EKT 441 MICROWAVE COMMUNICATIONS CHAPTER 3 MICROWAVE NETWORK
- Slides: 36
EKT 441 MICROWAVE COMMUNICATIONS CHAPTER 3: MICROWAVE NETWORK ANALYSIS (PART 1)
NETWORK ANALYSIS Many times we are only interested in the voltage (V) and current (I) relationship at the terminals/ports of a complex circuit. l If mathematical relations can be derived for V and I, the circuit can be considered as a black box. l For a linear circuit, the I-V relationship is linear and can be written in the form of matrix equations. l A simple example of linear 2 -port circuit is shown below. Each port is associated with 2 parameters, the V and I. l I 1 R Convention for positive polarity current and voltage I 2 + Port 1 V 1 C V 2 Port 2 -
NETWORK ANALYSIS l For this 2 port circuit we can easily derive the I-V relations. I 1 R I 2 I 1 V 1 j CV 2 V C V 2 2 l We can choose V 1 and V 2 as the independent variables, the I-V relation can be expressed in matrix equations I 1 Port 1 V 1 R I 2 C . I 1 V 2 Port 2 V 1 Network parameters (Y-parameters) I 2 2 - Ports V 2
NETWORK ANALYSIS l To determine the network parameters, the following relations can be used: or This means we short circuit the port l For example to measure y 11, the following setup can be used: I 1 V 1 I 2 2 - Ports V 2 = 0 Short circuit
NETWORK ANALYSIS By choosing different combination of independent variables, different network parameters can be defined. This applies to all linear circuits no matter how complex. l Furthermore this concept can be generalized to more than 2 I 1 ports, called N - port networks. l I 1 V 1 Linear circuit, because all elements have linear I-V relation V 1 I 2 V 2 2 - Ports I 2 V 2
ABCD MATRIX l Of particular interest in RF and microwave systems is ABCD parameters are the most useful for representing Tline and other linear microwave components in general. Take note of the direction of positive current! I 1 I 2 (4. 1 a) V 1 2 -Ports (4. 1 b) Open circuit Port 2 Short circuit Port 2 V 2
ABCD MATRIX l I 1 V 1 The ABCD matrix is useful for characterizing the overall response of 2 -port networks that are cascaded to each other. I 2 ’ I 2 V 2 I 3 V 3 Overall ABCD matrix
THE SCATTERING MATRIX l l l Usually we use Y, Z, H or ABCD parameters to describe a linear two port network. These parameters require us to open or short a network to find the parameters. At radio frequencies it is difficult to have a proper short or open circuit, there are parasitic inductance and capacitance in most instances. Open/short condition leads to standing wave, can cause oscillation and destruction of device. For non-TEM propagation mode, it is not possible to measure voltage and current. We can only measure power from E and H fields.
THE SCATTERING MATRIX l Hence a new set of parameters (S) is needed which l Do not need open/short condition. l Do not cause standing wave. l Relates to incident and reflected power waves, instead of voltage and current. • As oppose to V and I, S-parameters relate the reflected and incident voltage waves. • S-parameters have the following advantages: 1. Relates to familiar measurement such as reflection coefficient, gain, loss etc. 2. Can cascade S-parameters of multiple devices to predict system performance (similar to ABCD parameters). 3. Can compute Z, Y or H parameters from S-parameters if needed.
THE SCATTERING MATRIX l Consider an n – port network: Reference plane for local z-axis (z = 0) Each port is considered to be connected to a Tline with specific Zc. Port 1 Zc 1 Port n Zcn Port 2 Zc 2 T-line or waveguide Linear n - port network
THE SCATTERING MATRIX There is a voltage and current on each port. l This voltage (or current) can be decomposed into the incident (+) and reflected component (-). l V 1 + V 1 - Port 1 V 1 I 1 - Port V 1 n z=0 Port 2 Linear n - port Network V 1+ + Port 1 +z + V 1 -
THE SCATTERING MATRIX The port voltage and current can be normalized with respect to the impedance connected to it. l It is customary to define normalized voltage waves at each port as: l Normalized incident waves (4. 3 a) i = 1, 2, 3 … n Normalized reflected waves (4. 3 b)
THE SCATTERING MATRIX l Thus in general: V 1+ V 1 - Port 1 Vn+ Vn- Port n Zc 1 Port 2 V 2+ V 2 Zc 2 T-line or waveguide Linear n - port Network Zcn Vi+ and Vi- are propagating voltage waves, which can be the actual voltage for TEM modes or the equivalent voltages for non-TEM modes. (for non-TEM, V is defined proportional to transverse E field while I is defined proportional to transverse H field, see [1] for details).
THE SCATTERING MATRIX If the n – port network is linear (make sure you know what this means!), there is a linear relationship between the normalized waves. l For instance if we energize port 2: l V 1 - Port 1 Zc 1 Port 2 V 2+ V 2 - Zc 2 Linear n - port Network Vn- Port n Zcn Constant that depends on the network construction
THE SCATTERING MATRIX l Considering that we can send energy into all ports, this can be generalized to: (4. 4 a) l Or written in Matrix equation: or l (4. 4 b) Where sij is known as the generalized Scattering (S) parameter, or just S-parameters for short. From (4. 3), each port i can have different characteristic impedance Zci
THE SCATTERING MATRIX l Consider the N-port network shown in figure 4. 1. Figure 4. 1: An arbitrary N-port microwave network
THE SCATTERING MATRIX Vn+ is the amplitude of the voltage wave incident on port n. l Vn- is the amplitude of the voltage wave reflected from port n. l The scattering matrix or [S] matrix, is defined in relation to these incident and reflected voltage wave as: l [4. 1 a]
THE SCATTERING MATRIX or [4. 1 b] A specific element of the [S] matrix can be determined as: [4. 2] Sij is found by driving port j with an incident wave Vj+, and measuring the reflected wave amplitude, Vi-, coming out of port i. The incident waves on all ports except j-th port are set to zero (which means that all ports should be terminated in matched load to avoid reflections). Thus, Sii is the reflection coefficient seen looking into port i when all other ports are terminated in matched loads, and Sij is the transmission coefficient from port j to port i when all other ports are terminated in matched loads.
