Complex circuits
What do we do if we come across a circuit
more complex than the simple series configurations we've
seen so far? Take this circuit as an example:
The simple time constant formula (τ=RC) is
based on a simple series resistance connected to the
capacitor. For that matter, the time constant formula for an
inductive circuit (τ=L/R) is also based on the assumption of
a simple series resistance. So, what can we do in a
situation like this, where resistors are connected in a
series-parallel fashion with the capacitor (or inductor)?
The answer comes from our studies in network
analysis. Thevenin's Theorem tells us that we can reduce
any linear circuit to an equivalent of one voltage
source, one series resistance, and a load component through
a couple of simple steps. To apply Thevenin's Theorem to our
scenario here, we'll regard the reactive component (in the
above example circuit, the capacitor) as the load and remove
it temporarily from the circuit to find the Thevenin voltage
and Thevenin resistance. Then, once we've determined the
Thevenin equivalent circuit values, we'll re-connect the
capacitor and solve for values of voltage or current over
time as we've been doing so far.
After identifying the capacitor as the
"load," we remove it from the circuit and solve for voltage
across the load terminals (assuming, of course, that the
switch is closed):
This step of the analysis tells us that the
voltage across the load terminals (same as that across
resistor R2) will be 1.8182 volts with no load
connected. With a little reflection, it should be clear that
this will be our final voltage across the capacitor, seeing
as how a fully-charged capacitor acts like an open circuit,
drawing zero current. We will use this voltage value for our
Thevenin equivalent circuit source voltage.
Now, to solve for our Thevenin resistance,
we need to eliminate all power sources in the original
circuit and calculate resistance as seen from the load
terminals:
Re-drawing our circuit as a Thevenin
equivalent, we get this:
Our time constant for this circuit will be
equal to the Thevenin resistance times the capacitance
(τ=RC). With the above values, we calculate:
Now, we can solve for voltage across the
capacitor directly with our universal time constant formula.
Let's calculate for a value of 60 milliseconds. Because this
is a capacitive formula, we'll set our calculations up for
voltage:
Again, because our starting value for
capacitor voltage was assumed to be zero, the actual voltage
across the capacitor at 60 milliseconds is equal to the
amount of voltage change from zero, or 1.3325 volts.
We could go a step further and demonstrate
the equivalence of the Thevenin RC circuit and the original
circuit through computer analysis. I will use the SPICE
analysis program to demonstrate this:
Comparison RC analysis
* first, the netlist for the original circuit:
v1 1 0 dc 20
r1 1 2 2k
r2 2 3 500
r3 3 0 3k
c1 2 3 100u ic=0
* then, the netlist for the thevenin equivalent:
v2 4 0 dc 1.818182
r4 4 5 454.545
c2 5 0 100u ic=0
* now, we analyze for a transient, sampling every .005 seconds
* over a time period of .37 seconds total, printing a list of
* values for voltage across the capacitor in the original
* circuit (between modes 2 and 3) and across the capacitor in
* the thevenin equivalent circuit (between nodes 5 and 0)
.tran .005 0.37 uic
.print tran v(2,3) v(5,0)
.end
time v(2,3) v(5)
0.000E+00 4.803E-06 4.803E-06
5.000E-03 1.890E-01 1.890E-01
1.000E-02 3.580E-01 3.580E-01
1.500E-02 5.082E-01 5.082E-01
2.000E-02 6.442E-01 6.442E-01
2.500E-02 7.689E-01 7.689E-01
3.000E-02 8.772E-01 8.772E-01
3.500E-02 9.747E-01 9.747E-01
4.000E-02 1.064E+00 1.064E+00
4.500E-02 1.142E+00 1.142E+00
5.000E-02 1.212E+00 1.212E+00
5.500E-02 1.276E+00 1.276E+00
6.000E-02 1.333E+00 1.333E+00
6.500E-02 1.383E+00 1.383E+00
7.000E-02 1.429E+00 1.429E+00
7.500E-02 1.470E+00 1.470E+00
8.000E-02 1.505E+00 1.505E+00
8.500E-02 1.538E+00 1.538E+00
9.000E-02 1.568E+00 1.568E+00
9.500E-02 1.594E+00 1.594E+00
1.000E-01 1.617E+00 1.617E+00
1.050E-01 1.638E+00 1.638E+00
1.100E-01 1.657E+00 1.657E+00
1.150E-01 1.674E+00 1.674E+00
1.200E-01 1.689E+00 1.689E+00
1.250E-01 1.702E+00 1.702E+00
1.300E-01 1.714E+00 1.714E+00
1.350E-01 1.725E+00 1.725E+00
1.400E-01 1.735E+00 1.735E+00
1.450E-01 1.744E+00 1.744E+00
1.500E-01 1.752E+00 1.752E+00
1.550E-01 1.758E+00 1.758E+00
1.600E-01 1.765E+00 1.765E+00
1.650E-01 1.770E+00 1.770E+00
1.700E-01 1.775E+00 1.775E+00
1.750E-01 1.780E+00 1.780E+00
1.800E-01 1.784E+00 1.784E+00
1.850E-01 1.787E+00 1.787E+00
1.900E-01 1.791E+00 1.791E+00
1.950E-01 1.793E+00 1.793E+00
2.000E-01 1.796E+00 1.796E+00
2.050E-01 1.798E+00 1.798E+00
2.100E-01 1.800E+00 1.800E+00
2.150E-01 1.802E+00 1.802E+00
2.200E-01 1.804E+00 1.804E+00
2.250E-01 1.805E+00 1.805E+00
2.300E-01 1.807E+00 1.807E+00
2.350E-01 1.808E+00 1.808E+00
2.400E-01 1.809E+00 1.809E+00
2.450E-01 1.810E+00 1.810E+00
2.500E-01 1.811E+00 1.811E+00
2.550E-01 1.812E+00 1.812E+00
2.600E-01 1.812E+00 1.812E+00
2.650E-01 1.813E+00 1.813E+00
2.700E-01 1.813E+00 1.813E+00
2.750E-01 1.814E+00 1.814E+00
2.800E-01 1.814E+00 1.814E+00
2.850E-01 1.815E+00 1.815E+00
2.900E-01 1.815E+00 1.815E+00
2.950E-01 1.815E+00 1.815E+00
3.000E-01 1.816E+00 1.816E+00
3.050E-01 1.816E+00 1.816E+00
3.100E-01 1.816E+00 1.816E+00
3.150E-01 1.816E+00 1.816E+00
3.200E-01 1.817E+00 1.817E+00
3.250E-01 1.817E+00 1.817E+00
3.300E-01 1.817E+00 1.817E+00
3.350E-01 1.817E+00 1.817E+00
3.400E-01 1.817E+00 1.817E+00
3.450E-01 1.817E+00 1.817E+00
3.500E-01 1.817E+00 1.817E+00
3.550E-01 1.817E+00 1.817E+00
3.600E-01 1.818E+00 1.818E+00
3.650E-01 1.818E+00 1.818E+00
3.700E-01 1.818E+00 1.818E+00
At every step along the way of the analysis,
the capacitors in the two circuits (original circuit versus
Thevenin equivalent circuit) are at equal voltage, thus
demonstrating the equivalence of the two circuits.
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REVIEW:
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To analyze an RC or L/R circuit more
complex than simple series, convert the circuit into a
Thevenin equivalent by treating the reactive component
(capacitor or inductor) as the "load" and reducing
everything else to an equivalent circuit of one voltage
source and one series resistor. Then, analyze what happens
over time with the universal time constant formula.
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