.. _section-A5.7:

Appendix 5.7: Detailed Derivation of EQ EM Pump
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The equivalent circuits for the detailed and the simple EM pump models
can be reduced to the circuit shown in :numref:`figure-ReducedCircuit`, where :math:`Z_{i}` is the
impedance of each component in the path of the total phase current,
:math:`I`, being driven by the phase voltage, :math:`V_{s}`.

.. _figure-ReducedCircuit:
..  figure:: media/ReducedCircuit.png
	:align: center
	:figclass: align-center

	Reduced Equivalent Circuit for the EM pump models.

Because the equivalent circuits contain inductors, the circuit's total
impedance is a complex number.

.. math::
	:label: TotalImpedance

	Z_{T} &= Z_{1} + Z_{2} + Z_{3} \\
	&= Re\left( Z_{T} \right) + Im(Z_{T})j \\
	&= Re\left( Z_{1} \right) + Re\left( Z_{2} \right) + Re\left( Z_{3} \right) + Im\left( Z_{1} \right)j + Im\left( Z_{2} \right)j + Im\left( Z_{3} \right)j


Using the total impedance, the total phase current can be quantified
using Ohm's law.

.. math::
	:label: TotalCurrent

	I &= \frac{V}{Z_{T}} \\
	|I| &= \frac{|V|}{{|Z}_{T}|} \\
	|I| &= \frac{|V|}{\sqrt{\text{Re}\left( Z_{T} \right)^{2} + Im\left( Z_{T} \right)^{2}}}

The presence of the inductors introduces a phase difference between the
phase voltage and current. The phase difference is defined as

.. math::
	:label: PhaseDifference

	\tan\left( \phi \right) = \frac{\text{Im}\left( Z_{T} \right)}{Re(Z_{T})}

Simple Circuit Analysis
''''''''''''''''''''''''''

For the simple EM pump circuit, the impedance of each component is

.. math::
	:label: SimpleZ1

	Z_{1} = R_{1}

.. math::
	:label: SimpleZ2

	Z_{2} = X_{1}j

.. math::
	:label: SimpleZ3

	Z_{3} = \left( \frac{1}{X_{2}j} + \frac{s}{R_{2}} \right)^{- 1} = \frac{R_{2}X_{2}j}{R_{2} + \text{sX}_{2}j} = \frac{sR_{2}X_{2}^{2}}{R_{2}^{2} + s^{2}X_{2}^{2}} + \frac{R_{2}^{2}X_{2}}{R_{2}^{2} + s^{2}X_{2}^{2}}j

Using equation :eq:`TotalImpedance` and :eq:`TotalCurrent`, the current as a function of slip, frequency
and voltage can be determined.

.. math::
	:label: SimpleCurrent

	|I| = \frac{|V|}{\sqrt{\left( R_{1} + \frac{sR_{2}X_{2}^{2}}{R_{2}^{2} + s^{2}X_{2}^{2}} \right)^{2} + \left( X_{1} + \frac{R_{2}^{2}X_{2}}{R_{2}^{2} + s^{2}X_{2}^{2}} \right)^{2}}}

The current's dependence on slip is inherent, as :math:`X_{i} = 2\pi L_{i} f`.

Ohm's law can be used to determine the division of the total current
between the :math:`X_{2}` and :math:`R_{2}/s` branches

.. math::
	:label: SimpleDivision

	\left| I \right|\left| Z_{3} \right| = \left| I_{1} \right|\left| X_{2} \right| = \left| I_{2} \right|\frac{R_{2}}{s}

Therefore, the power dissipated in :math:`R_{2}/s` can be defined as

.. math::
	:label: SimplePower

	P_{R2} &= 3\left| I_{2} \right|\left| V_{R2} \right| \\
	&= 3\frac{\left| I_{2} \right|^{2}R_{2}}{s}

