Network Theorems Devices Science is emphatically an important part of culture today as scientific knowledge and its applications continue to transform the world and condition every aspect of the relations between men and nations Vannevar Bush Contents Educational Objectives Background Information Problem Formulation Procedures Educational Objectives In this lab you will test the predictions of two important network theorems as well as experimentally verify Thevenin s Theorem The Maximum Power Transfer Theorem Background Information 1 Thevenin s Theorem Any linear active one port network can be replaced by a single voltage source equal to the open circuit voltage of the one port in series with the network in which all independent sources are set to zero Proof Consider the linear circuit represented in the top panel of Figure 1 LAN linear active network which contains assorted sources and resistors and has a single port by which it can be connected to the outside world We attach a load resistor RL at the port as shown and observe that RL draws load current i1 and voltage v1 from the network Lab 4 1 Figure 1 Development of Thevenin s Theorem In Figure 1b a voltage source has been added in series with the load so as to oppose the current flow We have increased the voltage of this source until the current i2 is brought to zero When zero current is flowing at a port we say the port is open circuited Therefore in Figure 1b the port voltage not V2 is the open circuit voltage Voc of the LAN By Kirchoff s Voltage Law this is equal and opposite to the voltage of the external source we have applied In Figure 1c all independent sources inside the LAN have been turned to zero creating a linear passive network or LPN This means that voltage sources have been replaced by short circuits zero voltage and current sources have been replaced by open circuits zero current The only source now active is the external source of the center panel Voc Since this is the only active source we now expect the current to flow in the direction shown as i3 in the bottom panel According to Thevenin s theorem the load should receive the same current in the bottom panel 1c as it did in the top panel 1a To prove that this is the case apply the superposition principle In the top panel with the LAN sources active and the Voc source zero current i1 flows in the load In the bottom panel with the LAN sources all zero and the Voc source active the load current is i3 Now according to the superposition principle the current flow when all sources are active should be the sum of the currents produced by each acting alone But this sum is zero according to the center panel Therefore i1 and i3 must be equal in magnitude but opposite in direction If we reverse the polarity of the Voc source in the third panel the load receives exactly the current and voltage it did from the original network 2 Maximum Power Transfer Theorem To draw the maximum power from a resistive LAN the load resistance should be set equal to the Thevenin equivalent resistance of the LAN Proof Consider a LAN with Thevenin equivalent resistance Req to which a load resistor RL is connected as shown in Figure 2 The power to RL is P i2 RL Lab 4 2 i R eq Voc RL PRL i 2 RL Figure 2 RL drawing power from the Thevenin equivalent representation of a LAN We want to find the value of RL which will draw maximum power from the network To determine this we will first obtain an expression for the power P in terms of RL by substituting for the current i in the above expression the following fractional form i V Req RL Thus the square of the current can be written V2 i Req RL 2 2 Using this form the power P can be written V 2 RL P i RL Req RL 2 2 To find the value of the load resistance RL for which the power P is a maximum we differentiate P with respect to RL and set the result equal to zero This is just an application of Fermat s Theorem in Calculus which says that the derivative must be zero at the maxima and minima of a differentiable function After differentiating we obtain the result that 2 dP V Req RL 0 dRL Req RL 3 This equation shows that Req RL thus proving the maximum power transfer theorem This theorem has many practical applications One which you may be familiar with is the matching of components in an audio system For example if the output impedance of an amplifier is 8 Ohms it should be used with an 8 Ohm loudspeaker to achieve maximum power output Lab 4 3 Maximum Power Transfer Without Calculus The maximum power transfer theorem can also be derived without calculus by completing the square Recall P V 2 RL Req RL 2 This expression is a rational function in the variable RL when V and Req are considered fixed V which implies that RL V Req Substituting Req RL i these expressions into the equation for power gives We will work instead with the variable i Vi P Req i 2 Req Notice that the expression for power is a simple quadratic function of the current i and no longer a rational function The maximum of a quadratic function can be found by completing the square Applying this technique to the expression for power we obtain 2 V V2 P Req i 2 Req 4 Req 2 The parabola opens downwards and achieves a maximum of Pmax Since i V2 V when i 2 Req 4 Req V V we see by comparing the two denominators that the maximum power occurs Req RL 2 Req when RL Req Problem Formulation There are three parts to this experiment In part 1 you will first build a LAN and then determine experimentally the value of the load resistance which will draw the maximum power In part 2 you will determine through measurements the values of the Thevenin open circuit voltage and equivalent resistance of your LAN and then build a Thevenin equivalent circuit In part 3 you will verify the maximum power theorem and Thevenin s theorem using the results of parts 1 and 2 Lab 4 4 Procedures The circuit you will use as the LAN is shown in Figure 3 You will be given 5 resistors each of which can be connected in any of the positions except one resistor with longer leads which must be used to reach between non adjacent terminal posts The 10 volt source is the lab power supply set to 10V The load resistor RL is a resistor box You may want to use the color code on the resistors to try to calculate the expected Thevenin values However the required calculations are a little difficult for this configuration since no two resistors are in series or in parallel a RL b 10 V Figure 3 …
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