255 Final Flashcards
Length
Meters (meters)
Mass
Kilograms (kg)
Time
Seconds (s)
Thermodynamic temperature
Kelvin (K)
Amount of substance
Mole (mol)
Charge
Coulomb (C)
Frequency
Hertz (Hz) s^-1
Force
Newton (N) kgm/s^2
Energy or work
Joule (J) Nm
Power
Watt (J/s)
Electric current
Ampere (A) C/s
Electric potential
Volts (V) J/C
Electric resistance
Ohms V/A
Electric Conductance
Siemens (S) A/V or 1/ohm
Electric Capacitance
Farad (F) C/v
Magnetic flux
Weber (Wb) Vs
Inductance
Henry (H) Wb/A
Nano
10^-9
Micro
10^-6
Milli
10^-3
Kilo
10^3
Mega
10^6
Giga
10^9
Electric Circuit
An interconnection of electrical elements
Charge
What a person feels walking on a carpet then receiving a shock from a metal object. It is from electron exchange with the surroundings. Measured in coulombs
Coulomb
1 electron = 1.602x10^-19C
Electric current
Time rate of change of charge measured in amperes. (C/s)
Electric current mathematical definition
i=dq/dt
Charge mathematical definition
Q= integral from to to t of (i dt)
Direct Current (DC)
A current that remains constant in time and does not change direction
Alternating Current (AC)
A current that varies sinusoidally in time(does change direction)
Voltage (ie potential difference)
Energy required to move a unit charge through an element
Voltage mathematical definition
Vab= dw/dq (V= J/C = Nm/C)
Equivalent voltage
Vab = -Vba
Power
Time rate if change of expending/absorbing energy (W)
Power mathematical definition
P = dw/dt = dw/dq*dq/dt
Power (instantaneous)
P= vi (VA=W)
Power sign convention
If current enters through the positive terminal power is positive. If entering through the negative terminal, negative sign is used
Power conservation
+ power absorbed = - power supplied
Or
Sum = 0
Energy mathematical definition
W = integral from to to t of (p dt)
or
W = integral from to to t of (vi dt)
Independent voltage source
Voltage is set, current determined by circuit equations
Independent current source
Current is set voltage dictated by circuit equations
Dependent voltage source
Voltage is function of current/voltage elsewhere in circuit
Current determined by circuit equations
Dependent current source
Current is functin of current/voltage elsewhere in circuit.
Voltage determined by circuit equations
Problem solving steps
1) define problem
2) present everything known
3) establish potential solution paths
4) attempt a problem solution
5) evaluate solution, is it accurate
6) check to make sure problem has been solved satisfactory
Resistivity
p , the tendency if a materials to resist the flow of current. Unit is ohms
Resistance
R= pl/A (ohms)
p-material resistivity
l- length
A- cross sectional area
Ohms Law
v=iR
Short circuit
Resistance is 0
Open circuit
Resistance is infinite
Conductance
G = 1/R = i/V (Units is S of 1/ohm or A/V)
Power in a resistor
P = iv = i^2R = V^2/R = V^2G = i^2/G
Power sign
Power is always positive
Resistor color coding
Black (0), Brown (1), R(2), O(3), Y(4), G(5), B(6), V(7), Grey (8), white(9)
Branch
A single element such as a voltage source or a resistor
Node
Point of connection between two or more branches
Loop
Any closed path in a circuit
Kirchoffs Current Law
The current going into a node has to be equal to the current leaving the node.
Kirchoffs Voltage Laws
All voltages over a loop must sum to zero
Series equivalent resistance
Req = R1+R2…+Rn
Voltage division
Vn = Rn/ (R1+R2+…+Rn) *v
Or
Vn = Rn/Req *v
Parallel equivalent resistance
1/Req = 1/R1 + 1/R2 +…+1/Rn
Current division
in = Req/Rn *i
Series equivalent conductance
1/Geq = 1/G1 + 1/G2 + …+ 1/Gn
Parallel equivalent conductance:
Geq = G1+G2
Nodal Analysis Process
1) select node as reference node (often ground) and assign voltages to other nodes
2)Apply RCL to each non reference node. Use ohms law to express currents as node voltages
3) solve equations to get unknown node voltages
Supernode
A dependent or independent voltage source between two non reference nodes.
Mesh
Loop not containing another loop
Mesh analysis
Using kvl to solve for currents in a circuit with multiple meshes
Mesh analysis process
1) Assign mesh currents (typically clockwise)
2) Apply KVL to each mesh. Use ohms law to express voltages in terms of mesh currents.
