Circuit laws, circuit elements and their models in the complex frequency domain (s domain)

The two most important laws in circuits are Kirchhoff's current law (KCL) and Kirchhoff's voltage law (KVL), which are expressed as:

KCL: , KVL:

Performing Laplace transform on the equations of the two laws, we have:

KCL: , KVL:

The above two equations are the complex frequency domain (s domain) forms of Kirchhoff's laws. This shows that the image function of each branch current still follows KCL; the image function of each branch voltage in the loop still follows KVL.

The complex frequency domain (s domain) model of each circuit element, also called the operational circuit model, is introduced below.

1. Linear resistance element

Figure 9-4-1 (a) shows the time domain model of a linear resistor element. When the voltage and current reference directions are selected to be consistent, the relationship between the voltage and current is:

The relationship between voltage and current functions obtained by Laplace transformation is:

(Formula 9-4-1)

Therefore, the complex frequency domain (s domain) model of the resistor is shown in Figure 9-4-1 (b).

2. Linear Inductor Components

Figure 9-4-2

Figure 9-4-2 (a) shows the time domain model of a linear inductor. When the reference directions of voltage and current are consistent, the time domain relationship between voltage and current is:

After Laplace transformation, we get:

(Formula 9-4-2)

According to (Equation 9-4-2), the complex frequency domain model of the inductor element can be drawn, as shown in Figure 9-4-2(b), where sL is called the operational reactance of the inductor, which depends on the initial value of the inductor current and is called the additional operational voltage.

3. Linear capacitor components

Figure 9-4-3

Figure 9-4-3(a) shows the time domain model of a linear capacitor element. When the reference directions of voltage and current are consistent, the time domain relationship between voltage and current is:

After Laplace transformation, we get:

(Formula 9-4-3)

According to (Equation 9-4-3), the complex frequency domain model of the capacitor element can be drawn, as shown in Figure 9-4-3(b), where is called the operational capacitive reactance of the capacitor, and depends on the initial value of the capacitor voltage, which is called the additional operational voltage.

4. Independent power supply

For independent voltage sources and current sources, we only need to transform the corresponding voltage source voltage and current source current in the time domain through Laplace transformation to obtain the corresponding image function. For example: the DC voltage source voltage is transformed into ; the sinusoidal current source power is transformed into .

5. Controlled power supply

For a controlled power supply, if the control coefficient is a constant, then the complex frequency domain circuit model is the same as its time domain circuit, and the form remains unchanged. Figure 9-4-4 (a) is a VCVS in the time domain, and (b) is its complex frequency domain circuit model. The complex frequency domain circuit models of other forms of controlled power supplies can be obtained in the same way.

Figure 9-4-4