Impedance, Admittance and Equivalent Conversion of Sinusoidal AC Circuits

In the previous sections, the phasor relationship between voltage and current on resistors, inductors, and capacitors has been derived, and the concepts of reactance and capacitive reactance have been introduced. When the excitation source in the circuit is a single-frequency sinusoidal alternating current, the response voltage and current of each branch are also sinusoidal quantities of the same frequency. Therefore, in a sinusoidal steady-state circuit, any linear passive two-terminal network can be represented by a complex impedance and admittance.

Next, consider the case of an RLC series circuit. Assume that a sinusoidal voltage with an angular frequency of is applied to both ends of the RLC series circuit , as shown in Figure 3-8-1a. From the above analysis, it can be seen that the series circuit can generate the same voltage as the excitation voltage.

Figure 3-8-1

The excitation voltage is a sinusoidal alternating current i with the same frequency . According to Kirchhoff's voltage law, the voltage equation in phasor form can be obtained:

（3-8-1）

Let the current expression in the series circuit be , and the phase form be , according to the previous sections, the voltage equation can be expressed as

（3-8-2）

Where is the equivalent complex impedance of the series circuit, which is equal to the ratio of the terminal voltage phase to the current phase. The real part of the impedance Z is the resistance of the circuit, and the imaginary part is the reactance of the circuit. The reactance is equal to the difference between the inductive reactance and the capacitive reactance , and it is a signed algebraic quantity. The complex impedance can be expressed in polar coordinates

（3-8-3）

Where z is the impedance modulus, ; is the impedance angle, .

For any complex passive one-port network, when a sinusoidal voltage (or current) is applied to the port, the current (or voltage) of each branch in the network is a sinusoidal function with the same frequency as the excitation source. Similar to the linear resistance one-port network, which can be represented by an equivalent resistance, any linear passive one-port network can also be represented by an equivalent input impedance or admittance. The impedance Z of a one-port network is defined as the ratio of the input voltage phasor to the input current phasor , that is:

In the formula, voltage and current are taken as the associated reference direction. The input admittance Y is defined as the ratio of the input voltage to the input voltage , that is:

In the formula, the voltage and current also take the associated reference direction.

In actual circuit calculations, the conversion between impedance and admittance needs to be determined according to the series and parallel conditions of the circuit, as illustrated by the following example.

Example 3-8-1 In the circuit shown in Figure 3-8-3, given that ,,,, find the input impedance of the circuit. If an external voltage is applied , find the current in each branch.

Figure 3-8-3

Solution: First calculate the equivalent impedance on the right side of the cb terminal, the equivalent admittance of the impedance

but:

Equivalent impedance at the right end of cb :

Circuit input impedance:

Assume , then: