线性电路传递函数：快速分析技术导论

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"Linear Circuit Transfer Functions(2016)" 是一本由Christophe P. Basso编写的关于线性电路传递函数的书籍，专注于快速分析技术在开关电源领域的应用。开关电源是一种高效的电力转换设备，它通过控制开关元件（如MOSFET或IGBT）的开闭状态来调节输出电压。这种电源设计可以显著提高能源效率，并广泛应用于从消费电子到工业系统等各个领域。本书的内容可能涵盖了以下关键知识点： 1. **基本概念**：首先，书籍可能会介绍开关电源的基本工作原理，包括连续模式和断续模式的操作，以及DC-DC转换器的类型，如降压、升压、反相和SEPIC变换器。 2. **线性电路理论**：线性电路的传递函数是理解和分析电路动态行为的关键工具。书中可能会深入探讨阻抗、频率响应、拉普拉斯变换和傅里叶分析等概念，这些都是分析开关电源中滤波器、控制器和反馈网络性能的基础。 3. **快速分析技术**：书中提到的“快速分析技术”可能包括简化计算方法、近似技巧和图形解法，这些可以帮助工程师快速估算电路性能，而不必进行复杂的数学计算。 4. **控制系统的稳定性**：书中会讨论如何使用传递函数来评估开关电源的稳定性，包括闭环控制系统的波特图分析和奈奎斯特稳定性判据。 5. **电磁兼容性（EMC）**：由于开关电源的高速开关操作，EMC问题是个重要话题。书中可能涵盖了减小电磁干扰（EMI）的策略，如选择适当的滤波器设计和布局优化。 6. **实际应用案例**：为了使理论知识更具实践意义，书中的例子和练习可能会涉及实际开关电源设计的问题，让读者有机会应用所学知识解决真实世界的问题。 7. **版权与许可信息**：书籍的版权信息提醒读者，对内容的复制、存储、传播需遵守版权法，获取使用许可后方可进行。 "Linear Circuit Transfer Functions(2016)" 是一本适合电源工程师和电子工程专业学生的深度学习资料，通过详细阐述线性电路的传递函数及其在开关电源设计中的应用，帮助读者掌握关键的分析和设计技能。

Electrical Analysis – Terminology

and Theorems

This ﬁrst chapter is an introduction to some of the basic deﬁnitions and terms you must understand in

order to perform electrical analysis with efﬁciency and speed. By electrical analysis, I imply ﬁnding

the various relationships that characterize a particular electrical network. To excel in this ﬁeld, as in

any job, you need to master a few tools. Obviously, they are innumerable and I am sure you have

learned a plethora of theorems during your student life. Some names now seem distant simply because

you never had a chance to exercise them. Or you actually did but implementation was so obscure and

complex that you left quite a few of them aside. This situation often happens in an engineer’s life

where real-case experience helps clean up what you have learned at school to only retain techniques

that worked well for you. Sometimes, when what you know fails to deliver the result, it is a good

opportunity to learn a new procedure, better suited to solve your current case. In this chapter, I will

review some of the founding theorems that I extensively use in the examples throughout this book.

However, before tackling deﬁnitions and examples, let us ﬁrst understand what the term transfer

function designates.

1.1 Transfer Functions, an Informal Approach

Assume you are in the laboratory testing a circuit encapsulated in a box featuring two connectors: one

for the input, the second for the output. You do not know what is inside the box, despite the transparent

case in the picture! You now inject a signal with a function generator to the input connector and

observe the output waveform with an oscilloscope. Using the right terminology, you drive the circuit

input and observe its response to the stimulus. The input waveform represents the excitation denoted

u and it generates a response denoted y. In other words, the excitation variable propagates through the

box, undergoes changes in phase, amplitude, perhaps induces distortion etc. and the oscilloscope

reproduces the response on its screen.

The waveform displayed by the oscilloscope is a time-domain graph in which the horizontal axis x

is graduated in seconds while the vertical axis y indicates the signal amplitude (positive or negative).

Its dimension depends on the observed variable (volts, amperes and so on). The input waveform is

denoted in lower case as it is an instantaneous signal, observed at a time – the instant t – u(t). A similar

notation applies to the output signal, y(t). In Figure 1.1, you see a low duty ratio square-wave injected

in the box engendering a rather distorted waveform on the output.

Linear Circuit Transfer Functions: An Introduction to Fast Analytical Techniques, First Edition. Christophe P. Basso.

2 Linear Circuit Transfer Functions

Figure 1.1 A black box featuring an input and an output signal. What is the relationship linking output and input

waveforms?

This ringing signal tells us that the box could associate resonant elements, probably capacitors and

inductors but not much more than that. If we change the excitation, what type of shape will we obtain?

Knowing what is inside the box will let us predict its response to various types of excitation signals.

There are several available ways to characterize an electrical linear circuit. One of them is called

harmonic analysis. The input signal is replaced by a sinusoidal waveform and you observe how the

stimulus propagates through the box to form the response. This is shown in Figure 1.2:

The excitation level must be of reasonable amplitude – understand small – so that the response

signal is not distorted. The input signal dc bias must also be set accounting for the physical constraints

imposed by the active circuit so that upper- or lower-rail saturation is avoided. In other words, the box

internal circuitry is not overdriven and remains linear during the analysis. Linearity is conﬁrmed if the

output signal is sinusoidal with the same frequency as the input sine and only varies in amplitude and

phase while you ac-sweep the network. This is a so-called small-signal analysis. In the Laplace

domain, you perform such harmonic analysis when you set s  jω in which ω  2π f represents the

angular frequency expressed in radians per seconds (rads/s). Laplace analysis with s  jω applies to

linear circuits only.

