IMPEDANCE MATCHING for High-Frequency Circuit Design Elective by Michael Tse September 2003 Contents • • • • • The Problem Q-factor matching approach Simple matching circuits L matching circuits π matching circuits T matching circuits Tapped capacitor matching circuits Double-tuned circuits General impedance matching based on two-port circuits Immittance matrices and hybrid matrices ABCD matrix and matching Propagation equations from ABCD matrix Michael Tse: Impedance Matching 2 Impedance Matching • Impedance matching is a major problem in highfrequency circuit design. • It is concerned with matching one part of a circuit to another in order to achieve maximum power transfer between the two parts. max power transfer Circuit 1 Circuit 2 space max power transfer Michael Tse: Impedance Matching 3 The problem Given a load R, find a circuit that can match the driving resistance R¢ at frequency w0. R¢ R¢ ? R Obviously, the matching circuit must contain L and C in order to specify the matching frequency. Michael Tse: Impedance Matching 4 The Q factor approach to matching The Q factor is defined as the ratio of stored to dissipated power 2p ⋅ (max instantaneous energy stored ) Q= energy dissipated per cycle In general, a circuit’s reactance is a function of frequency and the Q factor is defined at the resonance frequency w0 . X w0 w As we will see later, the Q factor can be used to modify the overall resistance of a circuit at some selected frequency, thus achieving a matching condition. Michael Tse: Impedance Matching 5 Low Q circuit High Q circuit X X w w0 Definition: Q= w w0 w 0 dB w dX = 0 2G dw w =w 0 2R dw w =w 0 B = susceptance X = reactance R = resistance G = conductance Michael Tse: Impedance Matching It is easily shown that for linear parallel RLC circuits: Q = w0CR = R/(w0L) 6 Essential revision (basic circuit theory) R Resistance (Ω) Z IMPEDANCE (Ω) R Resistance (Ω) Y ADMITTANCE (S) G Conductance (S) L inductance (H) jwL = +jX reactance (Ω) 1 = –jB jw L susceptance (S) Michael Tse: Impedance Matching C capacitance (F) 1 = –jX jw C reactance (Ω) jwC = +jB susceptance (S) 7 Essential revision (basic circuit theory) Quality factor (Q factor) Series: X 1 G Q= = = R RB B L wL Q= R R † Parallel: R B Q = = RB = X G † C R † R 1 Q= wCR † L R Q= wL R C Q = wCR † L or C. Higher Q means†that it is closer to the ideal Michael Tse: Impedance Matching 8 Essential revision (basic circuit theory) Series to parallel conversion jX Z = R + jX Y= 1 1 R - jX R X = = 2 = j Z R + jX R + X 2 R 2 + X 2 R2 + X 2 R 1 1 R X = 2 + ÊÊ R ˆ 2 ˆ ÊXˆ 1+ Á ˜ jÁÁÁ ˜ + 1˜˜ Ë R¯ ËË X ¯ ¯ 1 1 = + Ê 1 ˆ R(1+ Q2 ) jX Á 2 + 1˜ ËQ ¯ † † Ê 1ˆ jX Á1+ 2 ˜ Ë Q ¯ R(1+Q2) or j R' Q † R j (1+ Q2 ) Q † Michael Tse: Impedance Matching † † = Ê 1 ˆ jRQÁ 2 + 1˜ ËQ ¯ 9 Essential revision (basic circuit theory) Parallel to series conversion jB G Z= 1 1 G - jB G B = = 2 = j Y G + jB G + B 2 G 2 + B 2 G2 + B2 Y = G + jB † † G(1+ Q2 ) † conductance (S) Ê 1 ˆ susceptance (S) jBÁ1+ 2 ˜ Ë Q ¯ Ê ˆ or j G' = j G (1+ Q2 ) = jGQÁ 12 + 1˜ Q Q Ë Q †¯ Michael Tse: Impedance Matching † 1 1 G B = 2 + ÊÊ G ˆ 2 ˆ Ê Bˆ 1+ Á ˜ jÁÁÁ ˜ + 1˜˜ ËG¯ ËË B ¯ ¯ 1 1 = + Ê 1 ˆ G(1+ Q2 ) jBÁ 2 + 1˜ ËQ ¯ 10 Example: RLC circuit (Recall Year 1 material) Resonant frequency is w 0 = R L C Q factor is Q = R 1 LC C L Z drops by 2 (3 dB) at w1 and w2. 