16.1: Electric Circuits Advanced Topics
Modern circuitry depends on much more than just resistors and capacitors. The circuits in your computer, cell phone, and iPod depend on circuit elements called diodes, inductors, transistors, and operational amplifiers, as well as on other chips. In particular the invention of the transistor made the small size of modern devices possible. Transistors and op amps are known as active circuit elements. An active circuit element needs an external source of power to operate. This differentiates them from diodes, capacitors, inductors and resistors, which are passive elements.
Key Concepts
 Inductors are made from coiled wires, normally wrapped around ferromagnetic material and operate according to the principles of magnetic induction presented in Magnetism. Inductors generate a backemf. Backemf is essentially an induced negative voltage which opposes changes in current. The amount of backemf generated is proportional to how quickly the current changes. They can be thought of as automatic flow regulators that oppose any change in current. Thus electrical engineers call them chokes.
In a circuit diagram, an inductor looks like a coil. The resistance \begin{align*}R\end{align*} and capacitance \begin{align*}C\end{align*} of an inductor are very close to zero. When analyzing a circuit diagram, assume \begin{align*}R\end{align*} and \begin{align*}C\end{align*} are precisely zero.
 Diodes are passive circuit elements that act like oneway gates. Diodes allow current to flow one way, but not the other. For example, a diode that “turns on” at \begin{align*}0.6\;\mathrm{V}\end{align*} acts as follows: if the voltage drop across the diode is less than \begin{align*}0.6\;\mathrm{V}\end{align*}, no current will flow. Above \begin{align*}0.6\;\mathrm{V}\end{align*}, current flows with essentially no resistance. If the voltage drop is negative (and not extremely large), no current will flow.
Diodes have an arrow showing the direction of the flow.
 Transistors are active circuit elements that act like control gates for the flow of current. Although there are many types of transistors, let’s consider just one kind for now. This type of transistor has three electrical leads: the base, the emitter, and the collector.
The voltage applied to the base controls the amount of current which flows from the emitter to the collector.
 For example, if the base voltage is more than \begin{align*}0.8\;\mathrm{V}\end{align*} above the collector voltage, then current can freely flow from the emitter to the collector, as if it were just a wire. If the base voltage is less than \begin{align*}0.8\;\mathrm{V}\end{align*} above the collector voltage, then current does not flow from the emitter to the collector. Thus the transistor acts as a switch. (This \begin{align*}0.8\;\mathrm{V}\end{align*} is known as a “diode drop” and varies from transistor to transistor.)
 Transistors have an infinite output resistance. If you measure the resistance between the collector and the base (or between the emitter and the base), it will be extremely high. Essentially no current flows into the base from either the collector or the emitter; any current, if it flows, flows from the emitter to the collector.
 Transistors are used in amplifier circuits, which take an input voltage and magnify it by a large factor. Amplifiers typically run on the principle of positive and negative feedback. Feedback occurs when a small portion of an output voltage is used to influence the input voltage.
Circuit element  Electrical symbol  Unit  Everyday device 

Voltage Source (\begin{align*}V\end{align*})  Volts \begin{align*}(V)\end{align*}  Batteries, electrical outlets, power stations  
Resistor (\begin{align*}R\end{align*})  Ohm \begin{align*}(\Omega)\end{align*}  Light bulbs, toasters, hair dryers  
Capacitor (\begin{align*}C\end{align*})  Farad \begin{align*}(F)\end{align*}  Computer keyboards, timers  
Inductor (\begin{align*}L\end{align*})  Henry \begin{align*}(H)\end{align*}  Electronic chokes, AC transformers  
Diode (varies by type)  none  Lightemitting diodes (LEDs)  
Transistor (varies by type)  none  Computer chips, amplifiers 
 An operational amplifier or opamp is an active circuit element that performs a specific function. The most common opamp has five leads: two input leads, one output lead, and two fixedvoltage leads.
