Dependent Sources Introduction and analysis of circuits containing dependent sources So far we have explored time independent resistive elements that are also linear We have seen that two terminal one port circuits can be modeled by simple circuits Thevenin or Norton equivalent circuits and that they have a straight line i v characteristic Here we introduce the idea of a dependent source We will see that the use of dependent sources permits the use of feedback Feedback can be used to control amplifiers and to build interesting transducers Dependent Sources A dependent source is one whose value depends on some other variable in the circuit An illustrative example of a dependent source is i black box equivalent of input circuit v1 g v1 black box equivalent of output circuit Here we see that there is an input circuit that develops a voltage v1 In a separate part of the circuit there is a linear voltage dependent current source that delivers a current given by i g v1 1 1 Where g is a constant with the units of A V So the current that flows into the output circuit depends on the measurement of a voltage on the input circuit Now clearly we could mimic a dependent source by looking at a meter and changing a potentiometer for example in relation to the reading Here we will introduce circuits that carry out this function without any intervention Notice that the above circuit is still linear since the output current depends linearly on the measured voltage For now we will concern ourselves with only linear dependent sources Later we will see examples of non linear dependent sources where the analysis will be somewhat more complex 6 071 22 071 Spring 2006 Chaniotakis and Cory 1 There are four general classes of linear dependent sources Their names acronyms and associated symbols are Voltage Controlled Voltage Source VCVS i1 v1 vs A v1 Current Controlled Voltage Source CCVS i1 v1 vs r i1 Voltage Controlled Current Source VCCS i1 v1 is g v1 Current Controlled Current Source CCCS i1 v1 is i1 The parameters A r g are real numbers and v1 i1 are voltages currents in some circuit 6 071 22 071 Spring 2006 Chaniotakis and Cory 2 Circuit Analysis with Linear Dependent Sources Linear dependent sources provide no new complications to circuit analysis Kirchhoff s laws still apply and formal circuit analysis goes ahead just as before The dependent source only introduces a constraint on the solution The simplest example is where the measurement and dependent source are in two isolated circuits Let s consider the current amplifier circuit shown on Figure 1 The circuit has one independent current source and one dependent current source The dependent current source is a CCCS We would like to determine the voltage vc as indicated ib Is Rs Rb ib Rc vc Figure 1 Current Amplifier Circuit The left hand circuit is a current divider and ib Rs Is Rs Rb 1 2 The right hand circuit is a current source The output voltage vc is given by vc ib Rc 1 3 So now we see that the output voltage vc depends on the measured current ib of the input circuit Combining Equations 1 2 and 1 3 we obtain RsRc vc Is Rb Rc 1 4 gain So the overall circuit behaves as an amplifier with the gain dependent on the resistors and the proportionality constant 6 071 22 071 Spring 2006 Chaniotakis and Cory 3 Let s now consider the slightly more interesting circuit shown on Figure 2 R1 R2 Is Vs 2 v3 v3 R3 Figure 2 Circuit with dependent voltage source Let s use nodal analysis to solve for the currents and voltages in this circuit Figure 3 shows the nodes of interest the variables and the polarities v1 node1 R1 R2 i1 Vs 2 v3 v2 node2 i2 Is R3 i3 v3 Figure 3 Nodal analysis of circuit with dependent sources KCL at node1 gives i1 Is i 2 0 Vs v1 v1 v 2 Is 0 R1 R2 1 5 KCL at node2 gives 6 071 22 071 Spring 2006 Chaniotakis and Cory 4 i 2 i3 0 v1 v 2 v 2 0 R2 R3 1 6 In matrix form Equations 1 5 and 1 6 become 1 1 R 2 R3 1 R2 1 Vs R 2 v1 Is R1 1 1 v2 0 R1 R 2 1 7 and the solution is given by v1 R 2 R3 IsR1 Vs R1 R 2 R3 1 8 R3 IsR1 Vs R1 R 2 R3 1 9 v2 Now need to include the constraints of the dependent sources These constraints are And v 2 v3 1 10 Vs 2v3 1 11 Substituting Equations 1 10 and 1 11 into Equations 1 8 and 1 9 we obtain v1 IsR1 R 2 R3 R1 R 2 R3 1 12 IsR1R3 R1 R 2 R3 1 13 v2 6 071 22 071 Spring 2006 Chaniotakis and Cory 5 Analysis of Circuits with Dependent Sources Using Superposition When employing the principle of superposition to a circuit that has dependent and independent sources we proceed as follows Leave dependent sources intact Consider one independent source at the time with all other independent sources set to zero Let s explore this with the following example For the circuit on Figure 4 calculate the voltage v Av R v Is Vs Figure 4 Circuit with dependent source Analysis using superposition We proceed by first considering the effect of the current source acting alone The circuit of Figure 5 shows the corresponding circuit for which the independent voltage source Vs has been suppressed A v1 R Is v1 Figure 5 Circuit with the voltage source suppressed By applying KVL we obtain v1 IsR Av1 0 1 14 And v1 becomes IsR 1 A 1 15 6 071 22 071 Spring 2006 Chaniotakis and Cory 6 v1 Next we evaluate the contribution to the output with the independent voltage source acting alone The corresponding circuit is shown on Figure 6 A v2 R Vs v2 Figure 6 Circuit with the current source suppressed Again applying KVL we have Av 2 v 2 Vs 0 1 16 Note that the voltage drop across R is zero since there is no current flowing in the circuit And v2 becomes Vs 1 17 v2 1 A And so the total voltage is written as the superposition of v1 and v2 v v1 v 2 1 Vs IsR 1 A 1 18 Let s now look at the slightly more complicated circuit shown on Figure 7 with multiple dependent and independent sources We will determine the voltage vo by using superposition Rs v1 Vs1 A v1 R1 A v2 v2 Vs2 vo Rs Figure 7 Circuit with dependent and independent sources 6 071 22 071 Spring 2006 Chaniotakis and Cory 7 The procedure is the same as before leave dependent sources intact calculate the contribution of each independent source acting alone Figure 8 shows the circuit with Vs2 suppressed The indicated output vo1 is the contribution of voltage source Vs1 Rs v1 Vs1 A v1 R1 A v2 …