(t) waveform must be zero under steady-state condition.

An equivalent form is obtained by dividing both sides of equation (6) by the switching period:

Equation (7) is recognized as the average value, or DC component, of vL(t). Equation (7) states that, under steady-state condition, the applied inductor voltage must have zero DC component. 

The inductor voltage waveform is reproduced in Figure 2.12 with the area, or net volt-seconds, under the vL(t) curve identified.

The total area lambda is given by the areas of the two rectangles:

The average value is therefore:

What lead us to state, using equation (2), that:

Considering Figure 2.3, the output voltage v(t) of a Buck Converter is essentially equal to the DC component of the switching voltage vs(t), and equation (10) states that the output voltage v(t) is less than or equal to the input voltage Vg, since 0<D<1.

La imagen tiene un atributo ALT vacío; su nombre de archivo es image-64.png

Source:

  •  Erickson R., Maksimovic D. Fundamentals of Power Electronics. Pp. 22-24.

Previously: The small ripple approximation – Chapter two

Literature review by Larry Obando – Electrical Engineering

Universidad Simón Bolívar – Universidad Central de Venezuela

Mathematical modeling and simulation services

Contacto – Whatsapp +34633129287

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Criterio de Nyquist, Digital Signal Processing, Procesamiento de señales digitales, Señales y Sistemas, Sistemas LDCID

Aliasing – sampling with a periodic impulse train

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Sampling – Chapter One

Sampling with a periodic impulse train.

The Periodic Sampling is the typical method of obtaining a Discrete-time representation of a Continuous-time signal, wherein a sequence of samples x[n] is obtained from a Continuous-time signal xc(t), according to the relation:

Where T is the sampling period and its reciprocal, Fs=1/T, is the sampling frequency. We refer to a system that implements the operation of equation (1) as an ideal continuous-to-discrete-time (C/D) converter.

To derive the frequency-domain relation between the input and output of an ideal C/D converter, first consider the conversion of xc(t) to xs(t) through modulation of the periodic impulse train:

We modulate the periodic impulse s(t) train with xc(t), obtaining:

Through the “sifting property” of the impulse function, xs(t) can be expressed as:

Figure 4.2(a) is strictly a mathematical representation that is convenient for gaining insight into sampling in both time domain and frequency domain. It is not a close representation of any physical circuit or system. This representation leads to a simple derivation of a key result and to a number of important insights that are difficult to obtain from a forma derivation based on manipulation of Fourier Transform formulas.

The sampling operation is generally not invertible; i.e., given the output x[n], it is not possible to reconstruct xc(t), the input to the sampler, since many continuous-time signals can produce the same output sequence. The inherent ambiguity in sampling is a fundamental issue in signal processing. Fortunately, it is possible to remove the ambiguity by restricting the input signals that go into the sampler.

Figure 4.2 (b) shows a continuous-time signal xc(t) and the results of impulse train sampling for two different sampling rates. Figure 4.2 (c) depicts the corresponding output sequences. The essential difference between xs(t) and x[n] is that xs(t) is, in a sense, a continuous-time signal that is zero except at integer multiples of T. The sequence x[n], on the other hand, is indexed on the integer variable n, which in effect introduces a time normalization; i.e., the sequence of numbers x[n] contains no explicit information about the sampling rate. Furthermore, the samples of xc(t) are represented by finite numbers in x[n] rather than as an area of impulses, as with xs(t).

Sampling principle.

A band-limited signal xc(t) with bandwidth Fo can be reconstructed from its sample values x[n]=xc(nT), if the sample frequency Fs=1/Ts is greater than twice the bandwidth Fo of xc(t):

Otherwise, aliasing would result in x[n]. The sampling rate of 2Fo  for an analog band-limited signal is called the Nyquist rate.

Note: according to Nyquist criteria, after xc(t) is sampled, the highest analog frequency that x[n] represents must be Fs/2 (i.e. ω=π).

Band-limited signal.

A continuous-time signal xa(t) is band-limited if there exist a finite radian frequency Ωo such that the CT-Fourier Transform Xa(jΩ) is zero for |Ω|> Ωo:

The frequency Fo is called the signal bandwidth in Hz.

Aliasing.

