Appendix A: Some Basic Circuit Theory
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Appendix A: Some Basic Circuit Theory Voltage and Current Polarities and Conventions Unless we are dealing specifically with the flow of electrons through a vacuum tube, we will assume current flows from positive to negative. This is called conventional current flow. Conventional current flow is used by virtually all manufacturers today on transistor and IC data sheets, application notes, etc. As shown in Fig. A.1, voltage drops across components will be designated using curved arrows. Think of these arrows as voltmeters that sense the voltage across a given component. The point of the arrow is equivalent to the positive lead of a voltmeter. Currents will be represented with arrow heads drawn on the wires in which the current is flowing. For the resistor and the diode shown in Fig. A.1,if current is flowing the direction indicated, the current is positive, and the voltage dropped across the components will be positive. Linear Circuits Ideally, resistors, capacitors, inductors, voltage sources, and current sources are linear circuit elements. If you graph the current vs. voltage characteristic for a linear component you get a straight line, as shown for the resistor in Fig. A.1. One of the nice things about linear components is that we can use Ohm’s law (V ¼ IR) to relate I, V, and R. Because the plot of current vs. voltage is a straight line, we can pick any point on the line and get the same result for R ¼ V/I. Diodes, transistors, and tubes are all nonlinear devices. An example is the graph of current vs. voltage for a typical diode shown in Fig. A.1. We can’t use Ohm’s law to characterize nonlinear devices because the quotient V/I is not a constant. Even though circuits that contain diodes, transistors, and tubes are nonlinear, in many cases we can treat them as if they are approximately linear and obtain useful analysis results. D.J. Dailey, Electronics for Guitarists, DOI 10.1007/978-1-4614-4087-1, 385 # Springer Science+Business Media, LLC 2013 386 Appendix A: Some Basic Circuit Theory Fig. A.1 Linear (resistor) and nonlinear (diode) devices Series Circuits A typical series circuit is shown in Fig. A.2. All circuit elements (resistors and a battery in this case) carry the same current in a series circuit. The various voltage drops across the circuit elements can be determined using Ohm’s law and Kirchhoff’s voltage law. Note that I have oriented the voltage-sensing arrows pointing counterclockwise on the resistors so that the voltage drops across the resistors will be positive. I did this because I like to work with positive numbers but the arrows could have been oriented the other direction, resulting in negative voltage values. Kirchhoff’s Voltage Law Kirchhoff’s voltage law (KVL) says that the algebraic sum of all voltages around a closed loop is zero. The series circuit of Fig. A.2 forms a closed loop. If we assume there are no other resistors than those shown, we sum the voltage drops moving clockwise from the battery to obtain the KVL equation 0 ¼ Vin À V1 À V2 À V3 ÀÁÁÁÀVn Or, alternatively we could write Appendix A: Some Basic Circuit Theory 387 Fig. A.2 A series circuit Fig A.3 A parallel circuit Vin ¼ V1 þ V2 þ V3 þÁÁÁþVn The general rule is that we add voltages whose sensing arrows are pointed in the direction we travel around the loop. We subtract voltages whose sensing arrows point against the direction we have chosen. Parallel Circuits A typical parallel circuit is shown in Fig. A.3. In a parallel circuit, all elements have the same voltage across them. The voltage drops across all elements in a parallel circuit are equal. That is, V1 ¼ V2 ¼ V3 ¼ÁÁÁ¼Vn ¼ Vin The current flowing through a given element may be determined using Ohm’s law, Kirchhoff’s current law, or a combination of the two. 388 Appendix A: Some Basic Circuit Theory Fig. A.4 Example of a node with five branches Nodes A node is a point where two or more circuit elements are connected together. For example, in Fig. A.3, the entire top line of the schematic is a node—the top terminal of all of the resistors could be drawn connected at a single point or node. The circuit in Fig. A.3 has a total of two nodes. In Fig. A.2, each junction between resistors is a node, so this circuit has n + 1 nodes. Branches A branch is a path through which current can flow. The circuit in Fig. A.1 has one branch. That is, there