1 Z= (1/R) + jwC - ( j /w L) Ê ˆ 1 1 ˜ w1,2 = w 0 ÁÁ 1+ ± 2 2Q ˜¯ 4Q Ë Bandwidth is Dw = w 2 - w1 = w1 w0 w2 1 RC Note: w1 and w2 are called 3dB corner frequencies. Their geometric mean is w0. For narrowband cases, their arithmetic mean is close to w0. Michael Tse: Impedance Matching 11 Practical components are lossy! = C RC Q factor = QC = w0CRC (unloaded Q factor) = RL L Q factor = QL = RL/w0L (unloaded Q factor) QLC = unloaded Q factor for the paralleled LC components 1 1 1 = + QLC QC QL (easily shown) Michael Tse: Impedance Matching 12 Simple matching circuits R¢ R¢ ? R L matching circuit (single LC section) p matching circuit T matching circuit Michael Tse: Impedance Matching 13 Design of L matching circuits Series L circuit: Objective: match Yin to R’ at w0 L Begin with Yin C R Yi n = jwC + 1 R + jwL È ˘ R w L ˙ = 2 + jÍw C - 2 2 2 ÍÎ R + (w L) R + (wL) ˙˚ Obviously, the reactive part is cancelled if we have L C= 2 R + w 20 L2 1 R2 - 2 where w 0 = LC L Michael Tse: Impedance Matching (#) 14 Thus, at w = w0, we have a resistance for Yin, which should be set to R’. R 2 + w 20 L2 R¢ = = R 1+ Q2 R ( ) (*) Here, Q is the Q-factor, which is equal to w0L/R (for series L and R). So, we can see clearly that Q is modifying R to achieve the matching condition. Design procedure: -Given R and R’, find the required Q from (*). -Given w0, find the required L from Q = w0L/R . -From (#), find the required C to give the selected resonant frequency w0. Michael Tse: Impedance Matching 15 Begin with Shunt L circuit: Zi n = jwL + L C Zin R 1 G + jw C È ˘ G wC = 2 + jÍwL - 2 ˙ 2 2 2 2˚ Î G +w C G +w C Reactive part is cancelled when C L= 2 G + w 20 C 2 1 G2 where w 0 = LC C 2 Finally, the matching condition requires that R¢ = 1/G R = 1+ (w 0 C /G)2 1+ Q2 (*) Design procedure is similar to the series case. Michael Tse: Impedance Matching 16 (#) Other L circuit variations C L Series: C R C L Shunt: R L C R L R Exercise: derive design procedure for all other L circuits. Michael Tse: Impedance Matching 17 General procedure for designing L circuits Series L circuit (suitable for R’>R) : jX1 R¢ R jX2 Shunt L circuit (suitable for R’<R) : jX2 R¢ jX1 R R¢ = R(1+ Q2 ) Ê 1 ˆ jR ¢ jX 2 = - jX 1ÁÁ1+ 2 ˜˜ = Q Ë Q ¯ X Q= 1 R R 1+ Q2 jX1 jX 2 = = - j R¢Q 1 1+ 2 Q B R Q= 1 = G X1 R¢ = Michael Tse: Impedance Matching 18 Advantages of L circuits: • Simple • Low cost • Easy to design Disadvantages of L circuits: • The value of Q is determined by the ratio of R/R’. Hence, • there is no control over the value of