THE SCATTERING MATRIX l For 2 -port networks, (4. 4) reduces to: (4. 5 a) (4. 5 b) Note that Vi+ = 0 implies that we terminate i th port with its characteristic impedance. l Thus zero reflection eliminates standing wave. l
THE SCATTERING MATRIX Vs V 1+ V 2 - Zc 1 Zc 2 2 – Port Zc 2 V 1 Measurement of s 11 and s 21: V 1 Zc 1 V 2+ 2 – Port Zc 2 V 2 - Measurement of s 22 and s 12: Zc 2 Vs
THE SCATTERING MATRIX Input-output behavior of network is defined in terms of normalized power waves l S-parameters are measured based on properly terminated transmission lines (and not open/short circuit conditions) l
THE SCATTERING MATRIX
THE SCATTERING MATRIX
THE SCATTERING MATRIX Reciprocal and Lossless networks l Impedance and admittance matrices are symmetric for reciprocal networks l A symmetric network happens when: (4. 6 a) l It is also purely imaginary for lossless network (no real power can be delivered to the network) (4. 6 b) l A matrix that satisfies the condition of (4. 6 b) is called a unitary matrix
THE SCATTERING MATRIX Transpose of [S], written as [S]t l Transpose of a Matrix (taken from Engineering Maths 4 th Ed by KA Stroud)
THE SCATTERING MATRIX If It is symmetrical when aij = aji When a [S] is symmetric, it is also reciprocal l Symmetrical Matrix (taken from Engineering Maths 4 th Ed by KA Stroud)
Reciprocal and Lossless networks (cont) l The matrix equation in (4. 6 b) can be re-written in; THE SCATTERING MATRIX For i = j (4. 7) l OR For i ≠ j Used to determine reciprocality for a 2 port network
THE SCATTERING MATRIX (Ex) l Find the S parameters of the 3 d. B attenuator circuit shown in Figure 4. 2: A matched 3 d. B attenuator with a 50 Ω characteristic impedance.
THE SCATTERING MATRIX (Ex) l From the following formula, S 11 can be found as the reflection coefficient seen at port 1 when port 2 is terminated with a matched load (Z 0 =50 Ω); l The equation becomes; On port 2
THE SCATTERING MATRIX (Ex) l To calculate Zin(1), we can use the following formula; Thus S 11 = 0. Because of the symmetry of the circuit, S 22 = 0. l S 21 can be found by applying an incident wave at port 1, V 1+, and measuring the outcome at port 2, V 2 -. This is equivalent to the transmission coefficient from port 1 to port 2: l
THE SCATTERING MATRIX (Ex) l From the fact that S 11 = S 22 = 0, we know that V 1 - = 0 when port 2 is terminated in Z 0 = 50 Ω, and that V 2+ = 0. In this case we have V 1+ = V 1 and V 2 - = V 2. l Where 41. 44 = (141. 8//58. 56) is the combined resistance of 50 Ω and 8. 56 Ω paralled with the 141. 8 Ω resistor. Thus, S 21 = S 12 = 0. 707
THE SCATTERING MATRIX (Ex) l A two port network is known to have the following scattering matrix: a) b) c) Determine if the network is reciprocal and lossless. If port 2 is terminated with a matched load, what is the return loss seen at port 1? If port 2 is terminated with a short circuit, what is the return loss seen at port 1?
THE SCATTERING MATRIX (Ex) l l Q: Determine if the network is reciprocal and lossless Since [S] is not symmetric, the network is not reciprocal. Taking the 1 st column, (i = 1) gives; So the network is not lossless. l Q: If port two is terminated with a matched load, what is the return loss seen at port 1? Usedcoefficient to determine l When port 2 is terminated with a matched load, the reflection seen at port 1 is Γ = S 11 = 0. 15. So the return loss is; reciprocality for a 2 port network
seen at port 1? l When port 2 is terminated with a short circuit, the reflection coefficient seen at port 1 can be found as follow l From the definition of the scattering matrix and the fact that V 2+ = - V 2(for a short circuit at port 2), we can write: THE SCATTERING MATRIX (Ex)
l Dividing the first equation by V 1+ and using the above result gives the reflection coefficient seen as port 1 as; THE SCATTERING MATRIX (Ex)
THE SCATTERING MATRIX (Ex) l The return loss is; l Important points to note: l Reflection coefficient looking into port n is not equal to Snn, unless all other ports are matched l Transmission coefficient from port m to port n is not equal to Snm, unless all other ports are matched l S parameters of a network are properties only of the network itself (assuming the network is linear) l It is defined under the condition that all ports are matched l Changing the termination or excitation of a network does not change its S parameters, but may change the reflection coefficient seen at a given port, or transmission coefficient between two ports
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