16.1: Electric Circuits Advanced Topics
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 back-emf. Back-emf is essentially an induced negative voltage which opposes changes in current. The amount of back-emf 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 one-way 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 | Light-emitting diodes (LEDs) | |
Transistor (varies by type) | none | Computer chips, amplifiers |
- An operational amplifier or op-amp is an active circuit element that performs a specific function. The most common op-amp has five leads: two input leads, one output lead, and two fixed-voltage leads.
The job of an op-amp is to use the voltage it is supplied to adjust its output voltage. The op-amp 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 vice-versa.
- 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 resistance, 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}{l} & \text{Inductance of a solenoid with N}\ \text{turns, area A,}\ \text{Henrys (H), length l} \\ X_L = 2\pi f L & \text{Inductive reactance for AC of frequency f}\\ X_C = \frac{1}{2\pi f C} & \text{Capacitive reactance for AC of frequency f}\\ Z = \sqrt{R^2+(X_L-X_C)^2} & \text{Impedance of an RC circuit (Pythagorean Theorem)}\\ \tan{\phi} = \frac{X_L-X_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 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\ \mu\mathrm{F}\end{align*} capacitors and a \begin{align*}40.0\ \mu\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?
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