Q. • the bandwidth is also not controllable. Solution: Add an element to provide added flexibility. fi p circuits and T circuits Michael Tse: Impedance Matching 19 p matching circuits Analysis by decomposing into two L circuit sections: jX2 R¢¢ jB3 R¢ + jX ¢ First section (from right): jB1 R jX’ Second section: j(X2–R’Q1) R¢¢ jB3 R¢ + jX ¢ R X ¢ = X 2 - R ¢Q1 2 1+ Q1 B Q1 = 1 = B1R G R¢ = Q2 = R¢ X ¢ X 2 - R ¢Q1 X = fi 2 = Q1 + Q2 R¢ R¢ R¢ R¢¢ = R ¢(1+ Q22 ) Q Q B¢¢ = B3 - 2 fi B3 = 2 R¢ R¢ Michael Tse: Impedance Matching 20 Impedance transformation in p matching circuits jX2 R¢¢ jB3 R¢ + jX ¢ jB1 R R¢¢ R 1 1+ Q12 R¢ 1+ Q22 Obviously, we have to set Q1 > Q2 if we want to have R”<R. Likewise, we need Q1 < Q2 if we want to have R”>R. Michael Tse: Impedance Matching 21 General procedure for designing p matching circuits For R¢¢ < R For R¢¢ > R 1. Select Q1 according to the max Q. 1. Select Q2 according to the max Q. 2. Find R’ using R¢ = R /(1+ Q12 ) 2. 3. Get Q2 using Q2 = Find R’ using R¢ = R¢¢ /(1+ Q22 ) R 2 -1 Get Q2 using Q1 = ¢ R 4. Obtain X2 using X2 = R’(Q1 + Q2). 4. Obtain X2 using X2 = R’(Q1 + Q2). 5. B1 = Q1/R 5. B1 = Q1/R 6. B3 = Q2/R” 6. B3 = Q2/R” 2 R¢¢ -1 ¢ R 3. Michael Tse: Impedance Matching 22 T matching circuits The analysis is similar to the p case. jX3 jX1 jB2 R¢¢ R R¢ + jX ¢ R¢ R 1+ Q12 The difference is that R is first raised to R’ by the series reactance, and then lowered to R” by the shunt reactance. The design procedure can be similarly derived. (Exercise) 1 1+ Q22 Michael Tse: Impedance Matching R¢¢ 23 General procedure for designing T matching circuits For R¢¢ > R For R¢¢ < R 1. Select Q1 according to the max Q. 1. Select Q2 according to the max Q. 2. Find R’ using R¢ = R(1+ Q12 ) R¢ 2 -1 Get Q2 using Q2 = ¢ ¢ R 2. Find R’ using R¢ = R¢¢(1+ Q22 ) R¢ 2 -1 Get Q1 using Q1 = R 4. Obtain X1 using X1 = Q1R. 4. Obtain X1 using X1 = Q1R. 5. B2 = (Q1+Q2)/R’ 5. B2 = (Q1+Q2)/R’ 6. X3 = Q2R” 6. X3 = Q2R” 3. 3. Michael Tse: Impedance Matching 24 Tapped capacitor matching circuit Ê 1+ Q2 ˆ p C2 ÁÁ 2 ˜˜ Ë Qp ¯ C1 L C2 R R 1+ Q2p Q factor Qp = w 0 C2 R Michael Tse: Impedance Matching 25 C1 R’ Ê 1+ Q2 ˆ p C2 ÁÁ 2 ˜˜ Ë Qp ¯ L Q1 = R’/w0L R R R¢ 1+ Q12 1+ Q2p 1+ Q2p required R¢ R = fi Qp = 2 2 1+ Q1 1+ Qp R 1+ Q12 - 1 R¢ ( Michael Tse: Impedance Matching ) 26 Ê 2 ˆ 1 1 1 Á Qp ˜ = + C C1 C2 ÁË1+ Q2p ˜¯ C R’ R 1+ Q2p L For a high Q circuit, w 0 ª 1 LC Also, we have the alternative approximation for Q1: Q1 ≈ w0R’C, which is set to w0 / Dw . Thus, we can go backward to find all the circuit parameters. Michael Tse: Impedance Matching 27 General procedure for designing tapped C circuits 1. 2. 3. 4. 5. 