Series-Parallel Circuits

Series-Parallel Circuits are circuits which are composed of series and parallel connections.

To analyse these we need to identify the series and parallel pieces and apply their rules to them.

Below are summaries of the key rules for series and parallel circuits which will help us to identify and simplify them.

Series Circuits

• Series circuit elements share a single common terminal between them
• The current is the same through series elements
• Voltage adds in series
• Resistance adds in series

Parallel Circuits

• Parallel circuit elements share two common terminals
• The voltage is the same through parallel elements
• Current adds in parallel
• Resistance reduces when elements are connected in parallel

Series-Parallel Circuit Analysis

Series-parallel circuit analysis differs from simple series and parallel circuits in that we need to first identify the series and parallel elements. We need to work out which circuit elements are in series and which are in parallel to then proceed with simplifying the circuit and applying the appropriate circuit rules.

The general process is as follows:

1. Identify the series and parallel connections in the circuit
2. Determine the equivalent resistances and continue to redraw the circuit and identify the resulting series and parallel connections
3. Once the circuit has been simplified to a single resistor, the total voltage or current can be found.
4. Use the above to go back and determine all of the voltages and currents in the circuit as required, it can be helpful to redraw the circuit back again in stages during this phase.

The identification phase can be challenging as the subcircuit connections are not always obvious on first inspection and the resulting analysis is also more complex than simple series and parallel circuits.

The best way to learn how to tackle series-parallel circuits is lots of practice, which is why the remainder of this topic is examples and questions :)

Examples:

1. a) What is the total current Is in the circuit below:

From the above circuit we can see that total current It splits into currents I1 and I2 through resistors R2 and R3 which share two common terminals at the end points which means they are connected in parallel. We can then simplify R2 and R3 into a single equivalent series resistor connected to R1 and we can calculate this resistance with the product over sum equation:

The circuit can now be drawn with the parallel resistance of R2||R3 in series with R1:

This can be simplified further by combining these two resistances in series to find the equivalent total resistance:

From here we can use Ohm's Law to determine the total current:

b) What is the voltage drop across R1?

As we now know the total current flowing through the circuit, we can use Ohm's law to determine V1:

c) What is the voltage drop across R2?

We now know the voltage drop across R1, hence there is only one unknown voltage in series so we can use the fact that voltages add in series to calculate it:

d) Calculate I1:

We can use Ohm's Law to calculate I1 using the voltage across R2:

e) Calculate I2:

Currents add in parallel circuits equal to the total current entering the parallel circuit, hence:

2. a) What is the total current It in the circuit below:

By inspection we can see that current It splits into 2 currents which flow through resistors R1 and R2 which recombine and then split again into 2 currents flowing through R3 and R4. The current then flows through R5 before returning to the voltage sources. This means that R1 is in parallel with R2 and R3 is in parallel with R4 and these equivalent resistors are in series with R5. Now to determine R1||R2 we could use the product over sum equation, however, recall that we also found that parallel resistances of the same value simplify to the resistance value divided by the total number of resistors. Hence the equivalent resistance is simply given by:

R3||R4 can then be calculated using the product over sum equation:

The circuit can then be redrawn with the equivalent parallel resistances connected in series to R5:

Adding the series voltage sources and resistances results in the final simplified circuit:

From here Ohm's law can be used to determine the total current:

b) What is the voltage drop across R5?

c) What is the voltage drop across R1?

Referring back to the first circuit simplification, we can now use It and R1||R2 to determine the voltage across R1 as voltages across components in parallel are equal:

d) What is the current flowing through R1?

Given the voltage across R1 from c) we can now use Ohm's Law to determine the current:

Thinking about this intuitively, there are two current paths of equal resistance so we would expect half the current to flow through each branch.

e) What is the current flowing through R2?

By inspection this would be 0.5mA which is equal to (It - I1). The current It splits into currents through R2 and R3.

f) What is the voltage drop across R4?

As we now know the voltages across R1 and R5, we can use the fact that the voltage drops in the series equivalent circuit sum up to the source voltage:

By inspection we also know that this must be 2V as R5 has the same resistance as R3||R4 and as the same current flows through both we have the same voltage drop.

g) What is the current flowing through R3?

Given the voltage across R4 from f) we can now use Ohm's Law to determine the current through R3 as it will have the same voltage across it as R4 being in parallel:

h) What is the current flowing through R4?

By inspection this would be 0.67mA which is equal to (It - I4)

3. a) What is the voltage provided by the current source in the circuit below?

This is an example of a circuit which can be trickier to identify the parallel and series sections and in such a case it can help to redraw the circuit. We can see that the total current flows into a network consisting of R1, R2, R3 and R4. To break this down further it is helpful to recall that series resistors share only a single terminal. From this we can see that resistors R1 and R2 share a terminal in common and that R3 and R4 share a common terminal which means these pairs of resistors are in series with each other. We can then redraw the circuit as follows to make this clearer:

From the above it is clear that the equivalent resistances Rx (R1 and R2) and Ry (R3 and R4) are in parallel and we can also see that the current It flows through R5 and R6 which means they are in series. We can calculate the parallel resistance and then add the result with R5 and R6 to determine the final resistance:

Hence the equivalent circuit is:

Now we can use Ohm's Law to calculate the current provided by the current source, Vs:

b) What is the current flowing through R1?

To determine the current flowing through R1 we need to calculate the voltage dropped across the branch. As we now know the voltage supplied by the current source, we know it will be dropped across the combination of R1,R2,R3,R4 and R5 and R6 all in series. In terms of numbers from the above we have 2.1kohm in series with 500ohm's and so the voltage across 2.1kohm is the total current multiplies by 2.1kohm's which is 10.5V. And so we now have 10.5V dropped across a 1k and 2k resistance, so we can workout the current flowing in the branch which is simply 10.5/3k = 3.5mA.

Questions:

1. a) What is the total current Is in the circuit below:

b) What is the voltage drop across R1?

c) What is the voltage drop across R2?

d) What is the current flowing through R2?

e) What is the current flowing through R3?

f) What is the current flowing through R4?

2. a) What is the source current Is in the circuit below?

b) What is the voltage drop across R1?

c) What is the voltage drop across R3?

d) What is the voltage drop across R4?

e) What is the current flowing through R3?

Lab:

I'd like to leave this as an exercise, its really important to try experimenting with various circuit configurations to get a feel for how series-parallel circuits in particular work and as everyone is different I think it's best that you fire up LTSpice and connect up interesting circuits to build intuition :)

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