Impedance and phasors Flashcards
What is symbol and unit for impedance?
Symbol: Z
Unit: Ohms, Ω
When does impedance apply?
Only for time dependent circuits.
What is the equation to find Impedance of a component in a time dependent circuit?
V/I = Z
How do you combine time dependent circuitry and non time dependent circuity into a single impedance?
Impedance = Resistance + j×Reactance
What is the real part of impedance?
Resistance
What is the imaginary part of impedance?
Reactance
What is the equation for the reactance of a capacitor?
Xc = -1/ωC
What is the equation for the reactance of an inductor?
Xl = ωL
What is the equation to combine impedances in parallel?
Z = Z1Z2/(Z1 + Z2)
What is the equation to combine impedances in series?
Z = Z1 + Z2
What is the equation for an impedance potential divider?
The same as a normal potential divider.
Vout = Vin×Z2/(Z1+Z2)
What is H(ω)?
The transfer function
What is the transfer function?
This is the ratio of the output voltage to the input voltage.
Given by the equation:
H(ω) = Vout/Vin
What is the maximum power transfer theorem?
Maximum power is transfered when the source impedance is the complex conjugate of the load impedance.
Zs* = Zl
With regards to the maximum power transfer theorem, what can be said about the source and loads resistance and impedance?
Rl = Rs Xl = -Xs
What is active power?
This is the actual power dissipated by the circuit
What is reactive power?
Useless power that moves back and forth between source and load
What is the equation for the active power delivered to the load when the Load impedance is matched with the source impedance?
Pl = (Vrms^2)/4Rs
Pl = Active power from the load Vrms = RMS voltage from the voltage source Rs = Resistance fo the source
What are the axis and directions of them on an Argand diagram?
X-axis: Real part
Y-axis: Imaginary part
After the imaginary and real part of the voltage or current of circuit components in a circuit are drawn onto an Argand diagram, How do you calculate the voltage or current of the power source?
By combining the imaginary and real part vectorally
What are the 2 equations for the impedance of a capacitor?
Z = jXc Z = -j/ωC
What are the 2 equations for the impedance of an inductor?
Z = jXl Z = jωL
Where there is an imaginary and real vector for voltage or current involving resistances and impedances, how can the impedance be calculated vectorally?
Z = √(R^2 + X^2 )
How would you calculate the angle of the magnitude of the voltage/current on an Argand diagram?
θ = tan^-1(Imaginary/Real)
How is an RC filter set up?
You have a voltage input (source) connected to a resistor and capacitor in series
What is the equation for the voltage across the resistor in an RC filter circuit?
Vr = Vin × ωτ / √( 1 + (ωτ)^2 )
What is the equation for the voltage across the capacitor in an RC filter circuit?
Vc = Vin × 1 / √( 1 + (ωτ)^2 )
How is an RC filter used as a lowpass filter?
The output voltage is across the capacitor.
How is an RC filter used as a highpass filter?
The output voltage is across the resistor.
For an RC lowpass filter, when the input frequency is a. High and b. Low what are the potential differences across the resistor and capacitor?
a. High: Potential difference across the resistor is high so capacitor (output) potential difference is low
b. Low: Potential difference across capacitor (output) is high and so the the potential difference across the resistor is low.
For an RC highpass filter, when the input frequency is a. High and b. Low what are the potential differences across the resistor and capacitor?
a. High: Potential difference across the resistor (output) is high so the potential difference across the capacitor is low.
b. Low: Potential difference across the capacitor is high so the potential difference across the resistor (output) is low.
How do you calculate the total input voltage from resistor and capacitor voltage? why?
Sum them together: Vin = Vr + Vc
Because Vc and Vr are complex voltages so we add them like vectors.
How do you calculate the phase difference of current or voltage in an LRC circuit?
By calculating the angle from the real axis on the aground diagram to the magnitude vector of the voltage/current.
θ = tan^-1(imaginary/real)
What is resonance and how does it occur?
This is a spike in amplitude that occurs when the periodic driving force oscillates a system at its natural frequency.
How do capacitors store energy?
In an electric field
How do inductors store energy?
In a magnetic field
What happens in an LC circuit when you charge up a capacitor and then allow current to flow around the circuit?
Closing the circuit causes sinusoidal current between the inductor and capacitor which does not dissipate (ideally)
What happens in an RLC circuit when you charge up the capacitor and allow the current to flow around the circuit?
Closing the circuit causes sinusoidal current between the inductor and capacitor which dissipates rapidly due to the resistor.
When does resonance occur?
When the imaginary part of the impedance is equal to zero.
Do the calculation to find the point of resonance for an RLC circuit
Z = jωL + 1/jωC + R Z = jωL - j/ωC + R Z = R + j(ωL - 1/ωC) ωL - 1/ωC = 0 ωL = 1/ωC ω^2LC = 1 ω^2 = 1/LC ω = 1/√LC
How is a band-pass filter made?
By having an inductor-capacitor-resistor in series.
And measuring voltage across the resistor.
What is the equation for an LCR band-pass filter?
Vout/Vin = R / (R+jωL + 1/jωC)
What is the equation for the natural frequency of a band-pass/stop filter?
ω0 = 1/√LC
What is the equation for the quality parameter?
Q = (1/R) × √(L/C)
What is the equation for an LCR band-pass filter when using the equations for natural frequency and quality?
Vout/Vin = 1/√(1+(ω/ω0 - ω0/ω)^2 Q^2)
What happens when ω0 = ω
Vout/Vin = 1
H(ω) = 1
This means that all the voltage is across the resistor
How is a band-stop filter made?
Same as a band-pass filter: By having an inductor-capacitor-resistor in series.
And measuring voltage across the Capacitor-inductor.
What is the equation for an LCR band-stop filter?
H(ω) = (√((ω/ω0 - ω0/ω)^2 Q^2))/√(1+(ω/ω0 - ω0/ω)^2 Q^2)
What happens when ω0 = ω
Vout/Vin = 0
H(ω) = 0
This means that all the voltage is across the resistor and none is across the capacitor.
What is a simple way to convert an inductor into natural frequency and quality factors?
√(LC) × √(L/C) = √L × √C × √L × √(1/C) = L × √(C/C) = L× √1 = L
√(LC) × √(L/C) = QR/ω0
L = QR/ω0
What is a simple way to convert 1/capacitor into natural frequency and quality factors?
(1/√(LC))×√(L/C) = √(1/LC) × √(L/C) = √(L/LC^2) = √(1/C^2) = 1/C
(1/√(LC))×√(L/C) = RQω0