6. Find Q1 from Q1 = w0 / Dw Given R’, find C using C = Q1/ w0R’ = 1 / 2π DwR’ Find L using L = 1 / w02C Find Qp using Qp = [ (R/R’)(1+Q12)–1 ]1/2 Find C2 from C2 = Qp / w0R Find C1 from C1 = Ceq C2 / (Ceq – C2) where Ceq = C2(1+ Qp2)/ Qp2 Michael Tse: Impedance Matching 28 Advantages of π, T and tapped C circuits: • specify Q factor (sharpness of cutoff) • provide some control of the bandwidth Disadvantage: • no precise control of the bandwidth For precise specification of bandwidth, use double-tuned matching circuits. Michael Tse: Impedance Matching 29 Double-tuned matching circuits Specify the bandwidth by two frequencies wm1 and wm2 . transmission gain GT wm1 wm2 w There is a mid-band dip, which can be made small if the pass band is narrow. Also, large difference in the impedances to be matched can be achieved by means of galvanic transformer. Michael Tse: Impedance Matching 30 The construction of a double-tuned circuit typically includes a real transformer and two resonating capacitors. • RG C1 L11 M • L22 C2 RL Transformer turn ratio n and coupling coefficient k are related by n= L1 1 k 2 L2 2 Michael Tse: Impedance Matching 31 Equivalent models: L22(1–k2) n:1 • RG C1 • L11 C2 ideal transformer Ê1 ˆ L11Á 2 -1˜ Ëk ¯ L2’ RG C1 L11 C2 ’ RL ’ † † RL Michael Tse: Impedance Matching Ê L11 ˆ Á 2 ˜R2 Ë k L22 ¯ Ê L11 ˆ Á 2 ˜C2 Ë k L22 ¯ 32 Exact match is to be achieved at two given frequencies: fm1 and fm2. L2’ RG C1 L11 R1 R2 C2 ’ RL ’ Observe that: • R1 resonates at certain frequency, but is always less than RG • R2 decreases monotonically with frequency So, if RL is sufficiently small, there will be two frequency values where R1 = R2. Michael Tse: Impedance Matching 33 resistance R2 R1 f fm1 fm2 Our objective here is to match RG and RL over a bandwidth Df centered at fo, usually with an allowable ripple in the pass band. Michael Tse: Impedance Matching 34 General Impedance Matching Based on Two-Port Parameters Two-port models i1 i2 + + v1 v2 – – Idea: we don’t care what is inside, as long as it can be modelled in terms of four parameters. Michael Tse: Impedance Matching 35 Two-port models i1 + port 1 i2 + v1 v2 – – z-parameters (impedance matrix): Èv1 ˘ Èz11 Í ˙=Í Îv 2 ˚ Îz21 z12 ˘Èi1 ˘ ˙Í ˙ z22 ˚Îi2 ˚ y-parameters (admittance matrix): Èi1 ˘ È y11 Í ˙=Í Îi2 ˚ Î y 21 Èv1˘ Èh11 Í ˙=Í Îi2 ˚ Îh21 y12 ˘Èv1 ˘ ˙Í ˙ y 22 ˚Îv 2 ˚ h12 ˘È†i1 ˘ ˙Í ˙ h22 ˚Îv 2 ˚ È i1 ˘ Èg11 Í ˙=Í Îv 2 ˚ Îg21 g12 ˘Èv1˘ ˙Í ˙ g22 ˚Îi2 ˚ † h-parameters (hybrid matrix): † g-parameters (hybrid matrix): † Michael Tse: Impedance Matching † port 2 v1 = z11i1 + z12i2 v 2 = z21i1 + z22i2 : : 36 Finding the parameters e.g., z-parameters v1 = z11i1 + z12i2 v 2 = z21i1 + z22i2 z11 = v1 v = 1 i1 i = 0 i1 port 2 open -circuited 2 z12 = v1 v = 1 i2 i = 0 i2 port 1 open -circuited 1 † z21 = z22 = v2 i1 v2 i2 = i2 = 0 = i1 = 0 Michael Tse: Impedance Matching † v2 i1 port 2 open -circuited v2 i2 port 1 open -circuited 37 Finding the parameters e.g., g-parameters i1 = g11v1 + g12i2 v 2 = g21v1 + g22i2 g11 = i1 i = 1 v1 i = 0 v1 port 2 open -circuited 2 g12 = † g21 = g22 = i1 i2 v2 v1 v2 i2 = v1 = 0 = i2 = 0 = v1 = 0 Michael Tse: Impedance Matching † i1 i2 port 1 short -circuited v2 v1 port 2 