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An introduction to Electricity and Strength of Materials with Peter Eyland
Lecture 3 (Simple circuits and networks)
In this lecture:
• Simple circuits are reduced to a single equivalent resistor by series and parallel substitution,
• Kirchhoff's rules for finding currents and potential differences in networks are demonstrated
Reduction by Series/parallel substitution
Example C7
For the circuit shown on the right, find the current through; the potential across; and the power dissipated in each resistor. 
answer C7
Step 1: reduce the parallel combination. 
The circuit now is now a series circuit 
Step 2: reduce the series combination. 
For the 50R 
Step 3: add the known current(s) and potential(s) to the circuit, and calculate any unknowns. 
The branch currents are: The branch powers are: 
Ratios in simple circuits
For any number of resistors connected in series, the potentials across each resistor are simply proportional
to the size of the resistor over the sum of them.
Take the case of three resistors connected in series, as shown in the diagram. 

Since the current is given by: 
The potential difference across resistor R_{1} is given by the ratio: 
For two resistors connected in parallel, the branch currents are proportional to the resistor in the other branch over the sum of them.
Take the case of two resistors connected in parallel, as shown in the diagram. 

First find the potential difference across the parallel combination. 
The branch currents are given by the ratios: 
Example C8
A potential difference of 120V is applied across three resistors (20R, 30R and 10R) connected in series. Find the potential difference across the 30R. 
Example C9
A current of 1mA flows into a parallel combination of 20R and 30R resistors. Find the current in the 20R resistor. 
Example C10.
For the resistor combination shown in the diagram, calculate the current through the 6Ω resistor. 
answer C10
Firstly, find the equivalent resistance for the three resistors in parallel. Check: the equivalent resistance is smaller than the smallest. 
Secondly, find the equivalent resistance for the new two top resistors in series. 
Thirdly, calculate the branch currents using the opposite resistance over the sum. The current in the 20Ω resistor = (5/25)×60 = 12A Check: the current through the smaller resistor is larger than the current through the larger resistor. 
Fourthly, the equivalent resistance of the 1246Ω parallel combination has been calculated as 2Ω.

Kirchhoff's rules for network circuits
Rule 1: In electric circuit networks, charge is conserved at junctions.
This means that the total current that flows into a junction
must equal the total current that flows out of that junction. Symbolically, this is written as:
Rule 2: In electric circuit networks, electrical energy is conserved around a closed circuit.
This means that the power supplied by the emf's is dissipated by the resistors. Symbolically, this is written as:
Dividing by the current, we see that the sum of the e.m.f.'s equals the sum of the potential drops across the resistors.
Kirchhoff's procedure.
(i) 
Label the salient points of the circuit. 
(ii) 
Apply rule 1 by going to one junction and assuming at least one current (i_{1}) goes into the junction and
at least one current (i_{2}) comes out. Make up other current directions and values as needed, e.g. i_{1}  i_{2}. 
(iii) 
Complete the current flows at all the junctions. Make sure at least one current goes into all junctions and one out, and make up current directions and values as needed. As before, wrong choices for directions just appear as negative values. 
(iv) 
Write the polarities on the resistors. Current flows down a potential difference so write a plus on the high side and minus on the low side. 
(v) 
Apply rule 2 to a number of loops (one loop for each unknown current). When you go down in potential write a minus, and when you go up, write a plus. 
(vi) 
Solve the equations for the unknown currents 
Example C11
For the circuit shown, find the size and direction of the current through each of the three resistors.     
At point C, choose that current i_{1} flows in from B to C. 

Current flows down a potential difference. 
There are only two independent currents. 
Example C12
For the circuit shown, find the size and direction of the current through each of the three resistors. 
Finding potential differences between any two points in circuit networks.
First find the currents by using Kirchhoff's rules, and then start at the first point and track the potentials to the second point.
Example C13
Find the potential difference between B and E.     
At point B, choose that current i_{1} flows in from E to B. 

The circuit has been redrawn to show the resistor polarities (current flows down a potential difference). 
The loop rule is applied for two independent loops
The circuit has been redrawn to show the current values and resistor polarities 
Now track from B to E V_{B }is 2V higher than V_{E} 
Summarising:
For simple circuits, currents and potential differences can be found by series and parallel substitutions.
Kirchhoff's rules are needed for networks, i.e.
Rule 1: In electric circuit networks, charge is conserved at junctions.
Rule 2: In electric circuit networks, electrical energy is conserved around a closed circuit.
(i) 
Label the salient points of the circuit. 
(ii) 
Apply rule 1 by going to one junction and assuming at least one current (i_{1}) goes into the junction and
at least one current (i_{2}) comes out. Make up other current directions and values as needed, e.g. i_{1}  i_{2}. 
(iii) 
Complete the current flows at all the junctions. Make sure at least one current goes into all junctions and one out, and make up current directions and values as needed. As before, wrong choices for directions just appear as negative values. 
(iv) 
Write the polarities on the resistors. Current flows down a potential difference so write a plus on the high side and minus on the low side. 
(v) 
Apply rule 2 to a number of loops (one loop for each unknown current). When you go down in potential write a minus, and when you go up, write a plus. 
(vi) 
Solve the equations for the unknown currents 
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