Manley Rowe Power or Energy Relations Derivation
Here we are going to derive Manley Rowe power / energy relations.Before derivation of Manley-Rowe relation little introduction as follows.
In 1956, J. M. Manley and H. E. Rowe derived a set of energy relations associated with nonlinear elements. These relations, now known as the M-R energy (or power) relations, have become quite useful in analyzing the operations of the paramp. The derivation of these relations can be done with the help of the below figure.
Figure shows signal sources vs of frequency ws and pump source vp of frequency wp connected across a nonlinear, lossless variable capacitance C. The R's are series resistances associated with the sources. The rectangular blocks represent band-pass filters (BPF's), which are tuned, respectively, to frequencies ws, wp, wp∓ws, 2wp∓2ws,......,mwp∓nws, where m and n are integers.
As shown in the figure, we find that vs and vp are applied to the nonlinear capacitor C through BPFs tuned to their respective frequencies. When excited, a nonlinear device will produce an output expressed in the general form
\[v_{o}=a_{1}v_{i}+a_{2}v_{i}^{2}+a_{3}v_{i}^{3}+.....+a_{n}v_{i}^{n}\]
-------------------(1)
where a1,a2,...an are constants associated with the nonlinear device, and
Assuming that
and
we find
Equation (5) shows that the nonlinear capacitor C produces all possible harmonics of frequencies ws and wp. Each one of these frequencies is isolated from the rest by the respective BPF tuned to that particular frequency, and allowed to dissipate its power in the respective load resistor connected in series with that BPF.
In the M-R relations, (3) and (4) are respectively written as
\[v_{s}=\frac{V_{s}}{2}(\epsilon^{jw_{s}t}+\epsilon^{-jw_{s}t})\]
------------------------------(6)
\[v_{p}=\frac{V_{p}}{2}(\epsilon^{jw_{p}t}+\epsilon^{-jw_{p}t})\]
------------------------------(7)
Then the overall voltage applied across C would be
\[v=v_{s}+v_{p}=\frac{V_{s}}{2}(\epsilon^{jw_{s}t}+\epsilon^{-jw_{s}t})+\frac{V_{p}}{2}(\epsilon^{jw_{p}t}+\epsilon^{-jw_{p}t})\]
-------------------------------(8)
The total charge in C is a function of the voltage, hence may be expressed as a Taylor's series, given by
\[Q=Q_{o}+v\frac{dQ}{dv}+v^{2}\frac{d^{2}Q}{dv^{2}}+v^{3}\frac{d^{3}Q}{dv^{3}}+......+v^{n}\frac{d^{n}Q}{dv^{n}}\]
------------------------------(9)
where all the derivatives are evaluated at v = 0. Since Q = f(v), which in turn is f(vp + vs), we may express Q in the form of a double-summation series as
\[Q=\sum_{n=-∞}^∞\sum_{m=-∞}^∞ Q_{nm}\epsilon^{j(nw_{p}+mw_{s})t}\]
------------------------------(10)
Since Q = f(v), we can also write v = f(Q). We may therefore express v in the form of a second double-summation series as
\[V=\sum_{n=-∞}^∞\sum_{m=-∞}^∞ V_{nm}\epsilon^{j(nw_{p}+mw_{s})t}\]
-----------------------------(11)
Differentiating Q with respect to t, we get the total current I as
\[I=\frac{dQ}{dt}=\sum_{n=-∞}^∞\sum_{m=-∞}^∞ j(nw_{p}+mw_{s})Q_{nm}\epsilon^{j(nw_{p}+mw_{s})t}\]
\[=\sum_{n=-∞}^∞\sum_{m=-∞}^∞ I_{nm}\epsilon^{j(nw_{p}+mw_{s})t}\]
----------------------------(12)
where we define the relation
We know that a pure capacitor is an energy storing device, and hence it cannot dissipate power and do work. Therefore, the net power into and out of it must be zero for power to be conserved (law of conservation of power). This means that the net power associated with a varactor must be zero, as we consider it as almost pure.
Now, the generalized average power at frequencies (mws + nwp) may be written as
Where I*nm is the complex conjugate of Inm and V*nm is the complex conjugate of Vnm. This derivation is carried out based on the basic power relation P = VI* = V*I.
