Prerequisites: Electrical Power Sources or simpler named, Sources are components that provide the power to an electrical circuit. The power coming from sources dissipate in the circuit forming different signals. Dependent and Independent Sources Sources are shown by two pretty standard symbols depending on if they are dependent or independent. They are shown below. Inside of the empty space of the circle or diamond in the source symbol will be filled with other symbols depending on their type, mentioned further below.
A Dependent Source has an output which has one or more of its parameters such as amplitude, offset or frequency dependent to any other entity in the circuit such as temperature, voltage or current. In contrast, an Independent Source generates its output completely independent to any other circuit parameter. Ideal and NonIdeal Sources To make calculations simpler, sources are divided into ideal an nonideal sources. Ideal sources are easier to deal with because engineers neglect some parameters in them. But to make the results closer to reality, using nonideal sources help much better. In nonideal sources, output of the source can be different depending on the circuit it is connected to, making them a little bit harder to include in calculations. It is always very important to consider the nonideality of a source as virtually there is no ideal source and this can significantly effect the final results. Voltage and Current Sources There are two different types of sources: Voltage Sources and Current Sources. Ideal voltage and current sources are defined as below: Ideal Voltage Source: is a source that provides a voltage signal (DC, AC or both) completely independent to the amount of current passing through it. Ideal Current Source: is a source that generates a current signal (DC, AC or both) completely independent to the level of voltage across it. Voltage and current sources are generally shown by symbols below no matter what their output waveform looks like. Usually the value of the source output is written beside the source to clarify its waveform. It can be a DC or AC signal or a combination of both, which is in terms of time. In figure below V_{s} or I_{s} stand for source voltage or current respectively.
Any circuit connected to a power source is called a Load in general, where the source should give power to. A high load is a load that consumes a larger power from the source and viseversa. Assume circuits below with ideal sources:
In the case of the voltage source connected to a resistor, the output voltage of the source is completely independent to the loading resistor as it is an ideal source. Only the current flowing through the voltage source and resistor can be variable depending on the resistor, which is equal to I_{o} = V_{s} / R_{L}. The power consumed in a resistor can be calculated from the general formula below:
This shows that a smaller resistor actually acts as a higher load as it increases the current. In the case of a current source connected to a loading resistor, the current flowing into the source and the resistor is completely independent to the loading resistor. Only the voltage across the source and resistor changes, which is equal to V_{o} = R_{L }. I_{s}. This time from the formula of power above it is seen that a larger resistor poses as a larger load as a larger resistor results in a higher voltage and therefore a larger power consumption. Therefore depending on the kind of source you have the meaning of a higher load can be different. A voltage source is a much more common source in circuits, such as in batteries or a wall power plug, and therefore mostly a lower resistance poses as a bigger load. To show non ideality of a voltage source, they add a series resistor to its output, therefore with more current drawn out of the source, more voltage drops across the source resistor and decreases the output level of the source. Below a nonideal voltage source is connected to a resistor.
It is seen that some voltage is dropped on the source output resistance, which is equal to:
Therefore the output voltage across the load resistor is equal to:
This shows that the output voltage drops by the value of V_{d}, and is dependent on the current flowing in the circuit. It is seen from the formula that when R_{s} gets smaller compared to R_{L} it gets less effective. When R_{s} approaches zero then V_{o} is almost equal to V_{s} and the source gets closer to ideal. An ideal voltage source has a R_{s} equal to zero. In case of a current source, they add a parallel resistor across the ideal current source. A larger R_{L} results in a larger voltage across the components. Under higher voltages across the source, more current is drawn by the source resistor, decreasing the current coming out of the nonideal source. Below a nonideal current source is connected to a load resistor.
The current flowing out of the nonideal current source is calculated using the formula below:
Here the formula shows that in oppose to an ideal current source, the output current of a nonideal current source is dependent to the output load resistor. The output current decreases when the output resistor value increases. When R_{s} gets larger compared to R_{L}, I_{o} gets closer to I_{s} making the source more ideal. An ideal current source has a R_{s} equal to infinity. The Source Resistance shown with R_{s} is the cause on this nonideality and is used to model the nonideality of sources. Types of Output Waveforms Some times to make the sources more specific and understandable they use symbols in a source depending on the type of waveform it generates. Four of the most common symbols are gathered in the table below:
They write the equation or value beside the source, which represents the output waveform and further in case of nonideal sources they add proper resistors to the source. Figure below is an example of a circuit that contains both dependent and independent sources in it.
As you can see there is an independent voltage source and a dependent current source, which its value is dependent to the voltage across R_{1} by a factor of a. It is a very good exercise to try and solve this circuit. You can assume a ground node somewhere, give numeric values to every parameter and solve for the voltages and currents of all the components using KVL and KCL laws.

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