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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, black boxequivalentof outputcircuitblack boxequivalentof inputcircuitv1+-ig v1 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 igv1= (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 1There are four general classes of linear dependent sources. Their names, acronyms and associated symbols are: Voltage Controlled Voltage Source: VCVS v1+-i1vs = A v1 Current Controlled Voltage Source: CCVS v1+-i1 vs = r i1 Voltage Controlled Current Source: VCCS. v1+-i1 is = g v1 Current Controlled Current Source: CCCS v1+-i1 βi1is = The parameters A, r, g, β, are real numbers, and v1 , i1 are voltages/currents in some circuit. 6.071/22.071 Spring 2006, Chaniotakis and Cory 2Circuit 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. IsibRsRbRcvc+- β ib Figure 1. Current Amplifier Circuit The left hand circuit is a current divider, and Rsib IsRsRb=+ (1.2) The right hand circuit is a current source. The output voltage vc is given by vc ib Rcβ= (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 gainRsRcvc IsRbRcβ=+  (1.4) So, the overall circuit behaves as an amplifier with the gain dependent on the resistors and the proportionality constant β. 6.071/22.071 Spring 2006, Chaniotakis and Cory 3Let’s now consider the slightly more interesting circuit shown on Figure 2. R1R2IsR3v3+-2 v3Vs= 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. R1R2IsR3v3+-2 v3i1i2i3v1v2+++---node1 node2Vs= Figure 3. Nodal analysis of circuit with dependent sources KCL at node1 gives 12112012iIsiVs v v vIs0RR+−=−−+−= (1.5) KCL at node2 gives 6.071/22.071 Spring 2006, Chaniotakis and Cory 423012 2023iivv vRR−=−−= (1.6) In matrix form, Equations (1.5) and (1.6) become 11 1123 2121110212VsvIsRR RRvRRR⎛⎞+−⎛⎞⎜⎟+⎛⎞⎜=⎜⎟⎜⎟⎜⎜⎟⎝⎠−+⎜⎟⎝⎠⎝⎠⎟⎟ (1.7) and the solution is given by (2 3)( 1 )1123RRIsRVsvRRR++=++ (1.8) 3( 1 )2123RIsR VsvRRR+=++ (1.9) Now need to include the constraints of the dependent sources. These constraints are 2vv3= (1.10) And 23Vs v= (1.11) Substituting Equations (1.10) and (1.11) into Equations (1.8) and (1.9) we obtain 1( 2 3)1123IsR R RvRRR+=++ (1.12) 13212IsR Rv3RRR=++ (1.13) 6.071/22.071 Spring 2006, Chaniotakis and Cory 5Analysis 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. RA vVs+-vIs 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. RA v1+-v1Is Figure 5. Circuit with the voltage source suppressed By applying KVL we obtain: 11vIsRAv0−+= (1.14) And v1 becomes 11IsRvA=+ (1.15) 6.071/22.071 Spring 2006, Chaniotakis and Cory 6Next we evaluate the contribution to the output with the independent voltage source acting alone. The corresponding circuit is shown on Figure 6. RA v2Vs+-v2 Figure 6. Circuit with the current source suppressed. Again applying KVL we have 22 0Av v Vs+−= (1.16) (Note that the voltage drop across R is zero since there is no current flowing in the circuit.) And v2 becomes 21VsvA=+ (1.17) And so the total voltage is written as the superposition of v1 and v2. (1211vv vVs IsRA)=+=++ (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. RsVs1+-voRsVs2A v1 A v2R1++--v1v2++-- Figure 7. Circuit with dependent and independent sources 6.071/22.071 Spring 2006, Chaniotakis and Cory 7The procedure is the same as before: leave dependent sources intact, calculate the contribution of


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MIT 6 071J - Lecture Notes

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