As stated above , law of conservation of power requires that
\[\sum_{n=-∞}^∞\sum_{m=-∞}^∞p_{nm}=0\]
-------------------------------(15)
Multiplying (15) with (nwp + mws) / (nwp + mws) (so that no change in the relation occurs),
we find
\[\sum_{n=-∞}^∞\sum_{m=-∞}^∞p_{nm}(\frac{nw_{p}+mw_{s}}{nw_{p}+mw_{s}})=0\]
------------------------------(16)
Splitting (16) into two parts, we have
\[\sum_{n=-∞}^∞\sum_{m=-∞}^∞p_{nm}(\frac{nw_{p}}{nw_{p}+mw_{s}})+\sum_{n=-∞}^∞\sum_{m=-∞}^∞p_{nm}(\frac{mw_{s}}{nw_{p}+mw_{s}})=0\]
-------------------------------(17)
Equation (17) may also be written in the form
\[w_{p}\sum_{n=-∞}^∞\sum_{m=-∞}^∞(\frac{np_{nm}}{nw_{p}+mw_{s}})+w_{s}\sum_{n=-∞}^∞\sum_{m=-∞}^∞(\frac{mp_{nm}}{nw_{p}+mw_{s}})=0\]
-------------------------------(18)
We observe that the choice of ws and wp is arbitrary, and that even if we interchange them the sum given by (18) should remain zero. This is possible if and only if each summation is separately zero. That is
\[\sum_{n=-∞}^∞\sum_{m=-∞}^∞(\frac{np_{nm}}{nw_{p}+mw_{s}})=0\]
-----------------------------(19a)
\[\sum_{n=-∞}^∞\sum_{m=-∞}^∞(\frac{mp_{nm}}{nw_{p}+mw_{s}})=0\] ----------------------------(19b)
Equations (19a) and (19b) represent the Manley-Rowe power (or energy) relations.
In 1956, J. M. Manley and H. E. Rowe derived a set of energy relations associated with nonlinear elements. These relations, now known as the M-R energy (or power) relations, have become quite useful in analyzing the operations of the paramp. The derivation of these relations can be done with the help of the below figure.
 |
| Equivalent circuit of paramp for derivation of M-R relations |
Figure shows signal sources vs of frequency ws and pump source vp of frequency wp connected across a nonlinear, lossless variable capacitance C. The R's are series resistances associated with the sources. The rectangular blocks represent band-pass filters (BPF's), which are tuned, respectively, to frequencies ws, wp, wp∓ws, 2wp∓2ws,......,mwp∓nws, where m and n are integers.
As shown in the figure, we find that vs and vp are applied to the nonlinear capacitor C through BPFs tuned to their respective frequencies. When excited, a nonlinear device will produce an output expressed in the general form
\[v_{o}=a_{1}v_{i}+a_{2}v_{i}^{2}+a_{3}v_{i}^{3}+.....+a_{n}v_{i}^{n}\]
-------------------(1)
where a1,a2,...an are constants associated with the nonlinear device, and
vi = vs + vp
----------------------------(2) Assuming that
vs = Vscoswst
-----------------------------(3)and
vp = vpcoswpt
------------------------------(4)we find
vo =a1(Vscoswst + Vpcoswpt) + a2(Vscoswst + Vpcoswpt)2 + ...... + an(Vscoswst + Vpcoswpt)n
---------------------------------(5)Equation (5) shows that the nonlinear capacitor C produces all possible harmonics of frequencies ws and wp. Each one of these frequencies is isolated from the rest by the respective BPF tuned to that particular frequency, and allowed to dissipate its power in the respective load resistor connected in series with that BPF.