open -circuited v2 i2 port 1 short -circuited 38 Input impedance: + i1 i2 + [Z] v1 – ZL v2 – Zin v1 = z11i1 + z12i2 v 2 = z21i1 + z22i2 † † fi† fi v1 i2 = z11 + z12 i1 i1 v2 i1 = -z21 - z22 -i2 i2 fi † Ê z12 z21 ˆ Z in = z11 - Á † ˜ Ë Z L + z22 ¯ Michael Tse: Impedance Matching † † i2 Z in = z11 + z12 i1 i1 Z L = -z21 - z22 i2 39 Similarly, we can find the input impedance at any port in terms of any of the two-port parameters, or even a combination of different twoport parameters. We will see that the matching problem can be solved by making sure that both input and output ports are matched. ZG i1 i2 [Z] ± + ZL v2 – ZIM1 ZIM2 matching: ZG = ZIM1 and ZIM2 = ZL image impedances Michael Tse: Impedance Matching 40 The ABCD parameters (very useful form) + v1 i1 i2 [ABCD] – + v2 – Here, voltage and current of port 1 are expressed in terms of those of port 2. So, this is neither an immittance matrix like Z and Y, nor a hybrid matrix like G and H. Èv1˘ ÈA B˘È v 2 ˘ Í ˙=Í ˙Í ˙ i C D Î 1˚ Î ˚Î-i2 ˚ Note: the sign of i2 in the above equation. This sign convention will make the ABCD matrix very useful for describing cascade circuits. † Michael Tse: Impedance Matching 41 + i’ i1 [ABCD]1 v1 – Èv1˘ ÈA1 B1 ˘È v' ˘ Í ˙=Í ˙Í ˙ i C D Î 1˚ Î 1 1˚Î-i'˚ i” + v’ – i2 [ABCD]2 Èv'˘ È A2 Í ˙=Í Îi"˚ ÎC2 + v2 – B2 ˘È v 2 ˘ ˙Í ˙ D2 ˚Î-i2 ˚ Since –i’ = i”, we have Èv1† ˘ ÈA1 Í ˙=Í Î i1 ˚ ÎC1 † B1 ˘È A2 ˙Í D1˚ÎC2 B2 ˘È v 2 ˘ ˙Í ˙ D2 ˚Î-i2 ˚ So, if more two-ports are cascaded, the overall ABCD matrix is just the product of all the ABCD matrices. † Michael Tse: Impedance Matching 42 To find the ABCD parameters, we may apply the same principle: A= B= C= v1 v2 = i2 = 0 -v1 i2 v i1 v2 = 2=0 = i2 = 0 -i1 D= i2 v v1 v2 2 = port 2 open -circuited z11 z21 -v1 z z -z z = 11 22 21 12 i2 port 2 short -circuited z21 i1 v2 = port 2 open -circuited 1 z21 -i1 z22 = = i2 port 2 short -circuited z21 =0 We can show easily that AD – BC = 1 † if z12 = z21, i.e., reciprocal circuit. Michael Tse: Impedance Matching 43 Matching problem ZG i1 i2 + v1 – ± [ABCD] + ZL v2 – ZIM1 Input image impedance v1 = Av 2 - Bi2 i1 = Cv 2 - Di2 † fi † v1 Av 2 - Bi2 = i1 Cv 2 - Di2 v2 A +B -i2 = v C 2 +D -i2 AZ L + B = CZ L + D Z in = Michael Tse: Impedance Matching † 44 ZG i1 i2 + v1 – + [ABCD] v2 – ZIM2 Output image impedance v1 = Av 2 - Bi2 i1 = Cv 2 - Di2 fi v 2 = Dv1 - Bi1 i2 = Cv1 - Ai1 fi because AD – BC = 1 † † † † Michael Tse: Impedance Matching † v 2 Dv1 - Bi1 = i2 Cv1 - Ai1 v D 1 +B -i1 = v C 1 +A -i1 DZG + B = CZG + A Z IM2 = 45 Under matched conditions, ZG = ZIM1 fi † † AZ L + B Z IM1 = ZG = CZ L + D fi Z IM1 = and ZL = ZIM2 and DZG + B Z IM2 = Z L = CZG + A AB † CD and Z IM2 = DB AC z11 y11 and Z IM2 = z22 y 22 Alternatively, we have † † Z IM1 = Michael Tse: Impedance Matching † 46 Note: image impedances are different from input and output impedances. 