No circuit
in this world is complete without a load, meaning a circuit is most of the
times designed to provide power to a load. While our quest is for minimizing
power losses in case of power transmission and distribution, still there are
areas where maximum power and not efficiency is the major concern. Power
transmission and distribution is always done keeping in mind the efficiency of
the system and whether the system proves to be economical or not.
But when it
comes to electronic and communication networks, very often the goal is to
either receive or transmit maximum power (though at reduce efficiency). This is
because, here the power involved is mostly a few mill watts or micro watts.
When applied
to DC networks, the theorem states that:
STATEMENT: A resistive load will abstract maximum power from a network when the load resistance is equal to the Thevenin resistance as seen from the load. That means when RL=Rth, power delivered to the load will be maximum.
Consider the
circuit given below:
Suppose we
obtain Vth and Rth of any arbitrary circuit and then
connect a load resistance RL across them. Then the power delivered
to the load is
\[P={{i}^{2}}{{R}_{L}}={{\left[
\frac{{{V}_{th}}}{{{R}_{th}}+{{R}_{L}}} \right]}^{2}}{{R}_{L}}\]
Now, for any
given circuit, thevenin resistance and voltage are fixed. Thus, the power delivered
to a load by varying the load resistance can be graphed as shown below:
PROOF OF THEOREM:
Power
consumed by the load =${{\left[ \frac{{{V}_{th}}}{{{R}_{th}}+{{R}_{L}}}
\right]}^{2}}{{R}_{L}}$
For power to
be maximum, $\frac{dP}{d{{R}_{L}}}=0$
\[\Rightarrow
\frac{d}{d{{R}_{L}}}\left[ {{\left( \frac{{{V}_{th}}}{{{R}_{th}}+{{R}_{L}}}
\right)}^{2}}{{R}_{L}} \right]=0\]
\[\Rightarrow
{{V}_{th}}^{2}\left[ \frac{{{\left( {{R}_{th}}+{{R}_{L}}
\right)}^{2}}-2{{R}_{L}}\left( {{R}_{th}}+{{R}_{L}} \right)}{{{\left(
{{R}_{th}}+{{R}_{L}} \right)}^{4}}} \right]=0\]
\[\Rightarrow
{{\left( {{R}_{th}}+{{R}_{L}} \right)}^{2}}=2{{R}_{L}}\left(
{{R}_{th}}+{{R}_{L}} \right)\]
\[\Rightarrow
{{R}_{th}}+{{R}_{L}}-2{{R}_{L}}=0\]
Thus, ${{R}_{th}}={{R}_{L}}$
This shows
that the maximum power transfer takes place when the load resistance RL
is equal to the thevenin resistance or the internal resistance of the circuit.
As, \[P={{\left[
\frac{{{V}_{th}}}{{{R}_{th}}+{{R}_{L}}} \right]}^{2}}{{R}_{L}}\]
Putting Rth=RL,
we get ${{P}_{\max
}}=\frac{{{V}_{th}}^{2}}{4{{R}_{th}}}=\frac{{{V}_{th}}^{2}}{4{{R}_{L}}}$
This is the
maximum power transferred to the load in case of DC circuits. But when an AC
source of internal impedance (R1+jX1) is supplying power
to load impedance (RL+jXL). Then, maximum power transfer
will take place when
\[\left|
{{Z}_{L}} \right|=\left| {{Z}_{i}} \right|\]
i.e. Modulus
of load impedance = modulus of internal impedance
Particularly,
the maximum power transfer in case of AC is observed when load impedance is the
complex conjugate of the source impedance i.e. If internal impedance = R1 +
jX1, then for maximum power, load impedance = R1 – jX1.
WHAT IS THE SIGNIFICANCE OF STUDYING MAXIMUM POWER TRANSFER THEOREM?
Studying any
topic theoretically without understanding and thinking about its applications
is like doing a crime to yourself. As far as, engineering is concerned, it’s
all about application.
So where is this theorem used?
Maximum
power transfer theorem is mostly utilized in electronic, as mentioned in the
introduction of this post. Most of the times, in electronics it is the maximum
power that is of more concern than efficiency.
For example,
have you seen ‘rabbit ear’ antennas? The one that used to be on TV sets. They
receive power from radio waves originating at a transmitter miles away. The
antenna does not collect much power. So, the TV receiver is designed to make
maximum use of the power provided by the antenna.
Maximum
power transfer theorem is used for optimization also. Following the above
example, the TV receiver is optimized when its input impedance is matched to
the output impedance of the antenna because this gives maximum power to the
receiver.
