To get an accurate picture of power consumption for an
AC system, we need to make frequent measurements,
preferably many times that of the supply frequency. In
this application, we use a sampling rate of 400 Hz,
which provides 8 samples per full cycle of the line
frequency (for AC supply frequency of 50 Hz). For a
sampling rate Fs, we get N samples in N/Fs seconds. By
multiplying this expression for time by average power,
we obtain an expression for energy consumed in terms
of wattseconds (the second expression in Equation 1).
From here, we can use simple math to calculate
Of course, it is difficult for a microcontroller to make
direct measurements when the supply voltage is
coming straight off the mains: say, 230V at up to 50A.
This makes it necessary to indirectly measure line
voltage and current at a level consistent with a micro-
controller, then rescale these measurements to arrive
at the original value. The best way to do this is to
reduce the voltage to a level and dynamic range that is
compatible with digital circuitry. (Measuring current
here is essentially the same as measuring voltage, in
that we will use a transducer that generates a voltage
proportional to the load current.) The actual voltage
and current readings can then be derived.
ENERGY FROM INDIRECT
Σ⎛ N ⎞
Fs • Kd2
We could accumulate a running total indefinitely and
directly interpret it for energy consumed over time. How-
ever, it’s more practical to accumulate up to some fixed
amount, then increment a counter to indicate energy
consumption. For our application, we will accumulate
10 Wh (0.01 kWh) before incrementing the counter. This
value represents the resolution limit of the meter. It is
equivalent to 36,000 wattseconds (10 Wh x 60 x 60); this
means that we increment the counter every time that the
right side of Equation 2 reaches 36,000.
We can also rearrange Equation 2 to define the power
consumed entirely in terms of Vd and Id. Since we have
already defined Fs, Kv, Ki and Kd in constant terms, we
can give the whole quotient on the right side of the
equation a constant value, D (Equation 3).
For this application, the derived voltage reading, Vd, is
related to the actual instantaneous line voltage Vi by
EQUATION 3: REDEFINING POWER IN
the expression, Vd = Vi Kd/Kv or Vi = Vd Kv/Kd, where Kd
TERMS OF Vd AND Id ONLY
is the digitization constant
tion and Kv is the voltage
for the ADC in
When 0.01 kWh is consumed:
the circuit design. For this particular application, Kd is
204.6, the digital value from the ADC that represents
1V. Kv is the factor by which the input line voltage is
reduced by a voltage divider; in our design, it is 300.
k = 1Vdk • Idk
3600 • Fs • Kd2
Kv • Ki
Similarly, the derived current reading, Id, is related to Ii
by the expression, Id = Ii Kd/Ki or Ii = Id Ki/Kd, where Ki
is the current proportionality constant specific to this
design; it is calculated by dividing the CT turn ratio by
the product of the current amplifier gain and the input
burden resistance. For this application, based on a
5000-turn CT, the value of Ki works out to be
approximately 8.7. Kd is the same as before.
The calculation of Ki when using a shunt is
somewhat different. The actual circuit
design for current measurement, and the
design considerations for using a shunt,
are discussed in more detail in “Hardware
Design”, starting on page 10.
In simple terms, any time that the accumulated sum of
the voltage and current products equals or exceeds D,
we increment the kWh counter. We also save any
remainder in excess of D to be used in the next round
Note that anything which might influence the value of
the constants may also affect the value of D and
requires changes to the amplifier design. This includes
the use of a shunt instead of a CT, or even changing the
CT turn ratio, both of which may change Ki.
By substituting the attenuated values of Vd and Id for
the Vi and Ii in the original power measurement
equation, we get an expression that relates the con-
sumed power directly to the indirect voltage and current
measurements, as shown in Equation 2.
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