Circuits 1 test 2, Circuits I Test 1

Find both square roots of the quantity (1 + j), both "by hand" and using a calculator.

(1+j)1/2 = (√2 ∠45o)1/2 = 21/4∠(45/2)o = 1.19∠22.5o or 21/4∠[(45+360)/2]o = 1.19 ∠202.5o.

How many initial conditions are needed in a circuit containing one capacitor or one inductor?

1

C in series

1/Ceq = 1/C1 + 1/C2 + ...

L in parallel

1/Leq = 1/L1 + 1/L2 + ...

S

1/Ω

p

10^-12

f

10^-15

a

10^-18

m

10^-3

u

10^-6

n

10^-9

T

10^12

k

10^3

M

10^6

G

10^9

S

A/V

Could the following voltage conceivably occur in some stable first-order circuit? v(t) = 2.3 + 0.6 et/(1 µs) V. Why or why not? If not, make the minimal change that would make it conceivably possible

Absolutely not (absent dependent sources, which might make the circuit unstable). This voltage "blows up" (becomes infinite) as t goes to ∞. An acceptable function might be v(t) = 2.3 + 0.6 e-t/(1 µs) V. (Note insertion of - sign).

C

As

Which type of analysis leads to matrix equations? Node analysis Mesh analysis Both of them

Both

J

C-V

A

C/s

C in parallel

Ceq = C1+C2+...

List ALL quantities associated with a capacitor that cannot change instantaneously (there are at least four)

Charge, voltage, stored energy, electric field between plates, electric displacement vector between plates

List ALL quantities associated with an inductor that cannot change instantaneously (there are at least four)

Current, magnetic field intensity, magnetic induction field, stored energy

C

FV

The voltage at the positive end of a 6.2 V voltage source is necessarily 6.2 V. T F

False! Voltage sources establish voltage differences, not absolute voltages. This is only true if the negative end is connected to ground.

You are relatively safe inside a car in a lightning storm (neglecting high winds) because your rubber tires ensure that you are well grounded. T F

False! You are (relatively) safe, but only because you are NOT grounded (rubber is of course an insulator). However, please don't try this experiment (stay inside)!

Using superposition, we can find the total voltage or current at any point in a linear circuit by adding the corresponding voltages or currents produced separately by EVERY source in the circuit, regardless of what kind of source it is. T F

False, this applies only to independent sources.

When performing nodal analysis, it is necessary and advisable to define a notation for every current entering or leaving each node. T F

False, this is unnecessary

A first order circuit can show "ringing." T F

False. First-order circuits always show simple exponential decays. Only underdamped second or higher order circuits can show "ringing" (damped oscillations).

Current enters a junction connected to two resistors. It is necessarily true that more current will flow through the smaller resistor. T F

False. It depends on what is connected further downstream.

A critically-damped 2nd order circuit can show "ringing." T F

False. Must be underdamped. (Of course no real circuit is ever exactly critically damped.)

The recommended "final" variables that should appear in the equations resulting from nodal analysis include node voltages and all control variables for dependent sources. T F

False. The control variables should preferably be eliminated, leaving only node voltages

The energy stored in a capacitor is directly proportional to the voltage across it.

False. W = (1/2) CV2, it is proportional to the square of the voltage

To find the Thévenin equivalent resistance of a resistive circuit containing sources, we can simply "kill" all independent and dependent sources in that circuit by setting their voltages or currents to zero, and then find the equivalent resistance of all resistors in the circuit. T F

False. You should only "kill" the independent sources.

What is the purpose of each of these solution processes (i.e., what do we get from it)?

From the analysis at t < 0 (or equivalently, at t = 0-), we get the value of the capacitor voltage or inductor current at that instant, which will be unchanged at t = 0+. From the analysis at t = ∞, we get the particular (steady-state) solution. From the analysis at t = 0+, where we replace the capacitor by an independent voltage source (or the inductor by an independent current source), we find the initial value of the sought variable of interest at t = 0+.

An arbitrary constant always multiplies which type of solution of an inhomogeneous linear differential equation: homogeneous, particular, or both?

Homogeneous only.

In what situation do we only need to solve a circuit at TWO different times, and which times are they?

If we are solving for the voltage of the capacitor as a function of time, or the current of the inductor as a function of time, we don't need to solve the "DC" circuit problem at t = 0+. We already know the value of the sought variable at that instant from the analysis done at t < 0.

Give a precise statement of the superposition theorem.

In any linear, time-invariant resistive circuit, we can find any voltage or current in the circuit by summing the corresponding voltages or currents that are caused by each independent source acting separately (i.e., after "killing" all other independent sources.) NOTE: To "kill" an voltage source, we replace it by a short circuit. To "kill" an independent current source, we replace it by an open circuit. One should never "kill" a dependent source! Also, one cannot sum powers due to each source to get the total power, because power is a nonlinear function of current & voltage.

When exactly is it necessary to use a test source at the terminals of a circuit to find its Norton equivalent circuit? (Same question for Thévenin equivalent).

In either case, whenever the circuit has no independent sources (or more generally, when its open-circuit voltage or short-circuit current is zero).

Likewise, what must be true about a current source to apply source transformation?

It must have a finite resistor connected in parallel with it. If its parallel resistance is infinite (no parallel resistance), it cannot be transformed.

What must be true about a voltage source in order to apply source transformation?

