## 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 v

As shown in the figure, we find that v

\[v_{o}=a_{1}v_{i}+a_{2}v_{i}^{2}+a_{3}v_{i}^{3}+.....+a_{n}v_{i}^{n}\]

-------------------(1)

where a

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(v

\[Q=\sum_{n=-∞}^∞\sum_{m=-∞}^∞ Q_{nm}\epsilon^{j(nw_{p}+mw_{s})t}\]

------------------------------(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 (mw

Where I

As stated above , law of conservation of power requires that

\[\sum_{n=-∞}^∞\sum_{m=-∞}^∞p_{nm}=0\]

-------------------------------(15)

Multiplying (15) with (nw

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 v

_{s}of frequency w_{s}and pump source v_{p}of frequency w_{p}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 w_{s}, w_{p}, w_{p}∓w_{s}, 2w_{p}∓2w_{s},......,*m*w_{p}∓*n*w_{s}, where m and n are integers.As shown in the figure, we find that v

_{s}and v_{p}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 a

_{1},a_{2},...a_{n}are constants associated with the nonlinear device, and
v

----------------------------(2) _{i}= v_{s}+ v_{p}Assuming that

v

-----------------------------(3)_{s}= V_{s}cosw_{s}tand

v

------------------------------(4)_{p}= v_{p}cosw_{p}twe find

v

---------------------------------(5)_{o}=a_{1}(V_{s}cosw_{s}t + V_{p}cosw_{p}t) + a_{2}(V_{s}cosw_{s}t + V_{p}cosw_{p}t)^{2}+ ...... + a_{n}(V_{s}cosw_{s}t + V_{p}cosw_{p}t)^{n}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(v

_{p}+ v_{s}), 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

I

-----------------------------(13)_{nm}= j(nw_{p}+ mw_{s})Q_{nm}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 (mw

_{s}+ nw_{p}) may be written as
P

--------------------------------(14)_{nm}= (V_{nm}I^{*}_{nm}+ V^{*}_{nm}I_{nm})Where I

^{*}_{nm}is the complex conjugate of I_{nm}and V^{*}_{nm}is the complex conjugate of V_{nm}. 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 (nw

_{p}+ mw_{s}) / (nw_{p}+ mw_{s}) (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.

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