Lecture Microwave Networs and cattering Parameters Optional Reading: teer ection 6.3 to 6.6 Pozar ection 4.3 ElecEng4FJ4 LECTURE : MICROWAE NETWORK AND -PARAMETER
Microwave Networs: oltages and Currents microwave-networ theory enables circuit-lie analysis methods which are simpler than field methods it also enables the integrated analysis of microwave structures with conventional lumped components (ICs, chip transistors, resistors, etc.) usual low-frequency definitions of voltage and current are not valid = E d L I = H dl C m if E is not conservative, voltage integral depends on the chosen integration path and is thus ambiguous if there are no metallic leads (dielectric guides), the current integral is meaningless and the voltage one is ambiguous networ parameters (Z, Y, ABCD) based on total voltages and currents at networ ports are thus inadequate ElecEng4FJ4 LECTURE : MICROWAE NETWORK AND -PARAMETER
Microwave Networ: oltage/current Waves at Ports incident and reflected voltage/current waves are defined at each port at nth port: = I I I n n n n = n n ( n =,, N) (omit ~ for simpler notation) current definitions In = n / Z I = / Z n n 0 0 [Pozar] N-port networ ElecEng4FJ4 LECTURE : MICROWAE NETWORK AND -PARAMETER 3
cattering Matrix: Common-impedance Definition the (scattering) matrix relates the voltage/current waves incident on all ports (incoming waves) to the voltage/current waves leaving all ports (outgoing or scattered waves) assume common characteristic impedance at all ports scattered Z0n = Z0, n=,, N N = N N NN N scattering parameters incident this definition is best suited for networs with TEM transmission-line ports (e.g., coaxial connectors) of common system impedance (e.g., Z 0 = 50 Ω) ElecEng4FJ4 LECTURE : MICROWAE NETWORK AND -PARAMETER 4
cattering Parameters: Physical Meaning ij = i j = 0 for all j to find ij, launch an incident wave on port j and measure the wave leaving port i meanwhile match all ports so that = 0 for all j ElecEng4FJ4 LECTURE : MICROWAE NETWORK AND -PARAMETER 5
Example: -parameters of 3-dB Attenuator Z 0 = 50 Ω Z 0 = 50 Ω Γ = 0 in, 0 = =Γ = 0 0, = 0 ZL = Z = Zin Z Z Z, 0 = 0 port matched, Z in, 4.8 58.56 = 8.56 50 Ω 4.8 58.56 0 = (symmetry) no reflection at port ElecEng4FJ4 LECTURE : MICROWAE NETWORK AND -PARAMETER 6
Example: -parameters of 3-dB Attenuator Z 0 = 50 Ω 4.8 58.56 50 = = 0.707 (8.56 4.8 58.56) 58.56 = 0 input power output power = 0 Z 0 = 50 port matched P av = Z P P av = = = = Z 0 0.707 P av 0 Z0 Ω ElecEng4FJ4 LECTURE : MICROWAE NETWORK AND -PARAMETER 7
Example: -parameters of 3-dB Attenuator 3 Z 0 = 50 Ω Z 0 = 50 Ω 0 0.707 0.707 0 transmission loss: TL 0log 0 3 db (under matched conditions) 0.707 ElecEng4FJ4 LECTURE : MICROWAE NETWORK AND -PARAMETER 8
0 Example: -parameters of hunt Element ( Y Y ) = 0 port matched, Zin, Z0 Y0 Y = Γ = = = Z Z Y Y = Y in, 0 Z = Z 0 in, Y = Y Z L, 0 L, 0 Y 0 0 = = Y0 ( Y0 Y ) Y0 Y circuit is symmetric port = Y Y Y0 0 0 Z in, = Y in, = ( Γ) Y Y0 = = = = = = 0 Y0 Y Y0 Y = in, ElecEng4FJ4 LECTURE : MICROWAE NETWORK AND -PARAMETER 9
Example: -parameters of hunt Element Y Y0 Y 0 Y Y0 Y = Y0 Y Y 0 Y Y0 Y Numerical Example: Let Y = Y 0 3 3 What value of Y would ensure port = matching? What is the TL in this case? 3 3 not matched to either port transmission loss: TL =0 log 0 3.5 db ElecEng4FJ4 0 /3
port Example: -parameters of eries Element Z port = port Z Z 0 Z Z Z Z = Γ = = = Z Z Z Z 0 Z = = Z 0 0 Z0 Z port matched, in, 0 0 0 in, 0 Z = Z 0 0 L, 0 Z Z Z 0 Z in, circuit is symmetric Z 0 Z Z Z 0 0 0 = = = ( Γ) = = ( Z Z0) ( Z Z0) Z 0 Z = 0 = ' ElecEng4FJ4 LECTURE : MICROWAE NETWORK AND -PARAMETER