If we assume that a fraction of the power is lost to the fluid, the
mechanical power dissipated in :math:`R_{2}/s` is

.. math::
	:label: SimpleMechanincal

	P_{R2} &= 3\frac{\left| I_{2} \right|^{2}R_{2}\left( 1 - s \right)}{s} \\
	&= 3\left| I \right|^{2}\left| Z_{3} \right|^{2}\left( 1 - s \right)\frac{R_{2}}{s} \\
	&= 3\left| I \right|^{2}\left( \frac{\left| Z_{3} \right|}{\frac{R_{2}}{s}} \right)^{2}\left( 1 - s \right)\frac{R_{2}}{s} \\
	&= 3\frac{s^{4}X_{2}^{4} + {s^{2}R}_{2}^{2}X_{2}^{2}}{\left( R_{2}^{2} + s^{2}X_{2}^{2} \right)^{2}}\left( 1 - s \right)\frac{R_{2}}{s}\left| I \right|^{2} \\
	&= 3\frac{s^{2}X_{2}^{2}}{\left( R_{2}^{2} + s^{2}X_{2}^{2} \right)}\left( 1 - s \right)\frac{R_{2}}{s}\left| I \right|^{2} \\
	&= 3\left| I \right|^{2}\frac{R_{2}(1 - s)}{\text{s}\left( \frac{R_{2}^{2}}{s^{2}X_{2}^{2}} + 1 \right)}

Using the relationship between pumping power and pump head, the
developed pump head for the simple EM pump model can be described as

.. math::
	:label: SimpleDeltaP

	\Delta P_{\text{pump}} &= \frac{P_{R2}}{Q} \\
	&= 3\left| I \right|^{2}\frac{R_{2}(1 - s)}{\text{Qs}\left( \frac{R_{2}^{2}}{s^{2}X_{2}^{2}} + 1 \right)}

Detailed Circuit Analysis
''''''''''''''''''''''''''

For the detailed EM pump circuit, the impedance of each component is

.. math::
	:label: DetailedZ1

	Z_{1} = R_{1}

.. math::
	:label: DetailedZ2

	Z_{2} = X_{1}j

.. math::
	:label: DetailedZ3

	Z_{3} &= \left( \frac{1}{X_{2}j} + \frac{1}{R_{D}} + \frac{1}{R_{J} + R_{m}} \right)^{- 1} \\
	&= \frac{\left( R_{J} + R_{M} \right)R_{D}X_{2}j}{\left( R_{J} + R_{m} \right)R_{D} + \left( R_{J} + R_{M} + R_{D} \right)X_{2}j} \\
	&= \frac{\left( R_{J} + R_{m} \right)\left( R_{J} + R_{m} + R_{D} \right)R_{D}X_{2}^{2} + \left( R_{J} + R_{m} \right)^{2}R_{D}^{2}X_{2}j}{\left( R_{J} + R_{m} \right)^{2}R_{D}^{2} + \left( R_{J} + R_{m} + R_{D} \right)^{2}X_{2}^{2}}

The magnitude of :math:`Z_{3}` is

.. math::
	:label: DetailedZ3Mag

	\left| Z_{3} \right|^{2} &= \frac{\left( R_{J} + R_{m} \right)^{2}\left( R_{J} + R_{m} + R_{D} \right)^{2}R_{D}^{2}X_{2}^{4} + \left( R_{J} + R_{m} \right)^{4}R_{D}^{4}X_{2}^{2}}{\left( \left( R_{J} + R_{m} \right)^{2}R_{D}^{2} + \left( R_{J} + R_{m} + R_{D} \right)^{2}X_{2}^{2} \right)^{2}} \\
	&= \frac{\left( R_{J} + R_{m} \right)^{2}R_{D}^{2}X_{2}^{2}}{\left( R_{J} + R_{m} \right)^{2}R_{D}^{2} + \left( R_{J} + R_{m} + R_{D} \right)^{2}X_{2}^{2}}