The job of an opamp is to use the voltage it is supplied to adjust its output voltage. The opamp will adjust its output voltage until the two input voltages are brought closer together. In other words, the output voltage will change as it needs to until \begin{align*}V_A  V_B = 0\end{align*}. This won’t happen unless the output voltage is somehow “fed back” into one of the inputs
 Digital circuits only care about two voltages: for example, \begin{align*}+5\;\mathrm{V}\end{align*} (known as “on”) and \begin{align*}0\;\mathrm{V}\end{align*} (known as “off”).
 Logic devices, which are active circuit elements, interpret voltages according to a simple set of mathematical rules known as Boolean logic. The most basic logic devices are the AND, OR, and NOT gates:
For an AND gate, the output will always be at an electric potential of \begin{align*}0\;\mathrm{V}\end{align*} (off) unless both the inputs are at \begin{align*}5\;\mathrm{V}\end{align*} (on), in which case the output will be at \begin{align*}5\;\mathrm{V}\end{align*} (on) as well.
For an OR gate, the output will always be at an electric potential of \begin{align*}0\;\mathrm{V}\end{align*} (off) unless either of the inputs are at \begin{align*}5\;\mathrm{V}\end{align*} (on), in which case the output will be at \begin{align*}5\;\mathrm{V}\end{align*} (on) as well.
For a NOT gate, the output will always be the opposite of the input. Thus, if the input is \begin{align*}5\;\mathrm{V}\end{align*} (on), the output will be \begin{align*}0\;\mathrm{V}\end{align*} (off) and viceversa.
 Alternating current changes direction of current flow. The frequency is the number of times the current reverses direction in a second. Household AC is \begin{align*}60\;\mathrm{Hz}\end{align*}. In AC circuits the current is impeded but not stopped by elements like capacitors and inductors.
 Capacitive Reactance is a measure of how a capacitor impedes the current flow from a given voltage in an AC circuit and is inversely proportional to capacitance. Inductive Reactance is a measure of how an inductor in an AC circuit impedes the current flow from a given voltage and is directly proportional to inductance.
 The total impedance of an AC circuit depends on resitance, capacitive reactance and inductive reactance.
 If the capacitive reactance and inductive reactance are both zero or unequal the voltage and current are out of phase. That is they peak at different times in the cycle. The phase angle measures the lag or lead of current over voltage.
Key Equations
\begin{align*}\begin{cases} \text{E} = \frac{L\Delta I}{\Delta t} & \text{Emf across an inductor with inductance L}\\ L = \frac{\mu_0 N^2 A}{I} & \text{Inductance of a solenoid with N}\text{turns,area A,}\ \text{Henrys (H)}\\ X_L = 2\pi f C & \text{Inductive reactance for AC of frequency f}\\ X_C = \frac{1}{X_L} & \text{Capacitive reactance for AC of frequency f}\\ Z = \sqrt{R^2+(X_LX_C)^2} & \text{Impedance of an RC circuit (Pythagorean Theorem)}\\ \tan{\phi} = \frac{X_LX_C}{R} & \text{Phase angle between peak current and voltage} \end{cases}\end{align*}
Advanced Topics Problem Set
 You purchase a circular solenoid with \begin{align*}100\end{align*} turns, a radius of \begin{align*}0.5\;\mathrm{cm}\end{align*}, and a length of \begin{align*}2.0\;\mathrm{cm}\end{align*}.
 Calculate the inductance of your solenoid in Henrys.
 A current of \begin{align*}0.5\end{align*} A is passing through your solenoid. The current is turned down to zero over the course of \begin{align*}0.25\end{align*} seconds. What voltage is induced in the solenoid?
 What is the voltage drop across an inductor if the current passing through it is not changing with time? Does your answer depend on the physical makeup of the inductor? Explain.
 Consider the transistor circuit diagram shown here. The resistor is a light bulb that shines when current passes through it.
