AC Circuits
• Pure AC voltage varies in time.
v ( t ) = V 0 cos ωt = V 0 e
jωt
• The AC current also varies in time with the same frequency but may have a different phase.
i ( t ) = I 0 cos ( ωt + φ ) = I 0 e
π/2 π
I0
j ( ωt + φ )
φ
2π
V0
Τ
• The RMS voltage and current, Vrms , Irms, are related to the amplitude.
V rms =
I rms =
⟨ v 2⟩ =
⟨ i 2⟩ =
V0
⟨ V 20 cos 2ωt⟩ = ------2
I0
⟨ I 20 cos 2( ωt + φ )⟩ = ------2
• The phase doesn’t matter for rms measurement.
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Phasors
• A complex number can be represented as a magnitude and angle in the complex plane.
Im.
φ
Re.
• The imaginary unit in electronics is j to avoid confusion with current. j =
–1
• Euler’s formula for an exponential in jφ links to sinusoidal behavior.
e
jφ
= cos φ + j sin φ
• Trigonometric formulas can be replaced by exponential ones.
jφ
cos φ = Re ( e )
• AC signals are represented by a vector in the complex plane and are called phasors.
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Capacitors
• The voltage-current relationship for a capacitor in an AC circuit comes from the stored charge.
i
v = V0e jωt
C
Q
v = ---C
dv
---dt
=
---iC
• For a pure AC signal, there is a simple voltage-current relationship.
1
v = ---------- i
jωC
• This looks like a complex version of Ohm’s law.
• The capacitive reactance is a complex impedance.
1
Z C = ---------j ωC
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Inductors
• An inductor creates a voltage in response to a changing current.
i
v = V0e jωt
L
v = L
di
dt
• This gives rise to another complex current-voltage relationship.
v = jωLi
• The inductive reactance is a complex impedance.
ZL
=
jωL
• Capacitive reactances are negative on the imaginary axis and inductive reactances are positive on
the imaginary axis.
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Gain
• The unit of gain A is the decibel (dB).
• Decibels are a logarithmic measure,
Voltage and current, AdB = 20 log10 A
Power = voltage*current, AdB = 10 log10 A
• There are some useful approximations for amplitude gain.
A factor of 10 is a 20 dB measure
A factor of 2 is about a 6 dB measure
Neagtive dB is a reduction in magnitude
• A Bode plot displays gain versus the log of the frequency.
gain (dB)
log f (or ω)
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Complex Gain
• The complex gain includes the magnitude of gain and the phase shift.
• The absolute magnitude (a real) is the magnitude of the gain.
A = B + jC =
( B + jC ) ( B – j C ) =
2
B +C
2
• The angle in the complex plane is the phase shift.
tan φ = C ⁄ B
• Graphically:
2
B +C
2
φ
jC
φ = atan C
---B
B
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Low-pass Filter
• The low-pass filter is a common element in AC circuits.
• It is a voltage divider with complex impedance.
vin
R
1 ⁄ jωC
v out = --------------------------- v in
R + 1 ⁄ jωC
C
1
= ----------------------- v in
1 + jωRC
v out
1
A = --------- = ---------------------------------2 2 2
v in
1+ω R C
A = (1+ω2R2C2)-1/2; for large ω, A goes as 1/ω or 6 dB/octave.
φ = tan-1(−ωRC); for large ω, φ approaches −90 ° .
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Fourier Series
• A periodic signal with a period T can be expressed as f(t) = f(t + nT)
• A fourier series represents this as a sum of sine and cosine functions.
• The periodic function can be expressed with ω = 2π/T.
a0
f(t) = ----- +
2
∞
[ a n cos ( nωt ) + b n sin ( nωt ) ]
∑
n=1
• Instead of sines and cosines, the function can be represented in complex notation.
∞
f(t) =
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[ cn e
∑
n = –∞
jnωt
]
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Fourier Coefficients
• The Fourier coefficients are found by integrating the signal over any one period.
• The first term a0/2 is the DC component of f(t).
∫
T
a0T
f(t) dt = --------2
0
• The coeffiecients an, bn represent the strength of that frequency component.