1. Image impedances do not depend on the load impedance or the source impedance. They are purely dependent upon the circuit. Z IM1 = 2. z11 y11 and Z IM2 = z22 y 22 Input impedance (Zin) depend on the load impedance. Output impedance (Zout)†depends on the source impedance. For example, Ê z12 z21 ˆ Z in = z11 - Á ˜ Ë Z L + z22 ¯ Matching conditions: • Source impedance equals input image impedance • Load † impedance equals output image impedance Michael Tse: Impedance Matching 47 Example i1 Za Zc + v1 – port 1 We can easily see that i2 + Zb z11 = v1 = Za + Zb i1 port 2 open -circuited y11 = i1 1 = v1 port 2 short -circuited Z a + Z b Z c z22 = v2 i2 v2 – port 2 y 22 = i2 v2 = Zb + Zc port 1 open -circuited = port 1 short -circuited 1 Zc + Za Zb Thus, the image impedances are Z IM1 = (Z a + Z b )(Z a + Z b Z†c ) and Z IM2 = (Z c + Z b )(Z c + Z a Z b ) Michael Tse: Impedance Matching † 48 Matching a cascade of circuits 1 ZIM1 2 Z’IM1 = ZIM2 i2 + + v1 v2 – † † – † ZIM1 Z’IM2 = ZIM3 4 Z’IM3 = ZIM4 ZL Z’IM4 = ZL A wave or signal entering into circuit 1 from left side will travel without reflection through the circuits if all ports are matched. Convention i1 3 Propagation constant g ZIM2 input power v1i1 v1 e = = = output power v 2 (-i2 ) v 2 g † Michael Tse: Impedance Matching † Z IM2 Z IM1 49 Propagation equations v1i1 v = 1 v 2 (-i2 ) v 2 eg = In general, † Z IM2 Z IM1 † = A+ B Thus, † fi if the 2-port circuit is symmetrical v1 Av 2 - Bi2 B = = A+ v2 v2 Z IM2 † † v1 e = v2 g AC = BD i1 = CZ IM2 + D = -i2 eg = D A A D ( ( AD + BC AD + BC ) ) v1i1 = AD + BC -v 2i2 e-g = AD - BC Michael Tse: Impedance Matching † 50 Combining eg and e–g, we have eg + e-g cosh g = = AD 2 eg - e-g sinh g = = BC 2 Define n= † We have † Z IM1 = Z IM2 A D A = n coshg B = nZ IM2 sinh g sinh g nZ IM2 cosh g D= n C= Michael Tse: Impedance Matching † 51 From the ABCD equation, we have v1 = nv 2 cosh g - ni2 Z IM2 sinh g v i i1 = 2 sinh g - 2 cosh g nZ IM2 n Dividing gives † v1 Z L + Z IM2 tanh g 2 Z in = = n Z IM2 i1 Z L tanh g + Z IM2 For a transmission line, ZIM1 = ZIM2 = Zo, where Zo is usually called the characteristic impedance of the transmission line. Also, for a lossless†transmission line, g = jL is pure imaginary, and thus tanh becomes tan, sinh becomes sin, cosh becomes cosh. v1 Z L + jZ o tan L Z in = = Z o i1 Z o + jZ L tan L Michael Tse: Impedance Matching † 52 This is just the same transmission line equation. In communication, we usually express L as electrical length, and is equal to L = w l / v = 2p l / l wavelength frequency in rad/s length of transmission line velocity of propagation So, we can easily verify the following standard results: 1. If the transmission line length is l/2 or l, then the input impedance is just equal to the load impedance. 2. If the transmission line length is l/4, then the input impedance is Zo2/ZL. Impedance value for other lengths can be found from the equation or conveniently by using a Smith chart. Michael Tse: Impedance Matching 53