Let xa(t) be an analog (absolutely integrable) signal. The continuous-time Fourier Transform (CTFT) of xa(t) is:

Meanwhile:

In consequence, it can be shown that the discrete-time Fourier Transform (DTFT) of x[n] is a countable sum of amplitude-scaled, frequency-scaled, and translated version of the Fourier Transform Xa(jΩ):

This relation is known as the aliasing formula. The analog and digital frequencies are related through Ts:

While the sampling frequency Fs is given by:

When we want to express the sampling frequency in radians per second, we also use:

The graphical illustration of the aliasing phenomena is given by Figure 3.10:

Combining this we obtain this simple aliasing principle:

Example 3.17

The analog signal xa(t) is sampled at Fs to obtain the discrete-time signal x[n]. Determine x[n] and its corresponding DTFT.

Solution:

First, we must identify the highest frequency in xa(t). That will be F0:

Since  Fs <2 F0, there will be aliasing in x[n] after sampling. i.e., we will get the following case:

The sampling interval is:

Hence, we have:

That is:

Note that the highest digital frequency of x[n] is 1.75π, which is outside the Nyquist interval –π<ω<π, signifying that aliasing will occur. From the periodicity property of digital sinusoidal sequences, we know that x[n] will be repeated every 2π. So, the alias of  x[n] can be determined by:

Using Euler’s identity:

From table 3.1 and the DTFT properties, the DTFT of x[n] is:

The plot of X(ejω) is shown in Figure 3.15.

Figure 3.15. The Fourier Transform of x[n].

Source:

Ingel V.; Proakis J. Digital Signal processing using Matlab (p 81)

Oppenheim A.; Schafer R.; Buck J. Discrete Time Signal Processing (p 141)

Book Review by Larry Obando – Electrical Engineering –

Universidad Simón Bolívar – Universidad Central de Venezuela

Mathematical Modeling and Simulation Services – whatsapp +34633129287

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Aproximación de ondulación pequeña (Small ripple approximation) – Buck Converter – Capítulo dos.

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Examinemos más de cerca las formas de onda del inductor y el capacitor en el convertidor reductor ilustrado en la figura 2-6.

En la práctica, es imposible construir un filtro de paso bajo perfecto que elimine por completo los componentes de CA en las frecuencias de conmutación y sus armónicos. Por lo tanto, el filtro de paso bajo debe permitir que al menos una pequeña cantidad de los armónicos de alta frecuencia generados por el interruptor alcancen el voltaje de salida. En realidad, el voltaje de salida aparece como se ilustra en la Figura 2-7.

Por lo tanto, el voltaje de salida real v(t) consta del componente de CC deseado V, más un pequeño componente de CA no deseado vripple(t) que surge de la atenuación incompleta de los armónicos de conmutación por el filtro de paso bajo. Sin embargo, la magnitud de vripple(t) se ha exagerado en la figura 2-7. Casi siempre es una buena aproximación suponer que la magnitud de vripple(t) es mucho menor que el componente de CC, V:

Por lo tanto, el voltaje de salida v(t) está bien aproximado por su componente de CC, V:

Esta aproximación se conoce como Small ripple approximation (aproximación de ondulación pequeña) o la aproximación de ondulación lineal. Con esta aproximación, reemplazamos las expresiones sinusoidales amortiguadas o exponenciales para las formas de onda del inductor y el capacitor con formas de onda lineales más simples. Esta aproximación está justificada siempre que el período de conmutación sea más corto que las constantes de tiempo naturales del circuito. Además, esta aproximación debe aplicarse solo a variables continuas: la corriente del inductor y el voltaje del capacitor. No cambiar el voltaje, cambiar la corriente del voltaje del inductor. A continuación, analicemos la forma de onda de la corriente del inductor. Con el interruptor en la posición 1, el circuito se reduce a la figura 2.8a.

 La tensión del inductor vL(t) viene dada por:

Aplicando la aproximación de pequeña ondulación a la ecuación (3):

La corriente del inductor se puede encontrar mediante el uso de la definición:

Dado que el voltaje del inductor es esencialmente constante durante el primer intervalo (interruptor en la posición 1), la pendiente de corriente del inductor de la ecuación (6) también es esencialmente constante y la corriente del inductor aumenta linealmente, como en la Figura 2-10, donde podemos ver vL(t) frente a iL(t):

Se aplican argumentos similares en el segundo intervalo (interruptor en la posición 2). El lado izquierdo del inductor está conectado a tierra, lo que conduce al circuito de la figura 2.8b.

El uso de la aproximación de ondulación pequeña conduce a:

así:

En consecuencia, durante el segundo intervalo, la corriente del inductor disminuye linealmente en una ecuación de pendiente constante (8), como en la figura 2-10, posición 2 del interruptor.