Example: -parameters of eries Element port Z 0 Z port Z 0 Z Z0 Z Z Z Z 0 0 = Z0 Z Z0 Z Z0 Z Numerical Example: Let Z = Z 0 3 3 = What value of Z would ensure port matching? What is the TL in this case? 3 3 not matched to either port transmission loss: TL =0 log 0 3.5 db ElecEng4FJ4 /3
Example: Transmission-line egment port port = 0 port Z Z 0, γ 0s Z 0s Z 0 L port matched Z, γ 0s, Z in, L reflection -parameter Γ G Z L in Z Γ e γ Z s Z =Γ where Z, 0, = 0 in = Z Γ = = Z L in Z Γ e γ Z s Z Γ ( e γ L ) = = see L0 Γ e γ L Z 0s, 0 0 0, 0 0 0 note that is not the same as Γ special case of matched line: Z0 = Z0s, Γ = 0 = 0 ElecEng4FJ4 LECTURE : MICROWAE NETWORK AND -PARAMETER 3
matched port, = 0 port i Z γ Z 0s Z in, Example: Transmission-line egment 0, Z 0s L ElecEng4FJ4 LECTURE : MICROWAE NETWORK AND -PARAMETER 4 γ L eγ L = L e γ = = 0 = i ( Γ) reflected wave translated bac to -ꞌ incident wave translated bac to -ꞌ L ( ) L i eγ = = Γe γ ( L L ) i eγ γ Γe = ( Γ )( ) ( e γ L Γ Γ ) = = L γl γl eγ e e e = L e γ Γ Γ Γ ( Γ ) = Γ special case of matched line: Z0 = Z0s, Γ = 0 L = e γ
Properties of Matrix: Reciprocity reciprocal networs: exchange of source and measurement locations do not affect the measurement G i i linear circuit j j A I A I A A i i linear circuit j j G I i j = I i j in terms of Z- or Y-parameters, a networ is said to be reciprocal if Z = Z or Y = Y, i j ij ji ij ji Reminder: = ZI, I = Y networs composed of linear components are reciprocal reciprocal networs have symmetric matrices =, i j T or = ij ji ElecEng4FJ4 LECTURE : MICROWAE NETWORK AND -PARAMETER 5
Properties of Matrix: Loss-free Networs average power P av delivered to loss-free networ is zero (whatever power goes in, comes out) P av = = Re( TI ) Re ( T T)( * ) Z 0 Re[ T T * T T * ] Pav = Z 0 real imaginary real ( T T * ) 0 Pav = = Z 0 T = T * Z0 Z0 incident power scattered power (power conservation) ElecEng4FJ4 LECTURE : MICROWAE NETWORK AND -PARAMETER 6
Properties of Matrix: Loss-free Networs T = T * U T = ( ) T ( ) = T T T = U or ( ) = T the matrix of a loss-free networ is unitary since T = for a reciprocal AND loss-free networ U T H = U = = = U or = equivalent ElecEng4FJ4 LECTURE : MICROWAE NETWORK AND -PARAMETER 7
Properties of Matrix: Loss-free Networs 3 what does the unitary property of the matrix really mean? the dot product of any column of with its own conjugate is unity; with any other column it is zero N { i j = δij for all i, j δ, ij = 0, = loss-free -port networ: =, i = j = i = i What is the -parameter magnitude of a loss-free -port networ? Write down all equations for the -parameters of a loss-free - port networ. =, i = j = = 0, i =, j = = 0, i =, j = ElecEng4FJ4 LECTURE : MICROWAE NETWORK AND -PARAMETER 8 j j
Example: Properties of Matrix A -port networ has the following matrix: 0.5 0 0.85 45 =. 0.85 45 0.0 0 (a) Determine whether the networ is reciprocal and loss-free. (b) If port is terminated with a matched load, what is the return loss at port? (c) If port is terminated with a short circuit, what is the return loss at port? ElecEng4FJ4 LECTURE : MICROWAE NETWORK AND -PARAMETER 9
Generalized -parameters and Power Waves assumption of same port impedances (Z = Z 0, =,, N) is dropped to accommodate networs terminated with different TLs port waves do no have to be voltage or current waves so that the networ may terminate with non-tem waveguides let port support propagating mode given by the E and H vectors incoming field: E = ae, H field strength = ah and phase outgoing field: E = b e, H = b h a, b coefficients termed root-power waves (aa power waves) (e,h ) normalized traveling-wave vectors termed modal vectors field distribution in port s cross-section Re ( e h) ds= ElecEng4FJ4 LECTURE : MICROWAE NETWORK AND -PARAMETER 0 * a av, Re ( ) a Re ( ) E H s e h s P = d = d = =
