Electrical- DC and Transient Circuits

Question 1

Linear component

Answer 1

The current through it is directly proportional to the potential difference across it

Question 2

Passive component

Answer 2

Component can only dissipate or store energy, they cannot generate energy themselves

Question 3

Ideal component

Answer 3

The circuit element can be described by a simple equation which usually has a single variable. They are first order approximations to real-life components.

Question 4

Are resistors, capacitors and inductors linear and/or passive and/or ideal?

Answer 4

They are all linear and passive

Question 5

Define current relative to charge

Answer 5

i(t)=dq(t)/dt 
Where q(t) is net flow of charge through a notional cross-section

Question 6

What is the difference between upper and lower case symbols?

Answer 6

Upper case refer to constant values (don’t change with time) whereas lower case refer to time-varying quantities

Question 7

What is the sense of voltage or current through a component on a circuit diagram?

Answer 7

The direction of the current’s/voltage’s arrow

Question 8

Which way do current and potential difference travel through passive components?

Answer 8

In opposite directions

Question 9

What is the formula for potential difference between two plates of a capacitor in terms of charge?

Answer 9

v=q/C

Where C is capacitance

Question 10

Which direction are voltage and current drawn for ideal voltage sources?

Answer 10

In the same direction. Although only the voltage across it is specified and the current through it can be in either direction.

Question 11

Describe a current source

Answer 11

A component that defines the current through it but not the voltage which can be in either direction

Question 12

Which direction does the arrow of voltage point?

Answer 12

Towards the higher potential

Question 13

Describe the layout of a current divider

Answer 13

A current source connected to two resistors which are in parallel with each other. It shares the current between the 2 resistors

Question 14

How does superposition work in circuit analysis?

Answer 14

Used for analysing linear systems. Consider each source in turn and eliminate the other sources by replacing a voltage source with a piece of wire (V=0) and a current source with a gap (I=0). Solve the circuits formed to find the quantities of interest. Sum the answers of the desired quantities for each source considered.

Question 15

How does mesh (loop) analysis work?

Answer 15

Draw clockwise circulating currents in each loop of the circuit so all branches have at least one current. The currents through each component are the some of the circulating currents that pass through them. Apply Kirchoff’s voltage law to every loop starting with the top right node and working anti-clockwise. Rewrite the equations with unknown voltages in terms of current and resistance using Ohm’s law. If current source in loop, label its unknown voltage. Current it produces is the sum of circulating currents passing through it. Now have enough equations.

Question 16

How does nodal analysis work?

Answer 16

Choose a node to be 0 (draw as earthed). Label the unknown nodal voltages on each remaining node. Apply Kirchoff’s current law on each node using unknown currents coming out of each node. Consider for each node, the path to each of its adjacent nodes. Use Ohm’s law where the voltage is the difference between the nodal voltages. Substitute the branch equations into the node equations eliminating current. Solve for nodal voltages. If voltage source, use difference in voltages between its two nodes=voltage it produces.

Question 17

What is Thevenin’s theory?

Answer 17

That any one-port network connected to two terminals can be redrawn as a voltage source in series with a resistor connected to the same two terminals

Question 18

When does Thevenin’s theory apply?

Answer 18

Where there is only one external connection (earthed is an external connection). When the one-port network is made of resistors and sources only.

Question 19

How to find Thevenin equivalent circuit

Answer 19

The voltage of the voltage source will be the potential difference between the two terminals A and B. Keep A and B not connected to form an open circuit and use nodal analysis to find the potential difference between A and B. Redraw the circuit connecting A and B to form a short circuit and use mesh analysis to find the current in the wire connecting them. The voltage found before over this current will be the resistance of the resistor in the thevenin circuit.

Question 20

What is Norton’s theory?

Answer 20

Any one-port network connected to two terminals can be redrawn as a current source in parallel to a resistor connected to the same two terminals.

Question 21

How to find the Norton equivalent circuit

Answer 21

Using same methods as for Thevenin find the open circuit voltage using nodal analysis and find the short circuit current using mesh analysis. The current produced by the current source will be the short circuit current. The resistance of the resistor will be the open circuit voltage over the short circuit current.

Question 22

How to convert between Thevenin and Norton circuits

Answer 22

Resistance on resistor is the same. Voltage produced by voltage source is equal to the resistance of resistor times current produced by the current source.

Question 23

For DC, what are the relations for current and voltage in resistors, capacitors and inductors?

Answer 23

Resistors: V=IR
Capacitors: V is unknown, I is 0 (open circuit)
Inductors: V=0 (short circuit), I is unknown

Question 24

For transient circuits, what are the relations for current and voltage in resistors, capacitors and inductors?

