In this page, equations that describe the behaviour of some circuits are formed. Once you have formed the equations, you can calculate the coefficients based on the component values and then select the interactive page for the type of input to the circuit. In that page, you can key-in the coefficients. You can then obtain the solution step by step or see the solution.
For the in Fig. 27, equations that describe its behaviour are developed.
For the circuit in Fig. 27, the differential equation is obtained as shown below.
For this circuit, the initial conditions required are stated below.
We can obtain a Laplace transform function for the inductor current as follows. The transformed circuit is shown in Fig. 28.
Equations (1.6), (1.7) and (1.8) are self-explanatory.
Replace VC(s) in equation (1.7) by the expression for VC(s) in equation (1.8). Then you get equation (1.9). Re-arrange equation (1.9) and get equation (1.10).
Equation (1.11) presents the inductor current as a Laplace transform function. You can derive equation (1.11) from equations (1.4) and (1.5).
The state-equation, expressed by equation (1.14) can be expressed in terms of Laplace transforms.
We can get the following equation from equation (1.15).
Hence we get the Laplace transform function for the inductor current, as shown below.
It is the same function, obtained earlier.
Obtain the circuit equations for the circuit in Fig. 29.
Solution:
Differential Equations Approach
From the circuit in Fig. 29, we can obtain the following equations. The process is a bit tedious.
From equation (2.4), we can obtain a Laplace transform function for current i2(t).
Laplace Transforms Approach
From the transformed circuit in Fig. 30, we can obtain the following equation.
Equations (2.5) and (2.6) appear to be different. They are the same if equation is (2.7) is true.
It is seen that equations (2.5) and (2.6) are equal to each other.
State variableApproach
From the circuit in Fig. 29, we can obtain the following equations. It is relatively easy to form the state equation.
We can express the above equation in terms of Laplace transforms.
After simplification, we get the following equation.
Obtain the circuit equations for the circuit in Fig. 31.
Solution:
Differential Equations Approach
From the circuit in Fig. 31, we can obtain the following equations.
Replace iL in equation (3.1) by the expression in equation (3.3).
Laplace Transforms Approach
From the transformed circuit in Fig. 32, we can obtain the following equations.
Re-arrange equation (3.5).
It is quite tedious to obtain equation (3.6). We can verify whether we get the same expression, starting from the differential equation. From equation (3.4), we get that
It can be seen that equations (3.6) and (3.7) are equal to each other, as shown below.
State variableApproach
From the circuit in Fig. 31, we can obtain the following equations.
We can get the second equation as follows.
The state equation can then be obtained.
The examples have shown that it is tedious to deal with symbolic functions. Given numerical values, it is relatively easy to get the circuit equations. Hence only some examples are shown to illustrate the process of obtaining circuit equations in terms of symbols. Given any circuit, the process is similar. The next page presents an interactive example of a second-order circuit with a step input.