Electrical- AC Circuits and Filters Flashcards
How to express alternating current in terms of peak current and phase
i=IpkSin(ωt+φ)
Where ω is angular frequency
φ is phase
How to express AC voltage in terms of peak voltage and phase
v=VpkSin(ωt+φ)
Where ω is angular frequency
φ is phase
What does a phaser look like?
An argand diagram with a half line representing an AC voltage or current. The length of half line is peak value and the angle between it and the positive real axis is its phase relative to the reference phase.
When a capacitor is in series with an AC voltage source, what do you do to find the current using the input voltage?
The capacitor scales the input voltage by ωC and shifts it’s phase by π/2 (-π/2 in sin function) to get current. Comes from i=Cdv/dt for transient circuit and v=VpkSin(ωt+φ)
Means current leads voltage
What does I=5<45° mean?
Peak current is 5A and the current starts 45° late.
What is a capacitor in an AC circuit with voltage of -jI/ωC equivalent to as a resistor?
A resistor with impedance -j/ωC. Because V=IR still applies
What is impedance?
Symbol Z. The complex equivalent of resistance. Consists of resistance (R) and reactance (X)
Z=R+Xj
Resistance loses energy whereas reactance stores and releases it
Which parts of impedance do capacitors and inductors have?
Only reactance not resistance. So only imaginary part of impedance.
When an inductor is in series with an AC current source, what do you do to find the voltage across it using the input current?
An inductor scales the input current by ωL and shifts its phase by π/2 (+π/2 in sin function) to get voltage. Comes from v=Ldi/dt for transient circuit and i=IpkSin(ωt+φ).
Means current lags voltage
What is an inductor with impedance ωLIj in series with an AC current source equivalent to as a resistor?
A resistor with impedance ωLj. Because V=IR still applies
General formula for RMS voltage
Vrms=sqrt( (1/T)Sv^2dt )
Where T is time period and integral is from 0 to T
S means integral
Formula for instantaneous power
p=(v^2)/R
Formula for average power for a resistor
P=(Vrms^2)/R
Derive the formula for Vrms for sinusoidal AC current
Start with Vrms^2=(1/T)Sv^2dt. Sub in 2π/ω for T and for v VpkSin(ωt+φ). Take out constant Vpk^2ω/2π. Use double angle formula to rewrite sin^2 function. Integrate between 2π/ω and 0 and some bits cancel (4π is 0). Expands and simplify to Vrms^2=(Vpk^2)/2. Square root both sides.
For a resistor and either capacitor or inductor in series in an AC circuit, why is the power dissipated across the both components not the Vrms x Irms across both?
Because the reactance (X) of the capacitor or inductor doesn’t dissipate power, only resistance does. This calculation gives apparent power (S).
What is the correct formula for power dissipated across a resistor and inducto/capacitor in series in an AC circuit when resistance is known?
P=Irms^2 x R
Or P=IVR (both rms)
Where R in VR is subscripts and means Vrms across the resistor
Or P=Irms^2 x Re(Z)
Where Z is combined impedance of both components
Derive formula for active power (P) in terms of Irms and Vrms for a resistor and inductor in series and AC circuit
Know P=Irms^2 x Re(Z).
Also V=IZ so Z=V/I=magnitude of Vrms/magnitude of Irms
Describe the power triangle
Right angled triangle with bottom line angle θ from hypotenuse.
Hypotenuse is apparent power, S (simple formula)
Adjacent is active power, P (with the power factor)
Opposite is reactive power, Q (mag(Irms^2) x X)
What do the 4 types of filter do?
Low-pass: voltages at low frequencies pass, voltages at high frequencies are blocked
High-pass: high frequency voltages pass
Band-pass: voltages between 2 frequencies pass
Band-stop: voltages between 2 frequencies are blocked
What is important to remember about the frequencies when finding if a certain frequency voltage passes?
The frequency of the voltage is not an angular frequency so must use ω=2πf
What is gain?
The ratio of power out to power in. Measured in decibels (dB).