is only one path through which current can flow. The circuit of Fig. A.2 has n + 1 branches, However, we could have started our numbering with IT renamed I1, in which case there would be n branches. Kirchhoff’s Current Law Kirchhoff’s current law (KCL) states that the algebraic sum of all currents entering and leaving a node is zero. A node with five branches extending from it is shown in Fig. A.4. You can arbitrarily choose the polarity of the current sensing arrows. For example, if we assume that arrows pointing to the node are positive, while those pointing out of the node are negative, the KCL equation for Fig. A.4 is 0 ¼ I1 À I2 þ I3 þ I4 À Ix Referring back to Fig. A.3, application of KCL tells us IT ¼ I1 þ I2 þ I3 þÁÁÁþIn Appendix A: Some Basic Circuit Theory 389 The Superposition Principle The principle of superposition is used to analyze linear circuits that contain multiple voltage and/or current sources. The idea behind superposition is that we can determine the response for each source acting individually, then add these responses to get the complete response. For example, to analyze the circuit in 0 Fig. A.5, we can start by “killing” current source I1 and finding the output Vo caused by voltage source V1 acting alone. An ideal current source has infinite internal resistance, so when we kill a current source it is simply replaced with an open circuit. Redrawing the circuit with source I1 killed as shown in Fig. A.5,we 0 can find Vo using the voltage divider relationship. 0 R2 V o ¼ V1 R1 þ R2 2kO ¼ 12 V 1kO þ 2kO ¼ 6V Next, we kill voltage source V1 and determine the output voltage caused by current source I1 acting alone. The internal resistance of an ideal voltage source is zero, so when we kill a voltage source we replace it with a short circuit. Redrawing the circuit with voltage source V1 killed we find R1||R2 ¼ 667 O, so the second component of the output voltage is 00 V o ¼ I1ðR1jjR2Þ ¼ 3 mA(1 kOjj2kOÞ ¼ð3mAÞð667 OÞ ¼ 2V Fig. A.5 Example of the application of superposition in circuit analysis 390 Appendix A: Some Basic Circuit Theory The net output voltage is the sum, or superposition of the two components. 0 00 Vo ¼ V o þ V o ¼ 6Vþ 2V ¼ 8V Useful Formulas Ohm’s law V V V ¼ IR I ¼ R ¼ R I V2 P ¼ IV P ¼ I2RP¼ R n series resistances Req ¼ R1 þ R2 þÁÁÁþRn n parallel resistances 1 R ¼ eq 1 þ 1 þÁÁÁþ 1 R1 R2 Rn Frequency and period 1 1 2p o f ¼ T ¼ o ¼ 2pf ¼ f ¼ T f T 2p ÀÁpffiffiffiffiffiffiffi Inductive and capacitive reactance j ¼ À1 1 Àj 1 1 X ¼ ff90 X ¼ X ¼ jjX ¼ C 2pfC C 2pfC C j2pfC C 2pfC XL ¼ 2pfLff90 XL ¼ j2pfLjj XL ¼ 2pfL Bipolar transistor relationships V VBE T V IC ¼ bIB IE ¼ IC þ IB re ¼ IC ffi ISe T ICQ Triode relationships m gm 3=2 gm ¼ mrP m ¼ rP ¼ IP ¼ kVðÞP þ mVG rP gm JFET relationships 2 2IDSS jjVGS gm0 ¼ ID ¼ IDSS 1 À VP VP Appendix B: Selected Tube Characteristic Curves Figures B.1, B.2, B.3, B.4, B.5, B.6, B.7, and B.8 D.J. Dailey, Electronics for Guitarists, DOI 10.1007/978-1-4614-4087-1, 391 # Springer Science+Business Media, LLC 2013 392 Appendix B: Selected Tube Characteristic Curves Fig. B.1 12AT7 plate and transconductance curves Appendix B: Selected Tube Characteristic Curves 393 Fig. B.2 12AU7 plate and transconductance curves 394 Appendix B: Selected Tube Characteristic Curves Fig. B.3 12AX7 plate and transconductance curves Appendix B: Selected Tube Characteristic Curves 395 Fig. B.4 6AN8 pentode and triode plate curves 396 Appendix B: Selected Tube Characteristic Curves Fig. B.5 6L6GC pentode and triode mode plate curves Appendix B: Selected Tube Characteristic Curves 397 Fig. B.6 6L6GC and EL34 triode mode transconductance curves 398 Appendix B: Selected Tube Characteristic Curves Fig. B.7 EL34 pentode and triode mode plate curves Appendix B: Selected Tube Characteristic Curves 399 Fig. B.8 EL84 pentode mode plate and transconductance curves Appendix C: Basic Vacuum Tube Operating Principles Diodes The diode is the simplest vacuum tube. Recall that the vacuum tube diode operates by thermionic emission of electrons from a heated cathode within the evacuated glass envelope of the tube, as shown in Fig. C.1. Reverse Bias When the anode is at a negative potential with respect to the cathode VAK 0V, the diode is said to be reverse biased, and the current flow through the tube is approximately zero. This occurs because the free electrons around the cathode are repelled by the negative potential at the anode, as shown in Fig. C.1a.