In the M-R relations, (3) and (4) are respectively written as
\[v_{s}=\frac{V_{s}}{2}(\epsilon^{jw_{s}t}+\epsilon^{-jw_{s}t})\]
------------------------------(6)
\[v_{p}=\frac{V_{p}}{2}(\epsilon^{jw_{p}t}+\epsilon^{-jw_{p}t})\]
------------------------------(7)
Then the overall voltage applied across C would be
\[v=v_{s}+v_{p}=\frac{V_{s}}{2}(\epsilon^{jw_{s}t}+\epsilon^{-jw_{s}t})+\frac{V_{p}}{2}(\epsilon^{jw_{p}t}+\epsilon^{-jw_{p}t})\]
-------------------------------(8)
The total charge in C is a function of the voltage, hence may be expressed as a Taylor's series, given by
\[Q=Q_{o}+v\frac{dQ}{dv}+v^{2}\frac{d^{2}Q}{dv^{2}}+v^{3}\frac{d^{3}Q}{dv^{3}}+......+v^{n}\frac{d^{n}Q}{dv^{n}}\]
------------------------------(9)
where all the derivatives are evaluated at v = 0. Since Q = f(v), which in turn is f(vp + vs), we may express Q in the form of a double-summation series as
\[Q=\sum_{n=-∞}^∞\sum_{m=-∞}^∞ Q_{nm}\epsilon^{j(nw_{p}+mw_{s})t}\]
------------------------------(10)
Since Q = f(v), we can also write v = f(Q). We may therefore express v in the form of a second double-summation series as
\[V=\sum_{n=-∞}^∞\sum_{m=-∞}^∞ V_{nm}\epsilon^{j(nw_{p}+mw_{s})t}\]
-----------------------------(11)
Differentiating Q with respect to t, we get the total current I as
\[I=\frac{dQ}{dt}=\sum_{n=-∞}^∞\sum_{m=-∞}^∞ j(nw_{p}+mw_{s})Q_{nm}\epsilon^{j(nw_{p}+mw_{s})t}\]
\[=\sum_{n=-∞}^∞\sum_{m=-∞}^∞ I_{nm}\epsilon^{j(nw_{p}+mw_{s})t}\]
----------------------------(12)
where we define the relation
Inm = j(nwp + mws)Qnm
-----------------------------(13)We know that a pure capacitor is an energy storing device, and hence it cannot dissipate power and do work. Therefore, the net power into and out of it must be zero for power to be conserved (law of conservation of power). This means that the net power associated with a varactor must be zero, as we consider it as almost pure.
Now, the generalized average power at frequencies (mws + nwp) may be written as
Pnm = (VnmI*nm + V*nmInm)
--------------------------------(14)Where I*nm is the complex conjugate of Inm and V*nm is the complex conjugate of Vnm. This derivation is carried out based on the basic power relation P = VI* = V*I.
As stated above , law of conservation of power requires that
\[\sum_{n=-∞}^∞\sum_{m=-∞}^∞p_{nm}=0\]
-------------------------------(15)
Multiplying (15) with (nwp + mws) / (nwp + mws) (so that no change in the relation occurs),
we find
\[\sum_{n=-∞}^∞\sum_{m=-∞}^∞p_{nm}(\frac{nw_{p}+mw_{s}}{nw_{p}+mw_{s}})=0\]
------------------------------(16)
Splitting (16) into two parts, we have
\[\sum_{n=-∞}^∞\sum_{m=-∞}^∞p_{nm}(\frac{nw_{p}}{nw_{p}+mw_{s}})+\sum_{n=-∞}^∞\sum_{m=-∞}^∞p_{nm}(\frac{mw_{s}}{nw_{p}+mw_{s}})=0\]
-------------------------------(17)
Equation (17) may also be written in the form
\[w_{p}\sum_{n=-∞}^∞\sum_{m=-∞}^∞(\frac{np_{nm}}{nw_{p}+mw_{s}})+w_{s}\sum_{n=-∞}^∞\sum_{m=-∞}^∞(\frac{mp_{nm}}{nw_{p}+mw_{s}})=0\]
-------------------------------(18)
We observe that the choice of ws and wp is arbitrary, and that even if we interchange them the sum given by (18) should remain zero. This is possible if and only if each summation is separately zero. That is
\[\sum_{n=-∞}^∞\sum_{m=-∞}^∞(\frac{np_{nm}}{nw_{p}+mw_{s}})=0\]
-----------------------------(19a)
\[\sum_{n=-∞}^∞\sum_{m=-∞}^∞(\frac{mp_{nm}}{nw_{p}+mw_{s}})=0\] ----------------------------(19b)
Equations (19a) and (19b) represent the Manley-Rowe power (or energy) relations.