 If the base is raised to a voltage of \begin{align*}5\;\mathrm{V}\end{align*}, will the light bulb shine?
 If the base is lowered to a voltage of \begin{align*}0\;\mathrm{V}\end{align*}, will the light bulb shine?
 Why are transistors sometimes called electronic switches?
 Consider the opamp circuit diagram shown here. Note the fixedvoltage leads are omitted for clarity. (This is typical.) Let’s begin with an input voltage at point \begin{align*}A\end{align*} of \begin{align*}0.5\;\mathrm{V}\end{align*}. a. If the opamp is “doing its job,” what is the electric potential at point \begin{align*}B\end{align*}? b. What current is flowing through the \begin{align*}10\ \Omega\end{align*} resistor? c. Recall that no current ever flows into an opamp. What current must be flowing through the \begin{align*}100\ \Omega\end{align*} resistor? d. What must the output voltage be? Now let’s adjust the input voltage at point \begin{align*}A\end{align*} to \begin{align*}0.75\;\mathrm{V}\end{align*}. e. What is the output voltage now? f. By what factor is the opamp amplifying the input voltage? g. What are some practical applications for such a device?
 Consider the logic circuit shown here.
 If \begin{align*}A\end{align*}, \begin{align*}B\end{align*}, and \begin{align*}C\end{align*} are all off, what is the state of \begin{align*}D\end{align*}?
 If \begin{align*}A\end{align*}, \begin{align*}B\end{align*}, and \begin{align*}C\end{align*} are all on, what is the state of \begin{align*}D\end{align*}?
 Fill out the entire “logic table” for this circuit in the Table (below).
State of \begin{align*}A\end{align*}  State of \begin{align*}B\end{align*}  State of \begin{align*}C\end{align*}  State of \begin{align*}D\end{align*} 

on  on  on  
on  on  off  
on  off  on  
on  off  off  
off  on  on  
off  off  on  
off  on  off  
off  off  off 
 A series circuit contains the following elements: a \begin{align*}125\ \Omega\end{align*} resistor, a \begin{align*}175\;\mathrm{mH}\end{align*} inductor, two \begin{align*}30.0\ ?\mathrm{F}\end{align*} capacitors and a \begin{align*}40.0\ ?\mathrm{F}\end{align*} capacitor. Voltage is provided by a \begin{align*}235\ V_m\end{align*} generator operating at \begin{align*}75.0\;\mathrm{Hz}\end{align*}.
 Draw a schematic diagram of the circuit.
 Calculate the total capacitance of the circuit.
 Calculate the capacitive reactance.
 Calculate the impedance.
 Calculate the peak current.
 Calculate the phase angle.
 Resonance occurs at the frequency when peak current is maximized. What is that frequency?
Answers to Selected Problems

 \begin{align*}4.9 \times 10^{5} \;\mathrm{H}\end{align*}
 \begin{align*}9.8 \times 10^{5} \;\mathrm{V}\end{align*}
 Zero
 Yes
 No
 Because they turn current flow on and off.
 \begin{align*}0.5 \;\mathrm{V}\end{align*}
 \begin{align*}0.05 \;\mathrm{A}\end{align*}
 \begin{align*}0.05 \;\mathrm{A}\end{align*}
 \begin{align*}5.5 \;\mathrm{V}\end{align*}
 \begin{align*}8.25\mathrm{V}\end{align*}
 \begin{align*}11 \times\end{align*}
 \begin{align*}On\end{align*}
 \begin{align*}On\end{align*}
 \begin{align*}On, on, off, on, off, off, on, on\end{align*}
 b. \begin{align*}10.9\mu \;\mathrm{F}\end{align*} c. \begin{align*}195 \ \Omega\end{align*} d. \begin{align*}169 \ \Omega\end{align*} e. \begin{align*}1.39 \;\mathrm{A}\end{align*} f. \begin{align*}42^\circ\end{align*} g. \begin{align*}115\mathrm{Hz}\end{align*}
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