En la figura 2-10, la corriente del inductor iL(t) es simétrica con respecto a I. Dado que:

es el pico de ondulación (peak ripple), el pico a pico de ondulación (peak to peak ripple) es:

El pico a pico es también el cambio en la corriente, es igual a la pendiente por la longitud del primer intervalo DTs:

Solución para el pico a pico de ondulación conduce a:

Los valores típicos de la ondulación máxima se encuentran en el rango del 10 % al 20 % del valor a plena carga del componente I de CC. Por lo tanto, por diseño, la ondulación de corriente del inductor también suele ser pequeña en comparación con el componente I de CC. se justifica la aproximación:

La solución para L en la ecuación (10) produce:

La ecuación (12) se usa comúnmente para seleccionar el valor de la inductancia en el diseño del convertidor reductor. Es importante recalcar que las ecuaciones (1) a (12) se derivan de condiciones de estado estacionario.

Fuente:

  •  Erickson R., Maksimovic D. Fundamentals of Power Electronics. Pp18-22.

Previously: Convertidor Buck (Buck Converter)

Next: The inductor volt-second balance – Chapter three

Literature review by Larry Obando – Electrical Engineering

Universidad Simón Bolívar – Universidad Central de Venezuela

Mathematical modeling and simulation services

Contacto – Whatsapp +34633129287

dademuchconnection@gmail.com

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The small ripple approximation. The Buck Converter – Chapter two.

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Let us more closely examine the inductor and capacitor waveforms in the buck converter illustrated in Figure 2-6.

In practice, it is impossible to build a perfect low-pass filter that completely removes the AC components at the switching frequencies and its harmonics. So, the low-pass filter must allow at least some small amount of the high-frequencies harmonics generated by the switch to reach the output voltage. Actually, the output voltage appears as illustrated in Figure 2-7.

So, the actual output voltage v(t) consists of the desired DC component V, plus a small undesired AC component vripple(t) arising from the incomplete attenuation of the switching harmonics by the low-pass filter. However, the magnitude of vripple(t) has been exaggerated in Figure 2-7. It is nearly always a good approximation to assume that the magnitude of vripple(t) is much smaller than the DC component V:

Therefore, output voltage v(t) is well approximated by is DC component V:

This approximation is known as The small ripple approximation or the linear ripple approximation. With this approximation, we replace the exponential or damped sinusoidal expressions for the inductor and capacitor waveforms with simpler linear waveforms. This approximation is justified provided that the switching period is shorter than the natural time constants of the circuit. Also, this approximation must be applied just to continuous variables: the inductor current and the capacitor voltage. Not to switching voltage, switching current of inductor voltage. Next, let us analyze the inductor current waveform. With the switch in position 1, the circuit reduces to Figure 2.8a.

 The inductor voltage vL(t)  is given by:

Applying small ripple approximation to equation (3):

The inductor current can be found by use of the definition:

Since the inductor voltage is essentially constant during the first interval (switch in position 1), the inductor current slope of equation (6) is also essentially constant and the inductor current increase linearly, as in Figure 2-10, where we can see vL(t) vs iL(t):

Similar arguments apply in the second interval (switch in position 2). The left side of the inductor is connected to ground, leading to the circuit of Figure 2.8b.

Using the small-ripple approximation leads to:

So:

Consequently, during the second interval the inductor current decrease linearly at a constant slope equation (8), as in Figure 2-10 position 2 of the switch.

In Figure 2-10 the inductor current iL(t) is symmetrical about I. Since:

is the peak ripple, the peak to peak ripple is:

Which is also the change in current . It is equal to the slope times the length of the first interval DTs:

Solution for the peak ripple  yields:

Typical values of the peak ripple lie in the range of 10%-20% of the full-load value of the DC component I. So, by design, the inductor current ripple is also usually small compared to the DC component I. The small approximation is justified:

Solution for L in equation (10) yields:

Equation (12) is commonly used to select the value of the inductance at the design of the buck converter. Equations (1) to (12) are derived from steady-state conditions. Source:

  •  Erickson R., Maksimovic D. Fundamentals of Power Electronics. Pp18-22.

Previously: The Buck Converter (Introduction)

Next: The inductor volt-second balance – Chapter three

Literature review by Larry Obando – Electrical Engineering

Universidad Simón Bolívar – Universidad Central de Venezuela

Mathematical modeling and simulation services

Contacto – Whatsapp +34633129287

dademuchconnection@gmail.com

Power Electronics

Convertidor Buck (Buck Converter): Electrónica de Potencia-Introducción.