Using a similar analysis as was presented for the simple model, the
current passing through the fluid is determined to be

.. math::
	:label: DetailedCurrent

	\left| I \right|\left| Z_{3} \right| = \left| I_{3} \right|(R_{J} + R_{m})

and the mechanical power delivered to the fluid is

.. math::
	:label: DetailedMechanincal

	P_{m} &= 3\left| I_{3} \right|^{2}R_{m} \\
	&= 3\frac{\left| I \right|^{2}\left| Z_{3} \right|^{2}}{\left( R_{J} + R_{m} \right)^{2}}R_{m} \\
	&= 3I^{2}R_{m}\frac{R_{D}^{2}X_{2}^{2}}{\left( R_{J} + R_{m} \right)^{2}R_{D}^{2} + \left( R_{J} + R_{m} + R_{D} \right)^{2}X_{2}^{2}}

Therefore, the developed pump head in the detailed EM pump model is

.. math::
	:label: DetailedDeltaP

	\Delta P &= \frac{P_{m}}{Q} \\
	&= 3 \frac{I^{2}R_{m}}{Q}\frac{R_{D}^{2}X_{2}^{2}}{\left( R_{J} + R_{m} \right)^{2}R_{D}^{2} + \left( R_{J} + R_{m} + R_{D} \right)^{2}X_{2}^{2}}

Simple Circuit Parameters
''''''''''''''''''''''''''

In order to determine the resistance and inductance of the different
components in the simple EM pump model, measurable quantities are
provided at two distinct pump states. The first pump state, referred to
as the rated state, provides a pump head, :math:`\Delta P_{R}`, measured
flow rate, :math:`Q_{R}`, measured phase voltage, :math:`V_{R}`, measured phase
current, :math:`I_{R}`, measured slip, :math:`s_{R}`, measured efficiency,
:math:`\eta_{R}`, and measured frequency, :math:`f_{R}`. The second pump
state, referred to as the stalled state, provides the pump head at
stalled conditions, :math:`\Delta P_{S}`, for the measured frequency and
measured phase voltage. :math:`\Delta P_{P,R}` and :math:`\Delta P_{S,R}` are functions of the measured
pump head, :math:`\Delta P_{R}`, and measured pump head at stalled conditions,
:math:`\Delta P_S`.

When IDPOPT = 0, then the user-provided :math:`\Delta P_{R}` and :math:`\Delta P_{S}`
*include* pressure changes due to friction losses across the pump.
When IDPOPT = 1, then the user-provided :math:`\Delta P_{R}` and :math:`\Delta P_{S}` *do not
include* pressure changes due to friction losses across the pump.

To solve the initial conditions for the simplified pump model, the
following terms are defined.

.. math::
	:label: StallAdj

	\Delta P_{P,S} = \Delta P_{S}

.. math::
	:nowrap:

	\[
	\Delta P_{P,R} =
	\left\{
	\begin{array}{
		@{}% no padding
		l@{\quad}% some padding
		r@{}% no padding
		>{{}}r@{}% no padding
		>{{}}l@{}% no padding
	}
		\Delta P_R,&IDPOPT&\neq0 \\
		\Delta P_R + \Delta P_{f,R},&IDPOPT&=0
	\end{array}
	\right.
	\]

where :math:`\Delta P_{f,R}` is the frictional pressure drop that occurs within the pump at the rated flow rate and IDPOPT is a user-defined flag.
:math:`\Delta P_{f,R}` is internally calculated by SAS4A/SASSYS-1 using the geometry input.

