Actual Circuits and Circuit Models - Circuit Elements-EEWORLD

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The research object of circuit principle is not the actual circuit, but the idealized circuit model abstracted from the actual circuit. In order to facilitate the analysis and design of circuits, in circuit theory, it is necessary to establish their physical models based on the main physical properties of each component in the actual circuit. These abstract basic physical models are called ideal circuit elements, or circuit elements for short. Actual circuit devices are a combination of ideal circuit elements. The circuit composed of circuit elements is the circuit model of the actual circuit, which is an approximation of the actual circuit within a certain accuracy range. For an actual circuit, how to construct its circuit model based on its circuit characteristics requires rich circuit knowledge and the use of relevant professional knowledge.

Circuit components

1. Resistor

A resistor is a two-terminal component that reflects the conversion of electrical energy into other forms of energy. It is referred to as a resistor and is represented by the letter R. The reciprocal of resistance is called conductance, represented by the letter G. In the International System of Units, the unit of resistance is ohm, symbol "Ω", and the unit of conductance is siemens, symbol "S".

Any resistor whose terminal voltage is proportional to the terminal current is called a linear resistor. The symbol of a linear resistor is shown in Figure 1-2-1(a). The volt-ampere characteristic of a linear resistor is a straight line passing through the origin, and its slope is the resistance value, as shown in Figure 1-2-1(b):

Figure 1-2-1

The voltage u across the linear resistor and the current i passing through it satisfy Ohm's law. For the circuit shown in Figure 1-2-1, there is a mathematical expression: or (Formula 1-2-1)

The power and energy consumed in the linear resistor are: (Formula 1-2-2), (Formula 1-2-3)

In the International System of Units, the unit of power is watt, symbol "W", and the unit of energy is joule, symbol "J". The unit of measurement of the electric meter is kilowatt-hour (KW·h), also known as degree.

1 degree = 1kW·h = 1000·3600 J = 3.6×106 J

Any resistor whose terminal voltage and terminal current are not proportional is called a nonlinear resistor. The resistance value of a nonlinear resistor changes with the magnitude or direction of the current passing through it. It cannot be expressed by a definite resistance value, but by a volt-ampere characteristic.

2. Capacitor Components

Capacitor element is a two-terminal element that embodies electric field energy storage. It is referred to as capacitor and represented by the letter C. The symbol is shown in Figure 1-2-2. In the International System of Units, the unit of capacitance is farad, and the symbol is "F".

Figure 1-2-2

In actual circuits, as long as there is a physical phenomenon of electric field energy storage, the corresponding capacitor element can be abstracted. According to common physics knowledge, the terminal voltage of a capacitor has a certain relationship with the charge. If the charge on a capacitor is proportional to the terminal voltage, the capacitor is called a linear capacitor, and there is an expression: (Formula 1-2-4)

In the International System of Units, the unit of charge q is coulomb; the unit of voltage V is volt. If the charge on the capacitor is not proportional to the terminal voltage, and the size of the capacitor is related to the charge or voltage, then the capacitor is called a nonlinear capacitor. Nonlinear capacitance is represented by coulomb characteristics. If the coulomb characteristics of a capacitor (whether linear or nonlinear) change with time, it is called a time-varying capacitor, otherwise, it is called a non-time-varying capacitor.

The current in the capacitor is equal to the rate of change of charge. For the circuit shown in Figure 1-2-2, there is a mathematical expression: (Formula 1-2-5)

For linear non-time-varying capacitance, (Equation 1-2-5) can be written as: (Equation 1-2-6)

In a DC circuit, the rate of change of voltage V with respect to time t is zero, so the current I is zero. Therefore, DC current cannot pass through the capacitor, and the capacitor has the function of isolating DC.

Integrate (Formula 1-2-6) from to t, and we get: (Formula 1-2-7)

(Formula 1-2-7) shows that the capacitor voltage is related to the voltage at the time in addition to the charging current, that is, it has memory, so the capacitor is called a memory element. The voltage of the aforementioned resistor at any time is only related to the instantaneous current at that moment, and has nothing to do with the previous power-on condition, so the resistor is called a non-memory element.

Capacitor elements are energy storage elements. The energy storage of capacitors is:

(Formula 1-2-8)

3. Inductor components

Inductance is a two-terminal component that reflects magnetic field energy storage, referred to as inductance, represented by the letter L, and the symbol is shown in Figure 1-2-3. In the International System of Units, the unit of inductance is Henry, with the symbol "H".

Figure 1-2-3

In actual circuits, as long as there is a physical phenomenon of magnetic field energy storage, the corresponding inductor element can be abstracted. According to common physics knowledge, the magnetic flux of the inductor has a certain relationship with its terminal current. If the magnetic flux of the inductor is proportional to its terminal current, the inductor is called a linear inductor, and there is an expression:

(Formula 1-2-10)

In the International System of Units, the unit of flux is Weber and the unit of current is Ampere. If the flux of the inductor is not proportional to the current at its end, and the magnitude of the inductance is related to the flux or current, then the inductor is called a nonlinear inductor. Nonlinear inductance is represented by the Weber characteristic. If the Weber characteristic of the inductor (whether linear or nonlinear) changes with time, it is called a time-varying inductor, otherwise, it is called a non-time-varying inductor.

