Electric Circuits Lecture 2: Sources and Resistances By Sheharyar Zahid

More on series resistances Keeping KVL in mind, it states that the voltage rise must equal the sum of the drops in the following circuit [1]

Parallel Resistances For the Case of parallel resistances, we already know Now for this special case of two parallel resistances we can also use the following formula to make our analysis easier [1]

Parallel Resistances If we do a simple calculation we will see that the overall resistance is decreased This is because now current has more paths to flow through Also note that if one of the resistances is much larger then it can be ignored as the formula will show you

Potentiometer Consider a resistor that has a linear response The resistance can be varied if we change ‘l’ - the length of the resistive element (recall the expression for resistivity) Such a device which can be made to vary its resistance by the use of its ‘l’ along with a metal contact is called a potentiometer Essentially it is just a ‘variable’ resistance Various symbol conventions for a potentiometer are displayed >>> [1]

Voltage Divider A voltage divider does just that, it divides the voltage Function of this circuit is to take an input voltage (Vi) and provide a scaled down output voltage (Vo), since we know how the series resistance drops voltage we can say (Pg. 65) [1]

Voltage Divider Also called voltage attenuator as we can see this is used to drop the voltage hence attenuate it A two resistor voltage divider requires that the resistances be in series (i.e. carry the same current) and share a simple node If any other elements are connected to the shared node then we can no longer apply the given voltage divider formula Vo preserves the nature of Vi, i.e. AC will output AC and DC will output DC Vo is also (theoretically) linearly proportional to Vi Try an example from Franco pg. 66

Voltage Gain A relationship can be established between the output and input, We can see that the gain depends on the ratio of the resistors The choice of resistors will effect current and the power absorbed by the divider [1]

Variable gain attenuator Incorporating a potentiometer we can devise a variable gain attenuator Can be used to control the volume of audio devices [1]

Current Divider Produces an output current in response to an input current R1 and R2 share the same node therefore they are in parallel and have a common voltage drop. The current divides in the two resistances in an inversely proportional manner. [1]

Current gain We can also find the current gain of the divider by expressing the equation as Output is a fraction of input and is proportional to it Current gain depends on the ratio’s of the resistances Equal resistances split current equally (but practically, two identical resistors can not be the same, why?) The larger R2 is than R1, the lesser the output current will be and vice versa, if R2 >> R1 then output current will tend to zero [1]

Resistive Bridge Consists of 2 voltage dividers (also called bridge arms) with a common source Vs Now using the voltage gain formula on slide 8 we can derive an expression for output voltage [1] [1] [1] Try example on pg. 72 of Franco

More on the resistive bridge If the resistance ratios on each arm are identical i.e. Then output voltage = zero and the bridge is balanced A Wheatstone bridge is a resistive bridge with a variable resistance Used in null measurements when one resistance connected is unknown. Measurement is taken when output goes to zero [1] [1]

Resistive Ladders Look at the ‘Resistive Ladder’ below Using the knowledge you already possess it can be seen [1] [1]

An R-2R Ladder If we take the resistive ladder and set resistances equal to R and 2R, we get an R-2R ladder that gives progressively diminishing (halving) current and voltage levels [1]

Some Applications The Wheatstone bridge is used in instrumentation applications The R-2R ladder is used for making DAC’s – you will study about DAC’s and ADC’s in a later course

Practical Voltage Source Model We have already discussed how practical voltage sources have internal resistances V=Vs only when there is no current (i) The i-v characteristic can be modeled by [1] [1]

Practical Voltage Source Model It behaves like an ideal V source with a series resistance Compare this to the i-v characteristics of the ideal V source [1]

Practical Current Source Model Similarly a practical current source will exhibit a similar behavior The internal resistance is modeled as being in parallel because of how current divides in parallel (no current will divide in series) [1]

…About the i-v graphs We are assuming something in these i-v graphs, can you guess what it is? It has to do with the uniformity of the gradients …. [1]

The proportionality! For the voltage source we are assuming that the drop in voltage is linearly proportional to the load current For the current source we are assuming that current falls linearly proportional to the load voltage

Important Tasks! Do all the examples in the Franco book up till page 88 and page 92-95. They are very easy and you have studied everything you need for it. Quizzes can happen anytime NEXT WEEK WE WILL BEGIN: CIRCUIT ANALYSIS

References [1] Franco, S 1995, Electric circuit fundamentals, 2nd Edn, Saunders College Publishing