In order to determine the 4 circuit parameters and the current at
stalled conditions, five equations are required. Ohm's law provides us
with two of the five equations:

.. math::
	:label: ohmsRated

	\left( \frac{V_{R}}{I_{R}} \right)^{2} = \left| Z_{T}(s = s_{R}) \right|^{2}

.. math::
	:label: ohmsStalled

	\left( \frac{V_{R}}{I_{S}} \right)^{2} = \left| Z_{T}(s = 1) \right|^{2}

The rated and stall pressures provide an additional two equations:

.. math::
	:label: PressureRated

	\Delta P_{P,R} = \frac{3I_{R}^{2}R_{2}}{Av_{s}\left( \frac{R_{2}^{2}}{X_{2}^{2}s_{R}} + s_{R} \right)}

.. math::
	:label: PressureStalled

	\Delta P_{P,S} = \frac{3I_{s}^{2}R_{2}}{Av_{s}\left( \frac{R_{2}^{2}}{X_{2}^{2}} + 1 \right)}

The final equation comes from the total power dissipated by the pump

.. math::
	:label: TotalPower1

	P_{R} = 3V_{R}I_{R}\cos\left( \phi \right)

.. math::
	:label: TotalPower2

	\cos\left( \phi \right) = \frac{\text{Re}\left( Z_{T}\left( s = s_{R} \right) \right)}{|Z_{T}\left( s = s_{R} \right)|}

.. math::
	:label: TotalPower3

	P_{R} = \frac{\left( \Delta P_{R} \right)Q_{R}}{\eta_{R}}

.. math::
	:label: TotalPower4

	Re\left(Z_T(s=s_R)\right)^2 = \left(\frac{\left( \Delta P_{R} \right)Q_{R}}{3I_R^2\eta_{R}}\right)^2

Note, the synchronous speed of the EM pump can be determined using the
definition of slip and the rated conditions

.. math::
	:label: synchSpeed

	v_{s}A = \frac{Q_{R}}{1 - s_{R}}

Instead of solving the five equations at once, an iterative approach is
taken. By assuming a stall current, :eq:`PressureStalled` can be solved

.. math::
	:label: SolveX2

	\frac{R_{2}^{2}}{X_{2}^{2}} = \frac{3I_{s}^{2}R_{2}}{Av_{s}\Delta P_{s}} - 1

and plugged into :eq:`PressureRated`, providing an equation for :math:`R_{2}`

.. math::
	:label: SolveR2

	R_{2} = \frac{\left( \Delta P_{P,R} \right)\left( Av_{s}\frac{\left( s_{R}^{2} - 1 \right)}{s_{R}} \right)}{3I_{R}^{2} - \frac{3I_{S}^{2}}{\Delta P_{P,S}s_{R}}\Delta P_{P,R}}

with the value of :math:`R_{2}` known, :math:`X_{2}` can be determined using
:eq:`SolveX2` and :eq:`TotalPower4` can be solved to determine the value of
:math:`R_{1}`

.. math::
	:label: SolveR1

	R_{1} = \frac{\left( \frac{{\Delta P}_{R}Q_{R}}{3I_{R}^{2}\eta_{R}}\left( R_{2}^{2} + X_{2}^{2}{s_{R}}^{2} \right) - s_{R}R_{2}X_{2}^{2} \right)}{\left( R_{2}^{2} + X_{2}^{2}s_{R}^{2} \right)}

With :math:`R_{1}`, :math:`R_{2}`, and :math:`X_{2}` known :eq:`ohmsRated` can be used
to determine :math:`X_{1}`

.. math::
	:label: SolveX1

	X_{1} = \left\lbrack \left( \frac{V_{R}}{I_{R}} \right)^{2} - \text{Re}\left( Z_{T}(s = s_{R}) \right)^{2} \right\rbrack^{1/2} - \frac{R_{2}^{2}X_{2}}{\left( R_{2}^{2} + X_{2}^{2}s_{R}^{2} \right)}

Finally, :eq:`ohmsStalled` will be solved to determine the error in the
stall current. Using error in the stall current, a new stall current
will be estimated and :eq:`SolveX2` through :eq:`SolveX1` will be
repeated. This process continues until the error in the stall current is
within a user defined limit.