2. Ideal Sources and Circuit Laws

by HobbyGenius VIP on October 3, 2014 at 1:30 pm | 0 comments.


Voltage and Current

Before we start, we will introduce the idea of voltage and current – most of this was introduced in the physics tutorials, but for those who don’t want to read through all those tutorials then read on. Electric current is the movement of electrons in a material, in an insulator these electrons are tightly bound and aren’t easy to move but in a conductor they are free to move easily. Voltage is what makes the electrons flow; it can be thought of a potential difference between 2 points and shows the energy required for charge (the electrons) to move between these 2 points.

Ideal Voltage Source

An Ideal Voltage Source and a Resistance

An Ideal Voltage Source and a Resistance

An ideal voltage source is a power source that will supply a constant voltage level regardless of how much current is being drawn. In the circuit above, the V+ represents the voltage of the source and the small + next to the circle shows which end is positive. What’s significant about the ideal voltage source is that the current can be in either direction and it still maintains its voltage level; this is shown below.

Ideal Voltage Source

Ideal Voltage Source

Ideal Current Source

An Ideal Current Source and a Resistance

An Ideal Current Source and a Resistance

An ideal current source is a power source that will supply a constant current regardless of direction or magnitude of the voltage across it. In the circuit above, the i represents the current of the source and the arrow next to the two circles shows which direction the current is flowing in. The graph below shows how the direction of the voltage does not affect the current.

Ideal Current Source

Ideal Current Source

Current is the rate of flow of charge and be calculated using the following equation:

  • is current
  • is the total charge
  • is time

It is measured in Coulombs per Second () or Amps ().

Ohm’s Law

Voltage, current and resistance can all be related using Ohm’s Law:

  •  is the voltage
  •  is the current
  •  is the resistance

So we can always calculate the current passing through a component with a certain resistance if we know the voltage (and vice versa). Let’s use the ideal voltage source as an example:

Ideal-Voltage-Source

Imagine the voltage source was set at 5V and the resistor had a resistance of 100Ω, we could calculate the current flowing through the resistance by saying that:

Later on we will learn how to analyse a whole circuit to find the currents and voltages at certain points; this will rely heavily on ohm’s law.

Circuit Laws

Now that we know about ideal voltage and current sources and Ohm’s law, we will learn about equivalent circuits. We can simplify circuits in many ways, including by swapping a voltage source for a current source and vice versa and this is what we’ll be learning.

Resistors

The resistor laws are the easiest so we’ll start with them. Resistors can be in parallel or series and we can combine these combinations to form a single resistor with a new value. First look at the diagram below which should help you understand it a lot more easily.

Equivalent Resistor Circuits

Equivalent Resistor Circuits

As you can see from the diagram above, in series we just add up all the values of the resistors and replace it by the one resistor with the new value. Similarly in parallel we can replace the resistors with one by adding up 1 over the value of each resistor and then 1 over the whole lot. It’s easy to remember:

Kirchhoff’s Laws

Current Law

If we take a node – a place where several connections join – we can say that the sum of all these currents is equal to 0; or to simplify this we can say that the currents into the node is equal to the currents out of the node.

Kirchhoff’s Current Law

Kirchhoff’s Current Law

Let’s look at the circuit above; as you can see there are 3 resistors joining at a node and we want to find out the value of . Since we don’t know the direction we simply choose any directions and if this direction is correct then the calculated value will be positive and if not then it will be negative, and we can re-draw it in the opposite direction.

We’ll then use the equation . Since these currents are all flowing into the node then we will just add them, if they were all flowing out of the node then we could also add them, just as long as the currents stay consistent – so all currents flowing in are positive and all currents flowing out are negative (and vice versa).

Current Law Example

Kirchhoff’s Current Law Example

Kirchhoff’s Current Law Example

On the left we have drawn the circuit but we do not know the value of  so we add this arrow facing whichever way we want. We can then calculate:

We have calculated that  is equal to -1000mA, since this is negative we know that we drew the direction the wrong way round. On the right of the diagram is the redrawn circuit with it the correct way round.

  • Always remember that a negative current is simply current in the opposite direction
Negative Current

Negative Current

Voltage Law

Kirchhoff’s Voltage Law states that the sum of the voltages in a closed loop will equal zero. In the example below, if we add all the voltages (, ,  and ), the total will equal zero. Like the current law, any voltages that are opposing the direction should have the opposite sign; so in the image below, all the voltages are clockwise so it makes sense for them to be positive (because the majority are clockwise). If we had any voltages in the opposite direction, these would be negative.

Kirchhoff’s Voltage Law

Kirchhoff’s Voltage Law

Voltage Law Example

Kirchhoff’s Voltage Law Example

Kirchhoff’s Voltage Law Example

In the example above, we have 3 resistors with the following values:

  • R1 = 12Ω
  • R2 = 4Ω
  • R3 = 12Ω

The voltage source is outputting 12V.

The first thing we must do is identify the closed loops within the circuit. As mentioned before, it doesn’t matter which way the arrows face: if the arrow is facing the wrong way, like current, the voltage will be negative.

Kirchhoff’s Voltage Law Example

Kirchhoff’s Voltage Law Example

We now have 2 loops which we will be finding. Let’s first apply KVL to loop 1:

 (This is due to Ohm’s Law)

We now have 2 unknowns: i1 and i2 which are not equal; let’s make an equivalent circuit to make this easier to look at:

Kirchhoff’s Voltage Law Example

Kirchhoff’s Voltage Law Example

We have now combined resistor 2 and 3 to produce an equivalent resistor. We can now say that the total resistance is equal to:

By applying Ohm’s Law we can calculate the current coming out of the voltage source (which is equal to i1):

Now that we know the value of i1 we can substitute it back into our first calculation:

We now have a value for i1 and i2 so let’s apply KVL to our 2nd loop:

Substitute in the values we already know:

Divider Circuits

Conductance

Instead of using Resistance, we can also use Conductance, which is equal to . It is measured in per Ohms,  or Siemens, S. And then using this we can re-write ohms law as:

Current Divider

If we take several resistors in series and a current source, it can be shown that current is dropped after each resistor based on the total resistance in the circuit. If we then take an output after each resistor we can take output currents that are a fraction of the input current. Consider the circuit below:

Current Divider

Current Divider

We can say

The equivalent circuit can then be shown to be:

Current Divider Example

Current Divider Example

Current Divider Example

Let’s consider the example above:

From observation we can see that 1A has been split between two branches: i1 and i2. Since we know the value of i2 we can say:

Potential Divider

Similar to the current divider, there is also the potential divider which will divide a voltage up between resistors.

Potential Divider

Potential Divider

If we apply Kirchoff’s Voltage Law:

And then Ohm’s Law across R2:

Combine the two:

And therefore:

 

This is one of the easiest equations in electronics to learn and it is vital that you remember it.

Move onto the next tutorial where we will apply all this knowledge and analyse various circuits using some popular methods.