It must have a finite resistor connected in series with it. If its series resistor is zero (no series resistor), it cannot be transformed.

W

J/s

L in series

Leq = L1 + L2 + ...

A 10 kΩ resistor dissipates more power than a 1 kΩ resistor. T F

May or may not be true. If they both have the same voltage, the smaller resistor dissipates more power (V2/R). If they both have the same current, the larger resistor dissipates more power (I^2R).

J

N-m

Mesh analysis (as we have studied it) is restricted to what kind of circuits?

Planar (can be drawn so that no wires cross without connecting).

Give the formulas for adding resistors in series, resistors in parallel, conductances in series, and conductances in parallel.

Req = R1 + R2 + ... (series). 1/Req = 1/R1 + 1/R2 + ... (parallel). 1/Geq = 1/G1 + 1/G2 + ... (series). Geq = G1 + G2 + ... (parallel).

Give a precise statement of Thévenin's theorem (as studied to date), including all qualifications about when it applies or doesn't apply. Any linear, time-invariant, resistive circuit is equivalent at its terminals to the series combination of an ideal voltage source and a resistor (the Thévenin equivalent resistance).

Same question for Norton's theorem: Any linear, time-invariant, resistive circuit is equivalent at its terminals to the parallel combination of an ideal current source and a resistor (the Thévenin equivalent resistance).

When breaking a circuit to use Thévenin's theorem to solve it, we must never do what?

Separate a controlled source from its controlling variable.

KCL

The algebraic sum of all currents leaving [or entering] a node (or any other closed surface) in a circuit is zero.

KVL

The algebraic sum of all voltage drops [or rises] around a closed loop in a circuit (which need not run entirely through conductors or circuit elements) is zero.

Carefully sketch the function 3 (1 - e-2t) for t ≥ 0, showing the tangent at t = 0 correctly.

The function is zero at t = 0 and increases exponentially (concave downward) to a final value at t = ∞ of 3. The time constant is 0.5 s. The tangent goes through the origin and passes through the point (0.5 s, 3).

When dependent sources are present and we are doing node or mesh analysis, the control variables for the dependent sources must be expressed in terms of what, exactly?

The node voltages (in nodal analysis), or the mesh currents (in mesh analysis).

In analyzing the transient behavior of a 2nd order circuit, we generally use the fact that what two things cannot change instantaneously

The voltage across a capacitor and the current through an inductor.

The Thévenin equivalent circuit of a circuit containing dependent sources could involve a negative value of RTh. T F

True

When applying KVL to sum voltage drops, we (conventionally) add a minus sign to the value of a voltage source when the direction we are following leads us into the negative terminal, and the terms involving voltage drops across resistors are all positive. T F

True

W

V-A

We can use superposition to find the total power absorbed by any circuit element by separately computing the powers absorbed there due to each independent source in the circuit and then adding those powers. T F

VERY false! Superposition only works for voltages & currents (try it!).

list at least one quantity that CAN change instantaneously

Voltage

H

Vs/A

J

W-s

In a switched circuit, one solves for the Thévenin equivalent resistance "seen" by the reactive element at what TIME? Why?

We do this for t > 0, as that as the time interval for which we are interested in the transient behavior. We are generally NOT interested in the value of RTh for t < 0.

Explain exactly when it is necessary to use a supernode when doing node analysis.

Whenever you have a voltage source with neither end connected to ground.

Explain exactly when it is necessary to use a supermesh when doing mesh analysis.

Whenever you have an "internal" current source (shared by two meshes).

The quantity K in the expression A sin (Kt) is the ____and necessarily has units of ____

angular time frequency, rad/s

In a first-order circuit with constant sources, the voltages and currents are always _______at t = ∞.

constant

An open circuit implies zero ________

current

Two circuit elements in series must have the same ____

current

In a first-order transient circuit, the voltages and currents always include _______ functions of time (insert the name of a function)

decaying exponential

The voltages and currents in a 2nd order circuit with constant sources might (depending on the amount of damping) contain what three (or four) types of functions of time?

decaying exponentials_, _constants_, and _damped sinusoids (sinusoids multiplied by decaying exponentials)_.

A good voltmeter has ____ input impedance_.

high

A linear relationship must have two properties,

homogeneity and additivity

What is it for an ideal current source?

infinite

N

kg-m/s^2

J

kg-m^2/s^2

A good current meter has ____ input impedance.

low

The currents of two voltage sources in series _________

must be the same!

In a first-order circuit with sinusoidal sources all at a frequency ω, the voltages and currents are always _________as we approach t = ∞.

sinusoidal at the same frequency ω

To find a sought voltage or current in a switched circuit as a function of time, one must generally solve an (effectively) DC circuit problem at what THREE times?

t < 0, t = 0+, and t = ∞.

The time constant in an arbitrary first-order circuit possibly containing many resistors and sources and containing one capacitor of value C is always C multiplied by

the Thévenin equivalent resistance seen from the capacitor terminals

The time constant in an arbitrary first-order circuit possibly containing many resistors and sources and containing one inductor of value L is always L divided by

the Thévenin equivalent resistance seen from the inductor terminals

A 2nd order circuit shows a transient response that has two zero crossings after the start of the transient. This circuit must be

underdamped

A short circuit implies zero _______

voltage

Two elements in parallel must have the same _____

voltage

What is the value of RTh (Thévenin equivalent resistance) for an ideal voltage source?

zero

V

Ω-A