3) solve the system of equations
Super mesh
Two meshes have a dependent or independent current source in common
Ohms law (conductance)
I = GV = V/R
Nodal analysis by inspection
Only applicable when circuit contains independent current sources.
-diagonal terms are aim of conductance’s connected directly to node 1 or 2
- off diagonal terms are the conductance’s connected between the nodes but negative.
-right hand side is the sum of the currents entering through the node
Mesh analysis by inspection
Only apply when circuit only contains independent voltage sources.
-diagonal terms are sum of resistance in mesh
-off diagonal terms are negative if resistance common to meshes
Right hand side is sum (taken clockwise) of independent voltage sources
Ohms law linearity
If v1=Ri1 and v2 = Ri2 then v = (i1+i2)R = i1R +i2R = v1+v2
Power current vs power voltage relationship
p=i^2R and p=R/v^2.
Are not linear relationships
Superposition
Voltage/current for an element is the sum of the voltage/current from each independent source
Applying superposition
1) turn of independent sources except one. Find the output v or i. (Voltage= short circuit, current = open)
2) repeat step 1 for each independent source
3) find total contribution by adding all the contributions from independent sources
Source transformation currently/voltage
Vs = isR or is= Vs/R
Can be used for dependent or independent sources
Thevenin theorem
A linear two part circuit can be replaced with a voltage source and resistor in series
Thevinin equivalent voltage
Vth = Voc
Thevinin resistance
Rth = Rin
Thevenin with dependent source options
1) put Vab = 1volt (test) find io into terminal a
2) put Iab = 1A (test) find Vo into a
Norton Theorem
A linear circuit can be replaced with a current source and resistor in parallel.
Norton equivalent current
In = Isc = Vth/Rth
Norton equivalent resistance
Rn = Rth
Power delivered to load
p= vi = i^2Rl = (Vth/Rth+Rl)^2*Rl = Vth^2 Rl/(Rth +Rl)^2
Maximum power is transferred when
Rl = Rth
Maximum power
Pmax = Vth^2/ 4Rth
Practical sources
Real world sources will not provide a consistent output and will vary depending on the load resistance.
Galvanometer
A type of ammeter
Wheatstone Bridge circuit
Designed to accurately measure resistance. The variable resistor is adjusted until there is no current through the galvanometer. Forms two voltage divider circuits
Wheatstone bridge measured resistance
Rx = R2R3/R1
What type of energy storers are resistors
They are passive elements that dissipate energy
Capacitor
Is a circuit element that is passive and stores energy. A capacitor has two parallel conducting plates that are separated by an insulator. Capacitors apply voltage v over the plates creating a positive charge q on the positive side and a negative charge q on the negative side. It is a linear relationship between charge and voltage
Capacitance definition
Ratio between charge (on one plate) and voltage on capacitor.
Capacitance definition mathematical
C = q/v (farads)
Alternative mathematical capacitance definition
C = EA/d = ErEoA/d
Capacitor current
Current can be found by taking the derivative of this charge with respect to time.
i=dq/dt = Cdv/dt
Capacitor voltage
Integrate the current relationship. Or V = 1/C integral from to to t of (i(z)dz) +vo(t)
Capacitor energy
W = 1/2 Cv^2 = 1/2 q^2/C
Capacitor under low DC frequency
Gives open circuit. I = 0
Capacitor under high AC frequency
Gives short circuit. I = infinity
Equivalent capacitance in parellel
Ceq = C1 + C2 +…+ Cn
Equivalent capacitance series
1/Ceq = 1/C1 + 1/C2 + …. 1/Cn
Inductance definition
Tendency to oppose change in current
Inductance mathematical definition
L = NuA/l
N- number of coils
u-permeability
A- cross section area
l- length
Inductor voltage
Proportional to the inductance and the derivative of the current with respect to time
V = L di/dt
Inductor current
i = 1/L integral from to to t of v(z)dz + i(to)
Inductor energy
W = 1/2 Li^2
Equivalent inductance series
Leq = L1 + L2 +… +Ln
Like resistors
Inductors in parallel
1/Leq = 1/L1 + 1/L2 + … + 1/Ln
General sinusoid
V(t) = Vm sin(wt)
Here Vm is the amplitude of the sinusoid w is the angular frequency and wt is the argument. The period is T = 2pi/w.
