2.1 Equivalent circuit of pulse transformer

In a linear modulator, a pulse transformer is designed as a step-up transformer used for transmitting pulsed power. We apply a typical equivalent circuit, as presented in Fig. 2, to analyze a pulse transformer.

Figure 2Full-screen View

Equivalent circuit of a typical pulse transformer

Term K denotes the coupling coefficient of the pulse transformer (K=1 in the ideal case), $C_{\text{d}1}$ denotes the distributed capacitance of the primary winding line, and $C_{\text{d}2}$ denotes the distributed capacitance of the secondary winding line. Term $C_{\text{w}}$ is the parasitic capacitance between the primary and secondary windings, $L_{\text{p}}$ indicates the primary inductance, $L_{\text{s}}$ indicates the secondary inductance, and $R_{\text{L}}$ indicates the load impedance. Term N denotes the step-up ratio of the pulse transformer, $C_{\text{t}}$ denotes the parasitic capacitance between the primary winding and ground, and $L_{\text{t}}$ denotes the lead inductance in series with $C_{\text{t}}$. Table 2 shows a comparison of the main parameters of the C-band pulse transformers in other facilities [14-20].

2.2 Shorten leading edge

In general, we can neglect unrelated parameters, such as L_p, L_t, and C_t when analyzing the leading edge [21]. In this case, the equivalent circuit for leading edge analysis is shown in Fig. 3. In a line-type modulator, the pulse forming network (PFN) impedance is usually designed to be equal to the load impedance. The leading edge can be calculated using Eq. (1). This depends on the distributed capacitance and leakage inductance [22].

Figure 3Full-screen View

Equivalent circuit for leading edge analysis

where $L_{\text{e}}$ indicates the leakage inductance, $C_{\text{d}}$ indicates the distributed capacitance, $R_{\text{L}}$ indicates the load impedance, $\text{σ}=\frac{Z_{\text{T}}}{2R_{\text{L}}}$, and $Z_{\text{T}}=\sqrt{\frac{L_{\text{e}}}{C_{\text{d}}}}$. Term y(σ) depends on the selected damping coefficient σ, and can be obtained from Table 3.

Fig. 4 shows the leading edge for a certain range of σvalues. As can be derived from the theories, for a specified value of distributed capacitance, a larger leakage inductance leads to a slower leading edge, and for a certain value of leakage inductance, a small distributed capacitance leads to a smaller overshoot. For instance, in the case of C_d = 20 nF and L_e = 2 µH, overshoot is approximately 2% compared with the case of C_d = 30 nF and L_e = 2 µH. The leading edge increases by approximately 200 ns for C_d = 30 nF and L_e = 3 µH compared with the case of C_d = 30 nF and L_e = 2 µH.

Figure 4Full-screen View

Leading edge with different values of damping factor σ

We use Eq. (2) to calculate the leakage inductance. We can also measure it using a precision LCR meter [23].

where ${\mu }_0$ is the permeability of free space (4π×10^-7 H/m), $C_{\text{L}}$ indicates the one-turn mean length of the primary coil, d indicates the distance between the primary and secondary windings, $L_{\text{c}}$ indicates the height of the secondary coil, and N_p indicates the number of primary winding turns.

We can apply corresponding methods to decrease the leakage inductance—for instance, using closed core, cone-shape windings, and close bifilar winding. The calculated and measured values of leakage inductance of this pulse transformer are 2.25 and 2.3 µH, respectively.

The distributed capacitance includes a stray capacitance between the primary and secondary windings and a stray capacitance in the klystron. The distributed capacitance between the primary and secondary windings can be calculated using Eq.(3). We can also measure the total distributed capacitance using a precision LCR meter.

where ${\epsilon }_0$ is the dielectric constant of vacuum. ${\epsilon }_{\text{r}}$ is the relative dielectric constant, $C_{\text{L}}$ is the one-turn length of the primary winding, $L_{\text{c}}$ is the secondary coil height, n is the transformer ratio, and d is the distance between the primary and secondary windings. The distributed capacitance between the primary and secondary windings is calculated to be 12 nF and the total distributed capacitance is measured to be 40 nF.

2.3 Flattop droop and primary inductance

Flattop droop depends on the inductance of the primary winding. The equivalent circuit used for flattop droop analysis is shown in Fig. 5

Figure 5Full-screen View

Equivalent circuit for flattop droop analysis

We use Eq. (4) to calculate the voltage across the resistance R_L.

where $L_{\text{p}}$ is the primary inductance, and $i_{\text{L}}$ is the current flowing through $L_{\text{p}}$.

Equation (5) can be derived from the principle of Kirchhoff laws.

where $E$ is the voltage of the pulser, and $R_{\text{PFN}}$ is the PFN characteristic impedance.

The initial conditions (t=0) are given by Eq.(6).

Based on Equations (4), (5), and (6), the voltage $U{\text{'}}_{\text{L}}$ can be calculated as follows:

where $\alpha =\frac{R_{\text{L}}}{R_{\text{L}}+R_{\text{PFN}}}$ is called the transmission factor from the primary to secondary windings of the pulse transformer, and $T_{\text{d}}=\frac{L_{\text{p}}\left(R_{\text{L}}+R_{\text{PFN}}\right)}{R_{\text{L}}{R}_{\text{PFN}}}$ is called the time constant of the pulse transformer.

The flattop droop D is defined in Eq. (8).

In a linear modulator, we usually design the PFN output impedance to be matched to the load impedance. The flattop droop D is determined by the primary inductance, where the relationship between primary inductance and flattop droop is given in Eq. (9).

Fig. 6 shows the primary inductance versus flattop droop for the C-band klystron pulse transformer. Fig. 7 presents a graphical view of flattop droop with different values of primary inductance in the simulation of the C-band pulse transformer. As shown in Fig. 7, a primary inductance of 100 µH results in approximately 10.8% flattop droop, a primary inductance of 300 µH results in approximately 3.6% flattop droop, and a primary inductance of 500 µH only results in approximately 2.2% flattop droop.

Figure 6Full-screen View

Primary inductance vs. flattop droop

Figure 7Full-screen View

Simulation result of flattop droop with different values of primary inductance

The primary inductance is associated with the core material permeability, winding turns, core size, and magnet flux pass length [24]. It can be calculated using Eq.(10).

where ${\mu }_0$ is the permeability of free space (4π×10^-7H/m), $u_{\text{r}}$ is the permeability of the core material, $l_{\text{m}}$ is the average magnetic pass length. $A_{\text{core}}$ is the core cross-section area. $$ N_p is the number of primary winding turns.

In general, core materials with high permeability are preferred. High permeability ensures the fewest turns to attain our desired magnetizing inductance [25]. Fewer turns lead to lower leakage inductance, smaller distributed capacitance, and smaller winding resistance. Silicon steel and ferrite are the most commonly used materials for the design of a high-power pulse transformer core. In our design, the primary inductance is 226 µH. Thus, we can calculate the flattop droop as 5%.

2.4 Reduce flattop ringing

Flattop ringing is determined by the parasitic capacitance between the primarily coil and ground C_t and the lead inductance $L_{\text{t}}$ in series with $C_{\text{t}}$. Fig. 8 shows the circuit used to analyze the flattop ringing. Fig. 9 shows the flattop ringing with different values of L_t and C_t. The ringing frequency can be calculated using Eq. (11).

Figure 8:Full-screen View

Equivalent circuit for analyzing flattop ringing

Figure 9:Full-screen View

Flattop ringing with different values of L_t and C_t

In summary, the main parameters of the C-band high-power pulse transformer are listed in Table 4.