• The cosine terms are symmetric and the sine terms are antisymmetric.
2 T⁄2
a n = --f(t) cos ( nωt ) dt
T –T ⁄ 2
∫
2 T⁄2
-bn = f(t) sin ( nωt ) dt
T –T ⁄ 2
∫
• The coefficients cn for the complex form are similar.
– jn ωt
1- T ⁄ 2
-cn =
dt
f(t)e
T –T ⁄ 2
∫
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Square Wave
• A symmetric square wave can be simply decomposed into a Fourier series.
V0
-V0
-T/2
T/2
• The function has been set up as a symmetric function around t = 0.
• The period is split into two symmetric halves.
v(t) = V 0
v(t) = – V 0
T
–T
--- < t < --4
4
T
t > --4
• Integration can be done on each half separately.
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Square Wave Coefficients
• Since the form is symmetric the sine terms will all be 0.
2- T ⁄ 2
-bn =
v(t) sin ( nωt ) dt = 0
T –T ⁄ 2
∫
• The symmetric terms will be 2 times the value of the positive side.
T⁄4
T⁄2
4
4
a n = --V 0 cos ( nωt ) dt + --– V cos ( nωt ) dt
T 0
T T⁄4 0
∫
∫
4V 0
4V 0
a n = ----------- sin ( nωT ⁄ 4 ) + ----------- sin ( nωT ⁄ 4 )
Tnω
Tnω
8V 0
4V 0
a n = ---------- sin ( 2πn ⁄ 4 ) = --------- sin ( nπ ⁄ 2 )
2πn
πn
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Low Order Expansion
• Consider the first few terms of the square wave series
4V 0
nπ
a n = --------- sin ⎛ ------⎞
⎝
πn
2⎠
a1 = 4V0/π
a2 = 0
4V0/π
a3 = 4V0/3π
a4 = 0
a5 = 4V0/5π
ω
2ω
3ω
4ω
5ω
6ω
7ω
8ω
ω
2ω
3ω
4ω
5ω
6ω
7ω
8ω
a6 = 0
0 dB
• The signal can be displayed in
terms of the frequencies and -10 dB
strengths.
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Bridge Circuits
• The Wheatstone bridge is a DC circuit.
I1
R1
R3
Vin
Iout
A
R2
R4
I2
• If the current meter shows Iout = 0, there are two balance equations.
I 1 R1 = I 2 R2
I1 R3 = I2 R4
• These can be combined and R4 can be found in terms of the other resistors.
R4 = R2 R3 ⁄ R1
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Bridge Circuits
• The Wheatstone bridge is typically used to measure an unknown resistance RX.
R2
R1
Vin
RX
R3
A
Iout
• If the user balances the current meter until Iout = 0.
• The unknown resistance comes from the balance equation.
RX = R2 R3 ⁄ R1
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Op-Amp Bridge
• An op-amps can be used to generate a balance condition.
• Negative feedback sets the voltages on the input of the
op-amp.
V inv = V non = V in – I 1 R i1 = I 2 R i2
• The resistr network sets the voltage on the output of the
op-amp.
V A = V in – I 1 R i1 – I 1 R f1 = ( I 2 + I out )R f2 + I 2 R i2
I1
Vin
Ri1
VA
Rf1
Rf2
−
+
Iout
I2
Ri2
R f2
R i2
------• The variable resistor can be adjusted until the resistors meet the following ratio:
= -------R f1
R i1
• The current only depends on Ri2 and Vin.
V in
I out = – -------R i2
• Measuring the current gives the value of the resistor Ri2.
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AC Bridges
• A DC bridge can be modified to measure complex impedances
i1
C3
R1
vin
R3
iout
A
R2
Z4
i2
• Match the real and imaginary parts at Z4 = a + jb.
R 1 ( a + jb ) = R 2 ( R 3 – j ⁄ ωC 3 )
• The imaginary part measures the capacitance or inductance.
b = – R 2 ⁄ ωR 1 C
3
• The real part measures any intrinsic resistance; and can be used to calculate the Q.
a = R2 R3 ⁄ R1
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