Answer 24

Resistors: v=iR
Capacitors: v=(1/C)Sidt + V0, i=Cdv/dt
Inductors: v=Ldi/dt, i=(1/L)Svdt + I0
Where S means integral

Question 25

For AC circuits, what are the relations for current and voltage in resistors, capacitors and inductors?

Answer 25

Resistors: V=IR
Capacitors: V=-jI/ωC, I=ωCVj
Inductors: V=ωLIj, I=-jV/ωL

Question 26

What is a transient state of a circuit?

Answer 26

A momentary event preceding the steady-state during a sudden change or start-up of a circuit. Certain current and voltage values can change over time.

Question 27

General formula for vout in a transient circuit

Answer 27

vout=Vfinal+(Vinitial-Vfinal)e^(-t/τ)
Where τ is time constant of circuit (either RC or L/R)
Find Vfinal and Vinitial by considering the current and voltage values in the capacitors and inductors

Question 28

What values are 0 for inductors and capacitors in transient circuits?

Answer 28

Capacitors: start at 0V, end at 0A
Inductors: start at 0A, end at 0V

Question 29

Formula for voltage across capacitor in transient circuits

Answer 29

vout=Vin(1-e^(-t/RC))

Question 30

Formula for voltage across resistor in series with inductor in transient circuits?

Answer 30

vout=Vin(1-e^(-Rt/L))

Question 31

Derive formula for vout across a capacitor in series with a voltage source and a resistor

Answer 31

By the voltage law: vin=vR+vout. Replace vR and vout with iR and (1/C)Sidt. Differentiate both sides with respect to time and times through by C. For constant vin, the Cd/dt(vin)=0. Sub for 0. Rearrange to make dt subject, then integrate both sides to get -RClnAi=t. Rearrange to make i subject so i=(1/A)e^(-t/RC). Back to voltage law equation. vin is constant Vin, replace i in iR, rearrange to make vout subject (-R/A=B). If vout=0 when t=0 find that B=Vin. Sub in and factorise.

Question 32

Deriving the formula for vout across a resistor in series with a voltage source and inductor Part 1

Answer 32

Voltage law: vin=Ldi/dt + iR.
Rearrange to make i+(L/R)di/dt=vin/R. Problem that vin won’t go.
Need a function dp/dt so p=(L/R)dp/dt because if times through by dp/dt
idp/dt+(L/R)(dp/dt)(di/dt)=(vin/R)dp/dt
So idp/dt+pdi/dt=(vin/R)dp/dt, notice d/dt(ip)=(vin/R)dp/dt (prod rule)
From before: dp/dt=(R/L)p. Rearrange and integrate to get p=e^(Rt/L)

Question 33

Deriving the formula for vout across a resistor in series with a voltage source and inductor Part 2

Answer 33

p=e^(Rt/L). Now have integrating factor dp/dt=(R/L)e^(Rt/L).
Sub in for dp/dt into first equation multiplied through by it.
L/RxR/L cancel so get (R/L)e^(Rt/L)i+(e^(Rt/L))di/dt=(vin/L)e^(Rt/L)
Notice d/dt(ie^(Rt/L))=(vin/L)e^(Rt/L). Say vin is constant Vin.
Integrate to get ie^(Rt/L)=(Vin/R)e^(Rt/L)+k (L cancel)
Divide by e^(Rt/L) to make i subject. Recall vout=IR.
Sub in for i to get vout=Vin+k e^(-Rt/L). If vout=0 for t=0 find kR=-Vin
Final expression: vout=Vin(1-e^(-Rt/L))

Question 34

How do voltages and current split when capacitors are in series or parallel with each other?

Answer 34

Multiple capacitors in series: voltages split according to 1/C so smallest C means highest voltage across that capacitor
Multiple capacitors in parallel: currents split according to C so largest C gets highest current

Question 35

How do voltages and current split when inductors are in series or parallel with each other?

Answer 35

Multiple inductors in series: voltages split according to L so largest L gets biggest voltage
Multiple inductors in parallel: currents split according to 1/L so smallest L gets biggest current

Question 36

Time constant for resistor capacitor/inductor circuits

Answer 36

Capacitor: RC
Inductor: L/R

Question 37

How to identify time constant for a circuit with inductor(s) or capacitor(s) in

Answer 37

All 1st order circuits can be simplified by superposition to either an inductor or capacitor in series with a resistor. Do this by doing superposition technique with current and voltage sources. Can simplify by combining components (eg R in series).