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El convertidor Buck se ha construido para reducir el voltaje de CC, utilizando solo interruptores, inductores y condensadores no disipativos. El interruptor produce una forma de onda rectangular vs(t) como se ilustra en la Figura 2-1.

En la práctica, el interruptor se realiza utilizando dispositivos semiconductores de potencia, como transistores y diodos, que se controlan para apagarse y encenderse según sea necesario para realizar la función del interruptor ideal. La frecuencia de conmutación fs está en el rango de 1 KHz a 1 MHz, dependiendo de la velocidad de conmutación de los dispositivos semiconductores. La relación de trabajo D es la fracción de tiempo que el interruptor pasa en la posición 1 y es un número entre cero y uno. El complemento de la relación de trabajo, D’, se define como (1- D). Como sabemos por la Transformada de Fourier, el componente DC de una función periódica como vs(t), viene dado por su valor promedio <vs(t)>:

Resolviendo la ecuación (1) obtenemos la siguiente solución como se ilustra en la Figura 2-2:

El filtro está diseñado para pasar la componente de CC de vs(t), y rechazar sus componentes de CA en la frecuencia de conmutación y sus armónicos. Para lograr esto, diseñamos el filtro de manera que su frecuencia de corte sea mucho más baja que la frecuencia de conmutación. Entonces, el voltaje de salida v(t) de la Figura 2-3 es esencialmente igual al componente de CC de vs(t):

Entonces, el convertidor reductor tiene una característica de control lineal como en la Figura 2-4. Los sistemas de retroalimentación a menudo se construyen para ajustar la relación de trabajo D para regular el voltaje de salida del convertidor, como en la Figura 1.11:

 

Simulación con Matlab-Simulink

Fuente:

  • Erickson R., Maksimovic D. Fundamentals of Power Electronics.Pp15-18.

SIGUIENTE: Aproximación de ondulación pequeña – small ripple aproximation – Capítulo dos

Literature review by Larry Obando – Electrical Engineering

Universidad Simón Bolívar – Universidad Central de Venezuela

Mathematical modeling and simulation services

Contacto – Whatsapp +34633129287

dademuchconnection@gmail.com

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The Buck Converter – Chapter one: Introduction.

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The buck converter has been built to reduce the DC voltage, using only nondissipative switches, inductor and capacitors. The switch produces rectangular waveform vs(t) as illustrated in Figure 2-1.

In practice, the switch is realized using power semiconductor devices, such as transistors and diodes, which are controlled to turn off and on as required to perform the function of the ideal switch. The switching frequency fs is in the range of 1KHz – 1MHz, depending on the switching speed of the semiconductor devices. The duty ratio D is the fraction of time that the switch spends in position 1, and is a number between zero and one. The complement of the duty ratio, D’, is defined as (1- D). As we know from Fourier Transform, the DC component of a periodic function as vs(t), is given by its average value <vs(t)>:

Solving equation (1) we obtain the following solution as illustrated in Figure 2-2:

Equation (2) confirms that the DC component of the voltage output vs(t) of the buck converter is equal to or lower than the input voltage vg. In the practice, vs(t) is not ideal, so it has harmonic components. In consequence, the converter has to have a filter integrated on it. What remains is to insert a low-pass filter as illustrated in Figure 2-3:

The filter is designed to pass the DC component of vs(t), but to reject its components at the switching frequency and its harmonics. To accomplish this, we design the filter such that its cutoff frequency is much lower than the switching frequency. So, the output v(t) voltage from Figure 2-3, is essentially equal to the DC component of vs(t):

The network of Figure 2-3 allows control of the output. Figure 2-4 is the control characteristic of the buck converter:

So, the buck converter has a linear control characteristic. Feedback systems are often constructed that adjust the duty ratio D to regulate the converter output voltage, such as in Figure 1.11:

Simulation with Matlab-Simulink

      Source:

  •  Erickson R., Maksimovic D. Fundamentals of Power Electronics.Pp15-18.

NEXT: The small ripple approximation – Chapter two

Literature review by Larry Obando – Electrical Engineering

Universidad Simón Bolívar – Universidad Central de Venezuela

Mathematical modeling and simulation services

Contacto – Whatsapp +34633129287

dademuchconnection@gmail.com