Formula of Gain(dB)=10log(Po/Pi)
Where Po is power out
Pi is power in
Both are magnitudes
It is also the square of voltage gain so =20log(Vo/Vi)
On a Bode plot, what is the maximum gain?
0dB
What is the corner frequency?
The critical frequency of voltage in an AC circuit where the gain is about -3dB. It’s when the term in the logarithm for finding gain multiplied by angular frequency equals 1.
Steps for drawing a Bode plot for high or low pass filter
The axes are gain(dB) against ω (logarithmic scale).
- Find corner frequency and plot it at 0dB
- Draw horizontal line from point to ω=0 (low pass) or ω=infinity (high pass)
- Draw line with gradient -20dB/decade from point to x axis for low pass (+ and to point for high pass)
- The actual line curves under the plotted point so the corner frequency is at -3dB
Describe an example of a series resonant circuit
All in series. AC voltage source, capacitor, inductor, resistor. Vout is measured across resistor.
How to get equation for gain for series resonant circuit
Treat inductor and capacitor as like one component and do potential divider formula. Get Vout=VinR/(R+j(ωL-1/ωC)). Rearrange to get voltage gain. Apply magnitude rules (square terms then root them). Get Gain(dB)=20log(R/sqrt(R^2+(ωL-1/ωC)^2)).
How does the series resonant circuit act as a band-pass filter?
At low frequencies, 1/ωC»_space;1 so gain is small
At high frequencies, ωL»_space;1 so gain is small
At the resonant frequency ωL=1/ωC so the gain is 0dB. This frequency is therefore 1/sqrt(LC) and the overall impedance at this frequency is just R. Bode plot looks like a hill (broad or sharp peak). The pass band is the area under the shape above gain=-3dB
What is the band width for a band-pass filter and specifically a series RLC circuit?
Band the width is the width of the pass band on the Bode plot in Hz.
For series RLC circuit it is R/L
What is quality factor?
A measure of the ‘peakiness’ of the gain response for a filter. Defined by the equation
q=resonant frequency/band width
It is also the magnification of the voltage or current
What is damping factor?
Defined by the equation
ζ=2/q
Where q is quality factor
Describe a parallel resonant circuit
An AC voltage source in series with a resistor and a separate branch containing an inductor and capacitor in parallel with each other.
How does the parallel resonant circuit act as a band stop filter?
At some frequency, the reactance of inductor will equal the negative of the reactance of the capacitor. The total current going into the parallel bit will equal the voltage on inductor/jXL + voltage on capacitor/jXC. At this critical frequency, the two terms equivalent to total current equal the negative of each other so cancel and total current is 0 in the circuit. The resonant frequency is 1/sqrt(LC). Bode plot has horizontal bits either side of resonant frequency and both lines go in to tend to -infinity gain
What is the quality factor of a parallel RLC circuit?
Rsqrt(C/L)
One over what it is for series
Examples of high pass filters
Series RC circuit where vout is across resistor
Series LC circuit where vout is across inductor
What does graph of phase against frequency look like for a high or low pass filter?
Phase in degrees or radians on linear y axis. Angular frequency on logarithmic x axis. The frequencies where there is unity gain, there is horizontal line at phase=0. The frequencies where there is roll-off gain, there is a horizontal line at phase=90°. The middle ends of these lines are equidistant from the corner frequency which is at 45° phase. There is a curvy line from each end through the corner frequency in a shape like a sideways tan graph from -90 to 90.
For a moving system with forces involved, how would you construct an analogous circuit?
Equate the forces using netF=ma. Write all accelerations, velocities and displacements in terms of velocity using integrals and derivatives. Say velocity is current. Match expressions in the equation to components for an AC circuit (di/dt is inductor, i is resistor, Sidt is capacitor). Say the expressions are potential differences across these components and draw the circuit so that the signs in the motion equation match those from voltage law equation.
What can analogous circuits be used to find?
Natural frequency of oscillating system. Damping factor of oscillating system.
