Linear Circuits Analysis Superposition Thevenin Norton Equivalent circuits So far we have explored time independent resistive elements that are also linear A time independent elements is one for which we can plot an i v curve The current is only a function of the voltage it does not depend on the rate of change of the voltage We will see latter that capacitors and inductors are not time independent elements Timeindependent elements are often called resistive elements Note that we often have a time dependent signal applied to time independent elements This is fine we only need to analyze the circuit characteristics at each instance in time We will explore this further in a few classes from now Linearity A function f is linear if for any two inputs x1 and x2 f x1 x 2 f x1 f x 2 Resistive circuits are linear That is if we take the set xi as the inputs to a circuit and f xi as the response of the circuit then the above linear relationship holds The response may be for example the voltage at any node of the circuit or the current through any element Let s explore the following example i Vs1 R Vs2 KVL for this circuit gives Vs1 Vs 2 iR 0 Or i Vs1 Vs 2 R 6 071 22 071 Spring 2006 Chaniotakis and Cory 1 1 1 2 1 And as we see the response of the circuit depends linearly on the voltages Vs1 and Vs 2 A useful way of viewing linearity is to consider suppressing sources A voltage source is suppressed by setting the voltage to zero that is by short circuiting the voltage source Consider again the simple circuit above We could view it as the linear superposition of two circuits each of which has only one voltage source i1 i2 Vs1 R R Vs2 The total current is the sum of the currents in each circuit i i1 i 2 Vs1 Vs2 1 3 R R Vs1 Vs 2 R Which is the same result obtained by the application of KVL around of the original circuit If the circuit we are interested in is linear then we can use superposition to simplify the analysis For a linear circuit with multiple sources suppress all but one source and analyze the circuit Repeat for all sources and add the results to find the total response for the full circuit 6 071 22 071 Spring 2006 Chaniotakis and Cory 2 Independent sources may be suppressed as follows Voltage sources Vs v Vs suppress short v 0 Current sources i Is Is i 0 suppress open 6 071 22 071 Spring 2006 Chaniotakis and Cory 3 An example Consider the following example of a linear circuit with two sources Let s analyze the circuit using superposition R1 R2 i1 i2 Is Vs First let s suppress the current source and analyze the circuit with the voltage source acting alone R1 R2 i1v i2v Vs So based on just the voltage source the currents through the resistors are i1v 0 Vs i 2v R2 1 4 1 5 Next we calculate the contribution of the current source acting alone R1 i1i v1 R2 i2i Is Notice that R2 is shorted out there is no voltage across R2 and therefore there is no current through it The current through R1 is Is and so the voltage drop across R1 is 6 071 22 071 Spring 2006 Chaniotakis and Cory 4 v1 IsR1 1 6 And so i1 Is Vs i2 R2 How much current is going through the voltage source Vs 1 7 1 8 Another example For the following circuit let s calculate the node voltage v R1 v R2 Vs Is Nodal analysis gives Vs v v Is 0 R1 R2 1 9 or v R2 R1R 2 Vs Is R1 R 2 R1 R 2 1 10 We notice that the answer given by Eq 1 10 is the sum of two terms one due to the voltage and the other due to the current Now we will solve the same problem using superposition The voltage v will have a contribution v1 from the voltage source Vs and a contribution v2 from the current source Is 6 071 22 071 Spring 2006 Chaniotakis and Cory 5 R1 Vs R1 v1 v2 R2 R2 Is v1 Vs R2 R1 R 2 1 11 v 2 Is R1R 2 R1 R 2 1 12 And Adding voltages v1 and v2 we obtain the result given by Eq 1 10 More on the i v characteristics of circuits As discussed during the last lecture the i v characteristic curve is a very good way to represent a given circuit A circuit may contain a large number of elements and in many cases knowing the i v characteristics of the circuit is sufficient in order to understand its behavior and be able to interconnect it with other circuits The following figure illustrates the general concept where a circuit is represented by the box as indicated Our communication with the circuit is via the port A B This is a single port network regardless of its internal complexity i A R4 Vn In R3 v B If we apply a voltage v across the terminals A B as indicated we can in turn measure the resulting current i If we do this for a number of different voltages and then plot them on the i v space we obtain the i v characteristic curve of the circuit For a general linear network the i v characteristic curve is a linear function i mv b 6 071 22 071 Spring 2006 Chaniotakis and Cory 1 13 6 Here are some examples of i v characteristics i i v R v In general the i v characteristic does not pass through the origin This is shown by the next circuit for which the current i and the voltage v are related by iR Vs v 0 or i 1 14 v Vs R 1 15 i i Vs R v Vs v Vs R Similarly when a current source is connected in parallel with a resistor the i v relationship is v i Is R open circuit i voltage 1 16 i Is R v 6 071 22 071 Spring 2006 Chaniotakis and Cory Is RIs v short circuit current 7 Thevenin Equivalent Circuits For linear systems the i v curve is a straight line In order to define it we need to identify only two pints on it Any two points would do but perhaps the simplest are where the line crosses the i and v axes These two points may be obtained by performing two simple measurements or make two simple calculations With these two measurements we are able to replace the complex network by a simple equivalent circuit This circuit is known as the Thevenin Equivalent Circuit Since we are dealing with linear circuits application of the principle of superposition results in the following expression for the current i and voltage v relation i m0 v m jV j b j I j j 1 17 j Where …