Power Waves vs. oltage Waves magnitude of root-power waves here is where the name comes from rms Note: a = a = P and b = P av, av, phase of root-power waves same as that of E-field at port a = E and b = E root-power waves and voltage waves in low-loss TL (Z 0, real) a = Pav, = and b = Pav, = Z Z 0, 0, a = and b = b rms = P P av, av, a, b 0, Z0, = = Z / W ElecEng4FJ4 LECTURE : MICROWAE NETWORK AND -PARAMETER
Example: Modal ectors and Power Waves (a) Find the TEM modal vectors e and h for the coaxial cable RG- 40U of inner-wire diameter d a = 0.9 mm, shield diameter d b = 3.0 mm and insulator relative permittivity ε r =.08. Ignore the losses. (b) What is the power wave if the cable carries 0 W of power? (c) What is the voltage wave (this is a 50-Ω cable)? ElecEng4FJ4 LECTURE : MICROWAE NETWORK AND -PARAMETER
Power Waves and Power Delivered to Port time-average power delivered to port Pav, = 0.5Re( I ) express total voltages and currents in terms of power waves = = Z( a b) a b I = I I = = Z Z av, 0.5 Re ( ) av, 0.5 0.5 imaginary P = a b ab ab P = a b power of incident wave power of scattered wave ElecEng4FJ4 LECTURE : MICROWAE NETWORK AND -PARAMETER 3
Generalized -parameter Definition definition of generalized matrix b= Ga G ij = b a i j a = 0 for all j matched port requirement generalized vs. commonimpedance -parameters G ij i = j Z 0 j Z 0i = 0 for all j G ij = ij Z 0 j Z 0i = 0 for all j a N, N N, b N Z 0N [Pozar] N ElecEng4FJ4 LECTURE : MICROWAE NETWORK AND -PARAMETER 4
Example: Power Delivered to Port Port of a networ is excited by a -W incident wave. What is the power delivered to the networ through port if: (a) = 0, (b) = 0.5? ElecEng4FJ4 LECTURE : MICROWAE NETWORK AND -PARAMETER 5
-parameters and Port Extension the position of the port plane impacts the phases of the waves (and their amplitudes, if the ports are lossy transmission lines) = e γ L,0 L,0 = e γ = L,0e γ L =, 0eγ z = L shifted position L z original position,0,0 z = 0 NL N,0 = N eγ NL N,0 = N e γ N N N = L N,0e γ NL N = N,0eγ N N N z N shifted position = L z = 0 N ElecEng4FJ4 LECTURE : MICROWAE NETWORK AND -PARAMETER 6 L N z N original position N,0 N,0 N 0, N
-parameters and Port Extension eγl 0 eγ L 0 = P ' where P = 0 eγ NLN eγl 0 eγ L 0 = P ' where P = 0 eγ NLN substitute in = = 0 0 0 P' = P' ' = ( P ) P' 0 0 ' = = e γ γ ili jlj PP 0 ij ij P P 0 = ( P ) ElecEng4FJ4 LECTURE : MICROWAE NETWORK AND -PARAMETER 7 ' P loss-free ports γ = jβ, =,, N
Example: -parameters and Port Extension The matrix of a -port networ is given by 0.5 0 0.80 45 =. 0.0 45 0.5 0 A TL of length λ/3 is added to port. tate the matrix of the modified device. ElecEng4FJ4 LECTURE : MICROWAE NETWORK AND -PARAMETER 8
ummary the -parameters are the complex reflection and transmission coefficients of a networ under conditions of impedance match the diagonal elements of the -matrix are the reflection coefficients the off-diagonal elements are the transmission coefficients the -parameter magnitudes give the square root of the output-toinput power ratio the -parameter phases give the change in the signal phase as it travels from the input to the output port of the device the -matrix of a reciprocal (linear) device is symmetric the -matrix of a loss-free device is unitary the generalized -matrix is necessary when the device ports have different characteristic impedances ElecEng4FJ4 LECTURE : MICROWAE NETWORK AND -PARAMETER 9