ECE 6345 Spring 5 Prof. David R. Jackson ECE Dept. Notes 6
Overview In this set of notes we calculate the power radiated into space by the circular patch. This will lead to Q sp of the circular patch.
Radiated Power of Circular Patch From Notes we have: E E r ah k h Q J k a ( ) = FF θ, θφ, ( )cosφtanc( z ) ( θ) π ( sin θ) η E E r ah k h P J k a ( ) = ( ) Assumption: ( ) E a, φ = cosφ FF φ, θφ, ( )sinφtanc z ( θ) π inc ( o sin θ) η z where TM Q( θ ) = Γ ( θ ) ( TE Γ ) P( θ) cosθ ( θ) J ( ) inc x J ( ) x x 3
Radiated Power of Circular Patch (cont.) The power density in the far field from the Poynting vector is Sr ( r, θφ, ) = Eθ + E η = η φ E ( ah) tanc ( kz h)( π ) η ( ). cos sin ( ) sin ( sin ) ( ) φ J θ Qθ + φ Jinc o θ Pθ Next, use E ωµ kη = = 4π r 4π r 4
Radiated Power of Circular Patch (cont.) We then have Sr r h kzh 8η (, θφ, ) = ( ) tanc ( ) ( ) ( ) cos ( ) sin + sin ( ) sin The space-wave power is r φ Q θ J ko a θ φ P θ Jinc θ ππ/ Psp = Sr ( r,, ) r sin d d θφ θ θ φ Performing the φ integrals, Psp = π ( ) h 8 η π / ( ) ( ) ( ) tanc kzh sin θ Q( θ) J sin θ + P( θ) Jinc sinθ dθ 5
Radiated Power of Circular Patch (cont.) Define π / Ic = C( θ) dθ where ( ) ( ) ( ) C( θ) = sinθ tanc kzh Q( θ) J sin θ + P( θ) Jinc sinθ We then have π Psp = ( ) h Ic 8η Note: We will get a CAD formula for I c later. 6
Calculation of Q sp The Q formula is Q sp U ω S = P sp U S = U E = ε εr E V z dv 4 = εε rh Ez ds S π a = εε rh Ez ρdρdφ = π εε rh Ez ρdρ a E z The electric field is ( ) ( kρ ) ( ) J ρφ φ, = cos J ( ) E a, φ = cosφ z Note: 7
Calculation of Q sp (cont.) The stored energy is then a r J ( ) US = h J ( k ) d εε π ρ ρ ρ Denote a I = J ( kρ) ρdρ = ρ J ( kρ) + ρ J ( kρ) ( kρ ) a = J ( ) + J ( ) ( ) a 8
Calculation of Q sp (cont.) Recall that k = x a x =.848 so ( ) ( ) J J x = = Hence a I J x x = ( ) We then have a US εε rhπ = J ( x ) J ( x) x or U S a εε rhπ = x 9
Calculation of Q sp (cont.) The formula for Q sp then becomes Q sp = ω εε rhπ x π 8η a h Ic ( ) This may be simplified by using the following expressions to eliminate ω and k : k µε a = x r r ω µε µε a = x r r = ω = a x µε r r x µε µε r r
Calculation of Q sp (cont.) We then have Q sp = ( x ) ε r x kh Ic Note that Q sp is proportional to the substrate permittivity and inversely proportional to the substrate thickness.
Calculation of Q sp (cont.) Summary (exact Q sp ) Q sp ( ) = x r x Ic kh ε π / ( θ) Ιc C dθ x =.848 ( ) ( ) ( ) C( θ) = sinθ tanc kzh Q( θ) J sin θ + P( θ) Jinc sinθ
The p Factor We can express the Q sp formula in terms of a p factor (which will eventually be approximated in closed form). Define: C ( ) C( ) θ θ a π / ( θ) Ι C dθ The term C ignores the patch array factor. Also, define: p I I c = π / π / C C ( θ) ( θ) dθ dθ The p term gives the ratio of the power radiated by the actual patch to the power radiated if we ignore the array factor, and collapse the magnetic current down to a single dipole. (See the end of the notes for a derivation of the equivalent dipole moment of the circular patch.) 3
Then we have Q sp The p Factor (cont.) ( ) = x r x p I kh ε Note that as x J inc ( x) J ( x) This allows us to express I in a simpler form without the Bessel functions: π / sin tanc Ι = θ ( k hn ( θ) ) P( θ) + Q( θ) dθ 4 4
The p Factor (cont.) For the p factor we have: p = π / ( ) ( ) ( ) + ( ) ( ) sinθ tanc khn ( θ) Qθ J sinθ Pθ Jinc sinθ dθ π / θ ( k hn θ ) P( θ) + Q( θ) dθ sin tanc ( ) 4 The term p depends on the patch radius a and the substrate parameters. (After making some approximations, it will depend only on the patch radius.) 5
Approximation for a Thin Substrate For a thin substrate, we have: ( k hn θ ) tanc ( ) so π / Ι sinθ P( θ) Q( θ) + dθ 4 p π / ( ) ( ) + ( ) ( ) sinθ Q θ J sinθ P θ Jinc sinθ dθ π / sinθ P( θ) + Q( θ) dθ 4 6
Approximation for a Thin Substrate (cont.) From Notes 9 we also have P( θ) = cos θ( Γ ( θ)) = Q TM ( θ) ( θ) TE = Γ = N ( k hn θ ) + j tan ( ) ε r ( θ)secθ cosθ µ r cosθ + j tan ( ) N( θ ) ( k hn θ ) For a thin substrate we then have P( θ) cosθ Q( θ ) 7
Approximation for a Thin Substrate (cont.) For the I term (the denominator of the p function) we then have: Ι π / sin θ 4 ( cos θ + ) dθ 4 π / ( ) = sinθ cos θ + dθ This yields I 4 3 8
Approximation for a Thin Substrate (cont.) The formula for the p function then becomes π / 3 p sinθ J ( sinθ) + cosθ Jinc ( sinθ) dθ 4 so that π / ( ) ( ) p 3 sinθ J sinθ + cos θ Jinc sinθ dθ The p factor now only depends only on the patch size. 9
Approximation for a Thin Substrate (cont.) Summary (approximate Q sp ) Q sp ( ) = x r x p I kh ε x =.848 I 4 3 π / ( ) ( ) p 3 sinθ J sinθ + cos θ Jinc sinθ dθ
Equivalent Dipole Moment of Circular Patch Consider an equivalent magnetic dipole that models the patch: y Patch a φ M s φ x h / λ M s φ = cosφ y As a the magnetic current sheet approaches an equivalent magnetic dipole. Equivalent dipole K x
Equivalent Dipole Moment of Circular Patch (cont.) The dipole moment of the equivalent magnetic dipole is calculated: π π s sφ S Kl = M yˆ ds = h M cosφa dφ = h cosφcosφadφ = ha π cos φadφ This yields Kl = πah
Equivalent Dipole Moment of Circular Patch (cont.) We can therefore physically interpret the p factor as follows: p = P patch rad dip rad P where patch P rad = power radiated by circular patch dip P rad = power radiated by magnetic dipole of equal moment ( Kl = πah ) 3
CAD Formula for p In the next set of notes we will obtain approximate closed-form CAD expression for p. 4