The induced voltage on the inductor is equal to the rate of change of the magnetic flux. For the circuit shown in Figure 1-2-3, there is a mathematical expression:

(Formula 1-2-11)

For linear non-time-varying inductance, (Equation 1-2-11) can be written as:

(Formula 1-2-12)

In a DC circuit, the rate of change of current I with respect to time is zero, so the voltage V is zero. Therefore, for DC, the inductor element is equivalent to a short-circuited wire.

Integrate (Formula 1-2-12) from to t, and we get:

(Formula 1-2-13)

Like capacitors, inductors are also memory elements. The magnetic field energy storage of inductors is:

(Formula 1-2-14)

Resistor R, capacitor C, and inductor L are the three most basic passive components in the circuit. The following introduces active components.

4. Independent power supply components

In actual circuits, there is generally a power source, which can be various batteries, generators, electronic power supplies, or tiny electrical signals. In circuit analysis, according to the different characteristics of the power source, two different circuit models can be established to characterize the power supply element: one is an ideal voltage source, and the other is an ideal current source.

(1) Ideal voltage source

Figure 1-2-4 shows three symbols of ideal voltage source. Figure is the commonly used symbol in Chinese textbooks, Figure is the commonly used symbol in British and American textbooks, and Figure is the symbol of battery pack. This book uses the symbol of Figure . represents the voltage drop from the positive electrode to the negative electrode of the voltage source as volts, represents the potential increase from the negative electrode to the positive electrode of the voltage source as volts.

Figure 1-2-4

An ideal voltage source provides a certain voltage to the outside world, and the magnitude of its voltage does not change with the magnitude of the current flowing through the voltage source. The volt-ampere characteristic of an ideal voltage source is shown by the solid line in Figure 1-2-5, which is a straight line parallel to the I axis with an intercept of .

Figure 1-2-5

Its volt-ampere characteristic shows that no matter what the size and direction of the current I flowing through the ideal voltage source is, the voltage across the two ends of the ideal voltage source is always .

The volt-ampere characteristic of a real voltage source is shown by the dotted line in Figure 1-2-5 . The linear equation describing the dotted line is:

(Formula 1-2-15)

Where:

The actual voltage source model can be drawn from (Equation 1-2-15), as shown in Figure 1-2-6. It consists of an ideal voltage source and an internal resistor in series.

Figure 1-2-6

(2) Ideal current source

Figure 1-2-7 shows two symbols of ideal current source. Figure is a commonly used symbol in Chinese textbooks, and Figure is a commonly used symbol in British and American textbooks.

Figure 1-2-7

The ideal current source provides a certain current to the outside world, and the magnitude of the current does not change with the magnitude of the voltage at both ends of the current source. The volt-ampere characteristic of the ideal current source is shown as the solid line in Figure 1-2-8, which is a straight line parallel to the U axis and perpendicular to the I axis . It can be seen from the figure that no matter whether the voltage at both ends of the ideal current source is positive or negative, large or small, the current I output by the ideal current source remains unchanged.

Figure 1-2-8

The volt-ampere characteristic of a real current source is shown by the dotted line in Figure 1-2-8. The equation of the dotted line is:

(Formula 1-2-16)

Figure 1-2-9

From (Formula 1-2-16), we can draw the actual current source model, as shown in Figure 1-2-9, which is composed of an ideal current source and a resistor in parallel. In (Formula 1-2-16), .

5. Controlled power supply components

A power source whose voltage of a voltage source or current of a current source is controlled by the voltage or current of other branches in the circuit is called a controlled source. A controlled source has two ports and is divided into four types, namely, voltage-controlled current source , voltage-controlled voltage source, current -controlled voltage source and current-controlled current source , as shown in Figure 1-2-10, where g, , r, and a are control coefficients. In Figure 1-2-10 , the controlled current source is proportional to the control voltage, and g is a proportional constant with the dimension of conductance, called transfer conductance. In Figure 1-2-10 , the controlled voltage source is proportional to the control voltage, and is a proportional constant without dimension, called transfer voltage ratio. In Figure 1-2-10 , the controlled voltage source is proportional to the control current, and r is a proportional constant with the dimension of resistance, called transfer resistance. In Figure 1-2-10, the controlled current source is proportional to the control current, and a is a proportional constant without dimension, called transfer current ratio.

Figure 1-2-10

A controlled source whose controlled quantity is proportional to the control quantity is called a linear controlled source, otherwise, it is called a nonlinear controlled source.

Actual components such as transistors, operational amplifiers, and transformers can be represented by circuit models containing controlled sources. For example , the triode shown in Figure 1-2-11 has a small signal circuit model of a current source controlled by current as shown in Figure 1-2-11. In short, controlled sources are often used when analyzing electronic circuits.