RC Series Circuit
Natural response
solve for
V0 is the capacitor voltage at time t = 0
Time required for the voltage to fall to
Figure 1 Series RC Circuit
is called the RC time constant:
The complex impedance, ZC (ohms) of capacitor with capacitance C (farads):
The complex frequency:
, where  is exp. decay,  is sinusoidal angular freq.
Voltage & Current:
In Sinusoidal Steady State:  = 0,
Time Domain (step input of amplitude V, LaPlace Transform;
s = j
):
Figure 2 Capacitor voltage step-response
RC Parallel Circuit
Figure 3 Resistor voltage step-response
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RL Series Circuit
Natural response
solve for
Figure 4 Series RL circuit
where
V0 is inductor voltage at time t = 0
Time required for the voltage to fall to
is called the RL time constant: τ = L / R
The complex impedance, ZL (ohms) of inductor with inductance L (henries):
The complex frequency:
, where  is exp. decay,  is sinusoidal angular freq.
Voltage & Current
In Sinusoidal Steady State:  = 0,
Time Domain (step input of amplitude V, LaPlace Transform;
s = j
):
RL Parallel circuit
Figure 5 Inductor voltage step-response
Figure 6 Resistor voltage step-response
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RLC Series Circuit
Figure 7 Damping Factor
Transient response: the differential equation for the circuit solves in 3 different ways
depending on the value of lambda, underdamped, overdamped and critically damped.
roots:
sol:
Overdamped:
Arbitary E(t):
Underdamped:
Arbitary E(t):
Critically Damped:
Arbitary E(t):
LaPlace:
Admittance: Poles:
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Figure 8 Plot showing underdamped and overdamped
responses of a series RLC circuit. The critical damping plot is
the bold red curve. The plots are normalised for L=1, C=1 and
Wo = 1
Figure 10 Sinusoidal steady-state analysis normalised to
R = 1 ohm, C = 1 farad, L = 1 henry, and V = 1.0 volt
Figure 9 Sinusoidal steady-state analysis normalised to R = 1
ohm, C = 1 farad, L = 1 henry, and V = 1.0 volt
RLC Parallel Circuit
Attenuation:
Damping Factor:
Q and Fractional Bandwidth:
and
Complex Admittance:
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Filters
In the filtering application, the resistor R
becomes the load that the filter is working
into. The value of the damping factor is
chosen based on the desired bandwidth of
the filter. For a wider bandwidth, a larger
value of the damping factor is required
RLC circuit as a low-pass RLC circuit as a high(and vice versa). The three components
give the designer three degrees of
filter
pass filter
freedom. Two of these are required to set
the bandwidth and resonant frequency.
The designer is still left with one which
can be used to scale R, L and C to
convenient practical values. Alternatively,
R may be predetermined by the external
circuitry which will use the last degree of
freedom.
Low-Pass Filter
An RLC circuit can be used as a low-pass
filter. The circuit configuration is shown in
figure 9. The corner frequency, that is, the
frequency of the 3dB point,
is given by
This is also the bandwidth of the filter.
The damping factor is given by
RLC circuit as a series
band-pass filter in series
with the line
RLC circuit as a parallel
band-pass filter in shunt
across the line
RLC circuit as a series
band-stop filter in shunt
across the line
RLC circuit as a parallel
band-stop filter in series
with the line
High-Pass Filter
The corner frequency is the same as the low-pass filter:
This is the stop-band width of the filter.
Band-Pass Filter
Centre Freq:
Bandwidth:
The shunt version of the circuit is intended to be driven by a high impedance source, that is, a
constant current source. Under those conditions the bandwidth is:
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