Periodic function
V(t) = (t+nT)
It repeats every nT where n is an integer
Sinusoid generall
V(t) = Vm sin(wt + theta)
Theta is an offset
Phasor
A complex number representing amplitude and phase of a sinusoid.
Rectangular form
Z = x +jy
Polar form
Z = r<theta
Exponential form:
Z = re^(jtheta)
Useful phasor conversion facts
r = sqrt(x^2 + y^2)
Theta = arctan(y/x)
X = rcos(theta) y = rsin(theta)
re^(jtheta) = rcos(theta) + j rsin(theta)
Euler identity
e^(jtheta) = cos(theta) +- j sin(theta) -> Re{e^jtheta} = cos(theta)
Note about phasor relationship
It is always for cosine. For sine a phase shift of -90 is included
Vmsin(wt+theta) = Vm<theta-90
Derivative of time domain
dv/dt (time domain) = jwV (phasor domain)
Or
Integral vdt = V/jw
Important notes about phasors
v(t) is the instantaneous time domain representation and V is the phasor domain representation.
v(t) is the time dependent and V is not
v(t) is real while V is complex
Impedance
The AC version of resistnce. It is the ratio of phasor, voltage, V to phasor current I.
The reciprocal of impedance
Is admittance
Admittance
Y = 1/Z = I/V (S)
And
Y = G +jB
Voltage/current relationship AC
Vth = ZnIn
Zn = Zth
Maximum power transfer AC
ZL should be Zth* = Rtb -jXth
P max = |Vth|^2/8Rth
Instantaneous AC power
p(t) = v(t) i(t) (W)
Instantaneous current
i(t) = Im cos(wt +thetai)
Instantaneous voltage
v(t) = Vmcos(wt +theta v)
Instantaneous power expanded
p(t) = 1/2 VmImcos(thetav -thetai) + 1/2 VmIm cos(2wt + theta v +theta i)
Instantaneous power average is
Positive
Average power
P = 1/T integral T to 0 p(t) dt
Or
P = 1/2 VmIm cos(thetav -theta i)
Or
P = 1/2 Real {VI}
Or
P = 1/2 VI
Power absorption of elements in AC
A resistive load (R) absorbs power at all times while a reactive load( L kr C) absorbs 0 average power
Maximum average power AC circuits
First set the impedance of the Thevenin equivalent and the load.
Zth = Rth +JXth
Zl = Rl + jXl
Need to find where Rl = Rth and Xth =-Xl
Max power transfer theorem
Zl = Rl + jXL = Rth -jXth = Zth*
Max avg power
Pave = |Vth|^2/8Rth
If the load is only real Rl = |Zth|
What is the effectiveness of a resistance load and how do we measure it?
The effective value of a periodic current is the dc current that delivers the same average power to a resistor as the periodic current.
Effective current
Ieff = sqrt(1/T integral to to t i^2 dt)
Effective Voltage
Veff = sqrt (1/T integral from t to to v^2 dt)
Reltionship between current and voltage for Rms and eff
Ieff = Irms
Veff = Vrms
Rms values
Xrms = sqrt(1/T integral to to t x^2 dt)
Effective vs root mean square
The effective value of a periodic signal os its root mean square rms value
Apparent power
The product of the rms value of current and rms value of voltage. The units are VA.
S = VrmsIrms
Power factor
The ratio between the power and apparent power
pf = P/S = cos(thetav -thetai)
Power factor angle
Thetav -theta i
Leading power factor
Means current leads ahead of voltage (like capacitor).
Power factor angle is negative
Lagging power factor
Means current is behind voltage( like inductor). Power factor angle is positive
Complex vs apparently power
S is complex power
|S| is apparent power
Complex power
S = IrmsZ = Irms^2 |Z|<thetav -thetai = Vrms^2/ Z = Vrms Irms
Also equal to P + Q
P = Re{S} = Irms^2R
Q = Im{S} = Irms^2X
Reactive power
Q, imaginary part
Q relationships for loads
Q= 0 for resistive load
Q < 0 for capacitive loads
Q> 0 for inductive loads
Power triangle
S is hypotenuse
P and Q are legs and angle is Thetav-Theta i
Impedance triangle
|Z| is hypotenuse and X and R are legs.
The angle is thetav -theta i
Conservation of power AC
Units are VARS.
S = S1+S2+…+Sn
Shunt Capacitance
C = Qc/wVrms^2
