An ideal dc voltage source produces a load voltage that is constant. The simplest example of an ideal dc voltage source is a perfect battery, one whose internal resistance is zero.

Figure 1(a) shows an ideal voltage source connected to a variable load resistance of 1 V to 10 MΩ. The voltmeter reads 10 V, exactly the same as the source voltage.

Figure 1(b) shows a graph of load voltage versus load resistance. As you can see, the load voltage remains fixed at 10 V when the load resistance changes from 1 Ω to 1 MΩ.

# Ideal Voltage Source

In other words, an ideal dc voltage source produces a constant load voltage, regardless of how small or large the load resistance is. With an ideal voltage source, only the load current changes when the load resistance changes.

An ideal voltage source is a theoretical device; it cannot exist in nature. Because when the load resistance approaches zero, the load current approaches infinity. No real voltage source can produce infinite current because a real voltage source always has some internal resistance.

## Second Approximation

The second approximation of a dc voltage source includes this internal resistance. Figure 2(a) illustrates the idea. A source resistance R_{S} of 1Ω is now in series with the ideal battery. The voltmeter reads 5 V when R_{L} is 1 Ω.

Why? Because the load current is 10 V divided by 2 Ω, or 5 A. When 5 A flows through the source resistance of 1 Ω, it produces an internal voltage drop of 5 V.

This is why the load voltage is only half of the ideal value, with the other half being dropped across the internal resistance.

Figure 2(b) shows the graph of load voltage versus load resistance. In this case, the load voltage does not come close to the ideal value until the load resistance is much greater than the source resistance. But what does much greater mean? In other words, when can we ignore the source resistance?

## Stiff Voltage Source

Now is the time when a new definition can be useful. So, let us invent one. We can ignore the source resistance when it is at least 100 times smaller than the load resistance.

Any source that satisfies this condition is a stiff voltage source. As a definition, Stiff voltage source:

R_{S} < 0.01R_{L} (equation 1)

This formula defines what we mean by a stiff voltage source. The boundary of the inequality (where < is changed to =) gives us the following equation:

R_{S} = 0.01R_{L}

Solving for load resistance gives the minimum load resistance we can use and still have a stiff source:

RL_{(min)} = 100R_{S} (equation 2)

In words, the minimum load resistance equals 100 times the source resistance.

Equation 2 is a derivation. We started with the definition of a stiff voltage source and rearranged it to get the minimum load resistance permitted with a stiff voltage source.

As long as the load resistance is greater than 100R_{S}, the voltage source is stiff. When the load resistance equals this worst-case value, the calculation error from ignoring the source resistance is 1 percent, small enough to ignore in a second approximation.

Figure 3 visually summarizes a stiff voltage source. The load resistance has to be greater than 100R_{S} for the voltage source to be stiff.

**Example**: The definition of a stiff voltage source applies to ac sources as well as to dc sources. Suppose an ac voltage source has a source resistance of 50 Ω. For what load resistance is the source stiff?

**Solution**: Multiply by 100 to get the minimum load resistance: R_{L} = 100R_{S} = 100(50 Ω) = 5 kΩ

As long as the load resistance is greater than 5 k Ω, the ac voltage source is stiff and we can ignore the internal resistance of the source. A final point.

Using the second approximation for an ac voltage source is valid only at low frequencies. At high frequencies, additional factors such as lead inductance and stray capacitance come into play.

## Current Sources

A dc voltage source produces a constant load voltage for different load resistances. A dc current source is different. It produces a constant load current for different load resistances. An example of a dc current source is a battery with a large source resistance [Fig. 4(a)]. In this circuit, the source resistance is 1 MΩ and the load current is:

When R_{L} is 1 Ω in Fig. 4(a), the load current is:

In this calculation, the small load resistance has an insignificant effect on the load current.

Figure 4(b) shows the effect of varying the load resistance from 1 Ω to 1 M Ω. In this case, the load current remains constant at 10 A over a large range. It is only when the load resistance is greater than 10 k Ω that a noticeable drop-off occurs in load current.

## Stiff Current Source

Here is another definition that will be useful, especially with semiconductor circuits. We will ignore the source resistance of a current source when it is at least 100 times larger than the load resistance. Any source that satisfies this condition is a stiff current source.

As a definition: Stiff current source: **R _{S} > 100R_{L}**

The upper boundary is the worst case. At this point: RS = 100R_{L}

Solving for load resistance gives the maximum load resistance we can use and still have a stiff current source:

**R _{L(max)} = 0.01R_{S}**

In words: The maximum load resistance equals 1 ⁄100 of the source resistance.

Above equation is a derivation because we started with the definition of a stiff current source and rearranged it to get the maximum load resistance. When the load resistance equals this worst-case value, the calculation error is 1 percent, small enough to ignore in a second approximation.

Figure 5 shows the stiff region. As long as the load resistance is less than 0.01R_{S}, the current source is stiff.

## Schematic Symbol

Figure 6(a) is the schematic symbol of an ideal current source, one whose source resistance is infinite. This ideal approximation cannot exist in nature, but it can exist mathematically. Therefore, we can use the ideal current source for fast circuit analysis, as in troubleshooting.

Figure 6(a) is a visual definition: It is the symbol for a current source. When you see this symbol, it means that the device produces a constant current I_{S}. It may help to think of a current source as a pump that pushes out a fixed number of coulombs per second. This is why you will hear expressions like “The current source pumps 5 mA through a load resistance of 1 k Ω.”

Figure 6(b) shows the second approximation. The internal resistance is in parallel with the ideal current source, not in series as it was with an ideal voltage source.

Following summary Table will help you understand the differences between a voltage source and a current source.

PrashantVery well explained. Only thing we will replace term resistance R with impedance Z.