Should you increase the input signal amplitude or change the operating bias point, slewing or

clipping may happen. In this case, you explore the box large-signal or nonlinear response. This is a

characterization different than the small-signal approach and it offers another insight into the circuit

operation. Let us keep linear and once the right input amplitude is found, i.e. a signal of comfortable

amplitude is observed on the oscilloscope screen, the frequency is varied step by step while output

amplitude/phase couples are recorded in an array. At each frequency point f, we store the ratio of the

response amplitude Yf in volts also. At each frequencyin volts to the excitation amplitude Uf

point f, we save the phase information linking both input and output waveforms. As U and Y are

complex variables affected by a magnitude and a phase, we can write:

Ys

s (1.1)

Us

represents a transfer function, a mathematical relationship linking a response signal Y to an

excitation signal U. Please note that the excitation signal U resides in the transfer function

Figure 1.2 The black box is now driven by a sinusoidal stimulus for a small-signal analysis.

3Electrical Analysis – Terminology and Theorems

denominator while the response Y sits in the numerator. It will always be this way throughout the

book.

The transfer function is a complex variable characterized by a magnitude noted

and an

argument, A

also noted arg A

. The ratios Yf we have stored correspond to thef f =Uf

transfer function magnitude (also called modulus) observed at a frequency f while the phase difference

between Y and U represents the transfer function argument or phase at the considered frequency. The

transfer function magnitude dimension depends on the observed variables as we will later see. Here,

because volts are involved for both variables, the transfer function magnitude is dimensionless or

unitless. Furthermore, A

can only be greater than or equal to zero. It is what makes the difference

between an amplitude which can take on any value, positive, null or negative and a magnitude which

can only be zero or positive. If it is 0, there is no output signal. If A

is less than 1, we talk about

attenuation. Now, if A

is greater than 1, it is designated as a gain. If the magnitude can only be a null

or positive number, what about a gain of -2 then? It simply characterizes a stage offering a gain of 2,

lagging or leading the excitation signal phase by 180°.

1.1.1 Input and Output Ports

It is convenient to represent our box as a two-port circuit. A port is a pair of connections that can input

or output signals such as voltage and current. Figure 1.3 shows an illustration of this principle where

you see two connecting ports, one input and one output.

Under some conditions, a port can take on the input and output roles at the same time. Imagine you

want to measure the output impedance of the box. To realize this measurement, you classically

implement Figure 1.4 where a current across the output terminals is injected while the voltage across

the same terminals is observed This is what is called a single injection, i.e. one stimulus and one

response. In this experiment, the box input port is shorted (see Appendix 1A). The excitation variable

is the current I

out

(s) injected into the port while the response is the voltage V

out

(s) collected across the

port’s terminals. The output impedance Z obtained from the ratio of the port voltage to the injected

Figure 1.3 The input port receives the excitation signal while the output port delivers the response.

Figure 1.4 A port can be both an input and an output at the same time. Here, an output impedance measurement.

4 Linear Circuit Transfer Functions

current is a transfer function. It has the dimension of an impedance expressed in ohms:

out

s

out

s (1.2)

out

s

out

, the excitation signal lies in the denominator while the response, V

out

, stands in the numerator. We

will come back on this important peculiarity.

If input and output connectors are ﬁxed, physical ports, which let you respectively inject and

observe signals, nothing prevents one from creating other observation ports as needed. Simply

remove a resistor, a capacitor or an inductor and its connecting points become a new port. This port

can now be used as a new input stimulus or as an output variable you want to observe. As already

mentioned, this newly created port can also play the role of an input and output port at the same time.

In that case, the box originally featuring one input and one output, becomes a two-input/two-output

system as illustrated in Figure 1.5 in which the inductor has been removed. Using adequate

terminology, we analyze the system by performing a double-injection: two stimuli – inputs 1 and

2 – giving two responses, outputs 1 and 2.

In this example, the voltage across the removed inductor terminals is the response while the

injected current is the excitation signal. By dividing the port voltage by the injected current, we have

the resistance offered by the port terminals when the element initially connected has been removed. In

other words, we ‘look’ at the resistance offered by the inductor port as shown in Figure 1.6 where the

symbol R? and the arrow imply this exercise. Expressed in a different manner, we ﬁnd the equivalent

output resistance exhibited by the port when ‘driving’ the inductor, hence the name driving point

resistance or driving point impedance abbreviated as DPI. Combining resistance and inductance

gives us a time constant τ (‘tau’) associated with this inductive element:

τ  (1.3)

To conduct this exercise and ﬁnd the resistance R , you can directly look at the sketch and infer the

resistive series-parallel arrangement without solving a single equation. This exercise is called network

inspection: you simply observe the network in certain conditions (for instance in dc, or when V

is set

to 0) and ﬁnd resistance values by observing how components are connected together. For example, in

Figure 1.6, what resistance do you ‘see’ looking into the inductor port while capacitor C is

Figure 1.5 If you remove a component from this circuit, its connections become a connecting port. You can bias

this port and consider it as a new input, or as a new output, or both of them at the same time.

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