Alternating Current Theory

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February 11, 2008 Theoretical Issues

Calculation and analysis presented here are used to resolve AC circuits in steady state. Waveforms present in this circuits are sinusoidal of the same frequency and may differ only by phase shift.

Theory:

Prior to calculation of the actual form of the signal presented below:

should be replaced by complex notation as follows:

Thus, the actual signal is equal the imaginary part of the complex signal:

Entered notation greatly simplifies the calculation, allowing the use advantages of complex numbers theory. Voltage and current in this notation has the form:

where

and

are the complex amplitudes of the voltage and current dependent on the signal frequency of

and phase

. It is worth to see that the factor

gives time dependence and its present in all components in linear equations, describing linear circuits. In result, factor

can be simplified in first calculation step. As a result, every voltage source or current source should be replaced by corresponding value of the complex amplitude, which leads to simple algebraic equations. The final result can by easily converted back to form dependent of time, multiplying it by the factor

and calculating the imaginary part of the so formed expressions.

Resistors

Resistors in the circuit alternating current are responsible for active power dissipation. Resistors don't store electricity. The complex impedance

is equal to the resistance R.

Capacitors

Capacitors can collect and then give electric charge. The current flowing through the capacitor gives the following formula:

where C is the capacitance of the capacitor. The unit of capacitance is 1F (Farad) Based on the above formula and equations mentioned earlier, the impedance of the capacitor

can be calculate as follows:

After simplifying by a factor of

we get:

Inductors

Inductor, similar to capacitors may store electricity. The voltage on the inductor is given by:

where L is the inductance of the coil. The unit of inductance is 1H (Henr) Based on the above formula and equations presented earlier - impedance of the coil

can be calculated as follows:

After simplifying by a factor of

we get:

Examples:

Example 1:

In the circuit of figure 1, the task is to calculate the voltage on the capacitor. The supply voltage is equal to

Figure 1:

After saving value of the voltage source in complex notation we get:

In the following calculations the factor

can by omitted, and for the calculation only the complex amplitude

is used. After calculations the result should be multiplied by the previously omitted factor. Final result is equal imaginary part of formula given by this product. In this example:

thus

. Using Ohm's Law we have:

and:

The two previous equations gives:

Multiplying

by the factor

and using the fact that

we get:

In the next step, this equation can be converted to an exponential form:

The final result in the real form is equal the imaginary part of

Example 2:

In the circuit of figure 2, the task is to calculate the voltage on the capacitor. The supply voltages are equal to

and

Figure 2:

After saving value of the voltage source in complex notation we get:

Using nodal method analysis we get following equation:

Further part of the calculation is similar to those in the first example. After calculations the result should be multiplied by the previously omitted factor